{"version":1,"lectureId":"01M14V00YG9JJ7KZM2ECASKC0D","attempt":0,"publication":{"slug":"time-dilation-from-a-light-clock","title":"Special Relativity: Light Clocks, Spacetime, and the Twin Paradox","subject":"physics","summary":"An algebra-first introduction to special relativity built from Einstein's two postulates and a light clock. The lecture derives time dilation with the Pythagorean theorem, obtains length contraction, makes the relativity of simultaneity concrete with a train lamp, interprets tilted spacetime axes, resolves the twin paradox through changing inertial frames, and closes with atmospheric muons as an experimental test.","metaDescription":"Derive time dilation and length contraction, understand simultaneity, resolve the twin paradox, and explain atmospheric muons.","transcript":"Special relativity begins with two statements that sound modest and turn out to reorganize space and time. We will keep one physical clock beside us while we follow their consequences. First, the laws of physics have the same form in every inertial frame. An inertial observer is simply one who is not accelerating. Second, every inertial observer measures the same speed c for light in a vacuum. The source may move, and the observer may move, but the measured light speed is still c. That second statement is the unfamiliar one. Ordinary speeds add. If you throw a ball forward from a train, someone beside the track combines the ball's speed with the train's speed. Light refuses that rule. We will not begin by guessing how clocks and rulers must change. We will ask what these two postulates force one very simple clock to do. Here is the clock. Two parallel mirrors face one another, separated by a fixed distance h. A pulse of light bounces between them. Let the pulse begin at the lower mirror. It travels upward at speed c, reaches the upper mirror, and returns. One complete up and down journey is one tick. The mirrors are separated by h, so the one-way journey has distance h. Speed is distance divided by time, which means the one-way time is h divided by c. A complete tick travels that distance twice. The clock therefore records delta tau equal to two h over c. I have called this interval delta tau because both endpoint events happen at the same place in the clock's own frame. The clock is there when the pulse leaves, and the same clock is there when the pulse returns. That is proper time: the time measured by a clock travelling with both events. Nothing relativistic has happened yet. We have only built a clock whose operation follows directly from the constant speed of light. Now let this entire clock move sideways. The passengers who carry it still see the same vertical pulse and the same tick delta tau. The interesting question is what someone beside the track must see. The passenger sees the clock at rest. The pulse goes straight up and straight down, and one tick lasts delta tau. The platform observer sees the whole apparatus move to the right. During the first half of the tick, the upper mirror moves sideways. The light still travels at speed c, so it follows a diagonal path to catch that mirror. During the second half, the clock keeps moving while the pulse returns to the lower mirror. The platform observer therefore sees two diagonal legs, not one vertical round trip. Both observers must use the same light speed c. The platform observer sees a longer light path, so that observer must assign more time to the tick. The geometry will tell us exactly how much more. Take only the upward half of the journey. The vertical separation of the mirrors is h. If the full platform time is delta t, this half lasts delta t over two. During that half tick, the clock moves horizontally by v times delta t over two. That is the bottom leg of a right triangle. Light covers the diagonal. Since its speed is c, the diagonal length is c times delta t over two. Now use only the Pythagorean theorem. The square of the diagonal equals the square of h plus the square of the horizontal leg. Multiply out the common factors and move the horizontal contribution to the left. Delta t squared over four multiplies c squared minus v squared. Solve for delta t. The answer is two h divided by the square root of c squared minus v squared. Carry that result forward. Factor c squared out of the square root. Two h over c is exactly the proper tick delta tau that the passenger measured. The remaining factor is one over the square root of one minus v squared over c squared. We call this factor gamma. So the platform time is gamma times the proper time. Since gamma is at least one, the moving clock accumulates less time between the same two meetings. Watch the triangle as the speed rises from forty percent to eighty percent of c. The mirror separation stays fixed, but the horizontal leg and the light path both grow. At low speed gamma is almost one, which is why ordinary experience hides relativity. At eighty percent of c, gamma is five thirds. Five platform seconds correspond to only three seconds on the moving clock. Time dilation is not a mechanical defect in the light clock. Every process carried with the moving observer follows the same spacetime geometry, including chemical reactions, heartbeats, and aging. Time dilation also forces a change in measured length. Turn a light clock so its pulse travels along the direction of motion. In the platform frame, call the moving mirror separation L. On the outward trip the front mirror runs away from the light, so the closing speed is c minus v. On the return trip the rear mirror moves toward the light. The closing speed is c plus v. Add the two travel times. The common denominator is c squared minus v squared. After simplifying, the platform round trip is two L gamma squared over c. Carry that platform result forward. In the clock's rest frame, the mirror separation is its proper length L zero, so the round trip is two L zero over c. Time dilation says the platform round trip equals gamma times the proper round trip. Equate those two expressions. Cancel the common factors. The moving length L equals the proper length divided by gamma. A moving object is shorter along its direction of motion. Length means the distance between two endpoints measured at the same time. That last phrase is essential. If the endpoints are moving, recording one endpoint now and the other endpoint later does not measure one length. To see why same time becomes delicate, put a lamp exactly at the centre of a train. The passengers know the rear and front walls are equally far from the lamp. The lamp flashes. Light travels left and right at the same speed c, through equal distances, so it strikes the rear and front simultaneously in the train frame. Now describe the same experiment from the platform. At emission, the lamp is between the moving rear and front walls. Both light pulses still travel at speed c in this frame. The rear wall moves toward the left-going pulse. The front wall moves away from the right-going pulse. Let the experiment run until the left pulse meets the rear. The rear has already been struck, but the right pulse still has farther to go. Remove the completed pulse and continue. Only later does the right pulse catch the front wall. The platform observer therefore says rear first, front second. This is not a disagreement about light taking time to reach someone's eyes. Each observer can correct for signal travel. They disagree about whether the two distant contact events share one time coordinate. The algebraic statement is this. The time separation in the moving frame equals gamma times delta t minus v delta x over c squared. If two events are simultaneous in one frame, delta t is zero. If they are separated in space, delta x is not zero, so another moving frame generally assigns a nonzero time separation. This is difficult because everyday objects move far too slowly for the disagreement to notice. It is also difficult because simultaneity feels like a fact about the events themselves. Relativity says it is a relation between the events and a chosen inertial frame. The train passengers and the platform observer agree on every event: the emission, the rear strike, and the front strike. What differs is the time coordinate assigned to separated events. A spacetime diagram makes that distinction visible. A spacetime diagram puts position on the horizontal axis and c times time on the vertical axis. Multiplying time by c gives both axes the same units. An object at rest stays at one x position, so its history is a vertical worldline. A moving object changes x as time passes, so its worldline tilts. Light is special. Since its speed is c, one unit of horizontal travel takes one unit of c times time. Light rays therefore follow these two forty-five-degree lines. Every inertial observer must draw light on those same lines. That shared light cone constrains how another observer's coordinate axes can sit in the diagram. The red c t prime axis is the moving observer's own worldline, x prime equal to zero. It tilts toward the right-going light ray. The red x prime axis is the set of events that the moving observer calls simultaneous with the origin. It tilts upward by the matching amount. So the moving observer's space and time axes tilt toward the light lines and toward each other. They are not a rotated pair of ordinary Euclidean axes. Their scaling is controlled by gamma. The coordinate equations are the Lorentz transformation. Position x prime equals gamma times x minus v t. The time equation mixes time with position. That mixed term, v x over c squared in ordinary time units, is the algebraic source of relative simultaneity. Set x prime equal to zero. The result is the tilted time axis, the moving observer's worldline. Set t prime equal to zero. The result is the tilted space axis, the moving observer's line of simultaneous events. Now choose two events E one and E two on one horizontal line. The stationary observer assigns them the same time. The moving observer's simultaneous line through E one is tilted. It reaches the same distant position at E three, not at E two. No event moves when we change coordinates. E one, E two, and E three stay where they are. What changes is which distant events one observer groups into a single present. The invariant quantity is not Euclidean distance on this page. Write it first. The time part is c delta t squared. The space part is delta x squared, and their difference has the same value for every inertial observer. For events on one clock's worldline, that invariant equals c delta tau squared. This is why different paths between the same meetings can contain different amounts of proper time. Two twins meet on Earth, synchronize clocks, and separate. One remains on Earth. The other travels outward at eighty percent of light speed, turns around, and returns at the same speed. The spacetime diagram makes the histories different immediately. The Earth twin follows one straight inertial worldline. The traveller follows one straight segment out and a different straight segment home. Let each leg last five Earth years. At speed zero point eight c, the turnaround is four light years away, and the reunion occurs after ten Earth years. For zero point eight c, gamma is five thirds. The Earth clock is present at departure and reunion, and it records ten years of proper time. On either travelling leg, five Earth-frame years correspond to five divided by gamma, which is three traveller years. Two legs therefore give six traveller years. At reunion the clocks can be compared at one place. The Earth twin has aged ten years and the travelling twin six. There is no dispute about that local comparison. The apparent paradox comes from saying that each inertial observer sees the other's clock run slowly. If the roles were perfectly symmetric, why would one twin be younger? The roles are symmetric only during any one inertial leg. Earth remains in one inertial frame throughout the experiment. The traveller changes inertial frames at the turnaround. Before turning, the traveller uses an outbound line of simultaneity. On these axes, one year of c times time has the same scale as one light year. Draw that line through the turnaround event. Where that line meets Earth's worldline, the outbound coordinates assign Earth a time of one point eight years at the distant turnaround moment. After turning, the traveller belongs to the inbound frame. Its line of simultaneity slopes the other way. That new line meets Earth's worldline at eight point two years. The inbound coordinates assign a much later Earth time to events simultaneous with the same turnaround. The difference between those coordinate assignments is six point four years. This is where the asymmetry lives: the traveller changes which distant Earth event belongs to the traveller's present. Earth does not physically age six point four years in an instant. Nothing discontinuous happens to Earth's clock. What changes abruptly, in an idealized instantaneous turnaround, is the traveller's coordinate rule for distant simultaneity. With a realistic gradual turn, the reassignment is gradual too. The final proper times remain essentially the same if the acceleration interval is short compared with the journey. Acceleration identifies which twin changes frames, but acceleration is not an extra aging penalty added by hand. The elapsed proper time comes from the complete spacetime path. Between the same departure and reunion events, the straight inertial worldline contains more proper time than the broken out-and-back path. Earth records ten years. The traveller records six. So the twin paradox is not a contradiction in time dilation. It is a warning that time dilation applies within one inertial comparison, while the traveller's full journey requires two different inertial frames. The last test comes from particles made high in the atmosphere. Muons are created roughly ten kilometres above the ground, but their proper mean lifetime is only two point two microseconds. Why do many reach detectors at sea level? Begin with the nonrelativistic expectation. Here is the ten-kilometre atmospheric depth. The muon's own lifetime is two point two microseconds. Even light travels only about zero point six six kilometres in that time. A slower particle should decay far above the ground. But atmospheric muons commonly travel near zero point nine nine eight c. At that speed gamma is about fifteen point eight. In Earth's frame, the moving muon's lifetime is dilated. Two point two microseconds multiplied by gamma becomes about thirty-four point eight microseconds. At almost light speed, that dilated lifetime corresponds to about ten point four kilometres. The muon can reach the ground. That is the Earth-frame explanation: the moving particle's internal clock accumulates less time while Earth clocks record the long flight. Now use the muon's own frame. The muon is at rest, so its lifetime is still the proper value, about two point two microseconds. The atmosphere and ground move toward it. The ten-kilometre atmospheric depth is moving, so it is length contracted. Divide ten kilometres by gamma fifteen point eight. The muon measures only about zero point six three kilometres between its creation event and the approaching ground. At zero point nine nine eight c, crossing that contracted distance takes about two point one microseconds of muon proper time. Let the ground approach. It covers the shortened distance before the muon's typical lifetime has elapsed. Earth says the muon's clock runs slowly. The muon says the atmosphere is short. These are not competing explanations. They are the time and space parts of one Lorentz transformation. Both descriptions predict the same local event: the muon and the detector meet. Experiments observe far more muons at the ground than a nonrelativistic lifetime calculation permits. Now collect the argument. First, every inertial observer measures the same light speed c. The moving light clock then forces moving clocks to accumulate less proper time between shared events. A longitudinal light clock and the same time-dilation law force moving lengths to contract along the motion. The train lamp shows the price of combining those results with constant light speed: distant simultaneity depends on the inertial frame. The spacetime diagram gathers all of this into tilted coordinate axes and invariant proper time. It also reveals the twin asymmetry: only the travelling twin changes inertial frames. Finally, atmospheric muons turn the geometry into an observation. Long lifetime in Earth's coordinates and short atmosphere in the muon's coordinates are two descriptions of the same arrival. Special relativity is not a story in which appearances deceive us. It is a precise rule for how different observers divide one spacetime into space and time, while agreeing on light, meetings, and every measurable event.","watch":{"version":1,"scenes":[{"title":"Two Postulates and One Clock","start":0,"end":135.0288125,"objects":{"card":"a Title that says \"Introductory Modern Physics — Special Relativity: Light Clocks, Spacetime, and the Twin Paradox\"","clock":"a Figure (x_range=(0.0, 4.0), y_range=(0.0, 4.0), aspect=(1.0, 1.0))","clock_heading":"a Heading that says \"A Clock Made Only of Light\"","height_mark":"a Line [green] labelled \"h\" drawn in clock (start=(0.55, 0.7), end=(0.55, 3.3))","light_path":"a Line [gray] drawn in clock (start=(2.0, 0.7), end=(2.0, 3.3), dashed=True)","lower_mirror":"a Line [blue] labelled \"upright(\"lower mirror\")\" drawn in clock (start=(0.8, 0.7), end=(3.2, 0.7))","postulate_heading":"a Heading that says \"Two Postulates\"","postulate_one":"a Text [text] that says \"1. 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We will keep one physical clock beside us while we follow their consequences.","live":[],"does":[[0,"card is shown on the screen, written out."],[1.5,"card: enter:write-left-to-right."],[10.6465,"card is hidden from the screen — left the board."]]},{"start":11.2465,"say":"First, the laws of physics have the same form in every inertial frame. An inertial observer is simply one who is not accelerating.","live":null,"does":[[11.2465,"postulate_heading is shown on the screen, written out."],[11.594999999999999,"postulate_one is shown on the screen, written out."],[13.487,"postulate_one (the \"same\" part) is emphasized."],[19.5005,"postulate_one (the \"same\" part) is no longer emphasized."]]},{"start":20.100499999999997,"say":"Second, every inertial observer measures the same speed c for light in a vacuum. The source may move, and the observer may move, but the measured light speed is still c.","live":["postulate_one","postulate_heading"],"does":[[20.401999999999994,"postulate_two is shown on the screen, written out."],[23.014999999999993,"postulate_two (the \"same\" part) is emphasized."],[29.968999999999994,"postulate_two (the \"same\" part) is no longer emphasized."]]},{"start":31.393499999999996,"say":"That second statement is the unfamiliar one. Ordinary speeds add. If you throw a ball forward from a train, someone beside the track combines the ball's speed with the train's speed. Light refuses that rule.","live":["postulate_one","postulate_two","postulate_heading"],"does":[]},{"start":45.309999999999995,"say":"We will not begin by guessing how clocks and rulers must change. 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A pulse of light bounces between them.","live":[],"does":[[54.66049999999999,"clock_heading is shown on the screen, written out."],[54.66049999999999,"clock is shown on the screen, written out."],[57.23799999999999,"lower_mirror is shown on the screen, written out."],[57.23799999999999,"upper_mirror is shown on the screen, written out."],[60.51199999999999,"height_mark is shown on the screen, written out."],[61.82299999999999,"pulse is shown on the screen, written out."],[62.56699999999999,"light_path is shown on the screen, written out."]]},{"start":64.45549999999999,"say":"Let the pulse begin at the lower mirror. It travels upward at speed c, reaches the upper mirror, and returns. 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The clock therefore records delta tau equal to two h over c.","live":null,"does":[[89.77999999999999,"rest_work is shown on the screen, written out."],[94.088,"rest_tick is shown on the screen, written out."]]},{"start":96.2665,"say":"I have called this interval delta tau because both endpoint events happen at the same place in the clock's own frame. The clock is there when the pulse leaves, and the same clock is there when the pulse returns.","live":["rest_tick","clock","clock_heading","lower_mirror","upper_mirror","height_mark","light_path","pulse"],"does":[[97.845,"rest_tick (the \"Delta tau\" part) is emphasized."],[107.133,"rest_tick (the \"Delta tau\" part) is no longer emphasized."]]},{"start":108.7315,"say":"That is proper time: the time measured by a clock travelling with both events. Nothing relativistic has happened yet. 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The geometry will tell us exactly how much more.","live":null,"does":[[179.71581249999997,"motion_heading is hidden from the screen — left the board."],[179.71581249999997,"moving_clock is hidden from the screen — left the board."],[179.71581249999997,"clock_body is hidden from the screen — moving_clock left the board."],[179.71581249999997,"moving_lower is hidden from the screen — moving_clock left the board."],[179.71581249999997,"moving_upper is hidden from the screen — moving_clock left the board."],[179.71581249999997,"moving_pulse is hidden from the screen — moving_clock left the board."]]},{"start":180.9158125,"say":"Take only the upward half of the journey. The vertical separation of the mirrors is h. If the full platform time is delta t, this half lasts delta t over two.","live":[],"does":[[180.9158125,"triangle_heading is shown on the screen, written out."],[180.9158125,"triangle is shown on the screen, written out."],[185.7808125,"vertical_side is shown on the screen, written out."]]},{"start":192.63831249999998,"say":"During that half tick, the clock moves horizontally by v times delta t over two. That is the bottom leg of a right triangle.","live":["triangle","triangle_heading","vertical_side"],"does":[[195.0178125,"horizontal_side is shown on the screen, written out."]]},{"start":201.75981249999998,"say":"Light covers the diagonal. Since its speed is c, the diagonal length is c times delta t over two.","live":["triangle","triangle_heading","vertical_side","horizontal_side"],"does":[[201.75981249999998,"right_angle is shown on the screen, written out."],[203.0368125,"light_side is shown on the screen, written out."]]},{"start":209.72031249999998,"say":"Now use only the Pythagorean theorem. The square of the diagonal equals the square of h plus the square of the horizontal leg.","live":["triangle","triangle_heading","vertical_side","horizontal_side","right_angle","light_side"],"does":[[211.1138125,"triangle moves to a new place on the board."],[211.1138125,"first_work is shown on the screen, written out."]]},{"start":218.51681249999996,"say":"Multiply out the common factors and move the horizontal contribution to the left. Delta t squared over four multiplies c squared minus v squared.","live":null,"does":[[218.86481249999997,"first_work is shown on the screen, written out."]]},{"start":229.15931249999997,"say":"Solve for delta t. The answer is two h divided by the square root of c squared minus v squared.","live":null,"does":[[229.50781249999997,"first_work is shown on the screen, written out."],[236.60131249999998,"first_work is hidden from the screen — left the board."]]},{"start":237.80131249999997,"say":"Carry that result forward. Factor c squared out of the square root. Two h over c is exactly the proper tick delta tau that the passenger measured.","live":null,"does":[[240.41381249999995,"d3 is shown on the screen, written out."],[243.44381249999998,"d3 (the \"2h/c\" part) is emphasized."],[245.42881249999996,"d3 (the \"2h/c\" part) is no longer emphasized."]]},{"start":249.08231249999997,"say":"The remaining factor is one over the square root of one minus v squared over c squared. We call this factor gamma.","live":["triangle","triangle_heading","vertical_side","horizontal_side","right_angle","light_side","d3"],"does":[[249.08231249999997,"gamma_formula is shown on the screen, written out."],[252.21681249999997,"gamma_formula (the \"1-v^2/c^2\" part) is emphasized."],[257.5058125,"gamma_formula (the \"1-v^2/c^2\" part) is no longer emphasized."]]},{"start":258.10581249999996,"say":"So the platform time is gamma times the proper time. Since gamma is at least one, the moving clock accumulates less time between the same two meetings.","live":["triangle","triangle_heading","vertical_side","horizontal_side","right_angle","light_side","d3","gamma_formula"],"does":[[259.56281249999995,"d4 is shown on the screen, written out."],[264.07881249999997,"time_law is shown on the screen, written out."],[267.6663125,"A box is drawn around time_law."]]},{"start":268.26631249999997,"say":"Watch the triangle as the speed rises from forty percent to eighty percent of c. The mirror separation stays fixed, but the horizontal leg and the light path both grow.","live":["triangle","triangle_heading","vertical_side","horizontal_side","right_angle","light_side","d3","gamma_formula","d4","time_law"],"does":[[270.15881249999995,"horizontal_side is redrawn as the numbers it depends on change."],[270.15881249999995,"right_angle is redrawn as the numbers it depends on change."],[270.15881249999995,"light_side is redrawn as the numbers it depends on change."],[270.15881249999995,"beta ticks to 0.8."]]},{"start":279.52431249999995,"say":"At low speed gamma is almost one, which is why ordinary experience hides relativity. At eighty percent of c, gamma is five thirds. Five platform seconds correspond to only three seconds on the moving clock.","live":null,"does":[[288.1048125,"gamma_formula is indicated — a transient flash."]]},{"start":294.5558125,"say":"Time dilation is not a mechanical defect in the light clock. Every process carried with the moving observer follows the same spacetime geometry, including chemical reactions, heartbeats, and aging.","live":null,"does":[[301.4408125,"time_law is indicated — a transient flash."],[307.35962499999994,"d3 is hidden from the screen — left the board."],[307.35962499999994,"d4 is hidden from the screen — left the board."],[307.35962499999994,"gamma_formula is hidden from the screen — left the board."],[307.35962499999994,"time_law is hidden from the screen — left the board."],[307.35962499999994,"triangle is hidden from the screen — left the board."],[307.35962499999994,"vertical_side is hidden from the screen — triangle left the board."],[307.35962499999994,"horizontal_side is hidden from the screen — triangle left the board."],[307.35962499999994,"right_angle is hidden from the screen — triangle left the board."],[307.35962499999994,"light_side is hidden from the screen — triangle left the board."],[307.35962499999994,"triangle_heading is hidden from the screen — left the board."]]}]},{"title":"Length and Simultaneity","start":308.4012916666666,"end":540.6758124999999,"objects":{"clock_length":"a Line [green] labelled \"L\" drawn in long_clock (start=(1.0, 0.35), end=(6.0, 0.35))","contraction":"a Math [text] that says \"$L = frac(L_0, gamma)$\"","dilation_relation":"a Math [text] that says \"$frac(2 L gamma^2, c) = gamma frac(2 L_0, c)$\"","front_event_rest":"a Point [green] labelled \"E_F\" drawn in train_frame (location=(7.2, 1.45))","front_mirror":"a Line [blue] labelled \"upright(\"front\")\" drawn in long_clock (start=(6.0, 0.5), end=(6.0, 1.9))","inbound":"an Arrow [red] labelled \"c+v\" drawn in long_clock (start=(6.0, 1.55), end=(1.0, 1.55))","lamp_rest":"a Point [yellow] labelled \"upright(\"lamp\")\" drawn in train_frame (location=(4.0, 1.45))","left_flash_rest":"an Arrow [yellow] drawn in train_frame (start=(4.0, 1.45), end=(0.8, 1.45))","left_pulse":"a Point [yellow] labelled \"L\" drawn in platform (location=((5.0 - platform_time), 1.45))","length_heading":"a Heading that says \"Turn the Light Clock Sideways\"","length_work":"a Derivation [text] that says \"$t_+ &= frac(L, c-v) \\ t_- &= frac(L, c+v) \\ Delta t &= frac(L, c-v)+frac(L, c+v) \\ &= frac(2 L c, c^2-v^2)=frac(2 L gamma^2, c)$\"","long_clock":"a Figure (x_range=(0.0, 7.0), y_range=(0.0, 2.4), aspect=(7.0, 2.4))","moving_front":"a Point [green] labelled \"upright(\"front\")\" drawn in platform (location=((8.0 + (0.5 * platform_time)), 1.45))","moving_lamp":"a Point [blue] labelled \"upright(\"lamp\")\" drawn in platform (location=((5.0 + (0.5 * platform_time)), 1.45))","moving_rear":"a Point [red] labelled \"upright(\"rear\")\" drawn in platform (location=((2.0 + (0.5 * platform_time)), 1.45))","moving_train":"a Polygon [gray] drawn in platform (vertices=(((2.0 + (0.5 * platform_time)), 0.6), ((8.0 + (0.5 * platform_…, filled=False)","outbound":"an Arrow [yellow] labelled \"c-v\" drawn in long_clock (start=(1.0, 1.2), end=(6.0, 1.2))","passenger_label":"a Tex [text] that says \"Passengers\"","platform":"a Figure (x_range=(0.0, 12.5), y_range=(0.0, 3.0), aspect=(12.5, 3.0))","platform_label":"a Tex [text] that says \"Platform observer\"","platform_time":"a VariableNumber","proper_roundtrip":"a Math [text] that says \"$Delta tau = frac(2 L_0, c)$\"","rear_event_rest":"a Point [red] labelled \"E_R\" drawn in train_frame (location=(0.8, 1.45))","rear_mirror":"a Line [blue] labelled \"upright(\"rear\")\" drawn in long_clock (start=(1.0, 0.5), end=(1.0, 1.9))","right_flash_rest":"an Arrow [yellow] drawn in train_frame (start=(4.0, 1.45), end=(7.2, 1.45))","right_pulse":"a Point [yellow] labelled \"R\" drawn in platform (location=((5.0 + platform_time), 1.45))","roundtrip":"a Math [text] that says \"$Delta t = frac(2 L gamma^2, c)$\"","sim_formula":"a Math [text] that says \"$Delta t' = gamma (Delta t - frac(v Delta x, c^2))$\"","sim_heading":"a Heading that says \"Simultaneity Depends on the Frame\"","sim_note":"a Panel that says \"Two separated events can be simultaneous in one inertial frame and occur at different times in another.\"","train_body_rest":"a Polygon [gray] drawn in train_frame (vertices=((0.8, 0.6), (7.2, 0.6), (7.2, 2.3), (0.8, 2.3)), filled=False)","train_frame":"a Figure (x_range=(0.0, 8.0), y_range=(0.0, 3.0), aspect=(8.0, 3.0))","train_heading":"a Heading that says \"A Lamp at the Centre of a Train\""},"beats":[{"start":308.4012916666666,"say":"Time dilation also forces a change in measured length. Turn a light clock so its pulse travels along the direction of motion.","live":[],"does":[[308.4012916666666,"length_heading is shown on the screen, written out."],[308.4012916666666,"long_clock is shown on the screen, written out."],[311.07129166666664,"clock_length is shown on the screen, written out."],[313.0452916666666,"rear_mirror is shown on the screen, written out."],[313.0452916666666,"front_mirror is shown on the screen, written out."]]},{"start":317.3487916666666,"say":"In the platform frame, call the moving mirror separation L. On the outward trip the front mirror runs away from the light, so the closing speed is c minus v.","live":["long_clock","length_heading","rear_mirror","front_mirror","clock_length"],"does":[[322.70129166666663,"outbound is shown on the screen, written out."],[326.9262916666666,"long_clock moves to a new place on the board."],[326.9262916666666,"length_work is shown on the screen, written out."]]},{"start":329.18679166666664,"say":"On the return trip the rear mirror moves toward the light. The closing speed is c plus v.","live":["long_clock","length_heading","rear_mirror","front_mirror","clock_length","outbound"],"does":[[329.9182916666666,"inbound is shown on the screen, written out."],[334.3652916666666,"length_work is shown on the screen, written out."]]},{"start":336.46279166666665,"say":"Add the two travel times. The common denominator is c squared minus v squared.","live":["long_clock","length_heading","rear_mirror","front_mirror","clock_length","outbound","inbound"],"does":[[336.81129166666665,"length_work is shown on the screen, written out."]]},{"start":343.79679166666665,"say":"After simplifying, the platform round trip is two L gamma squared over c.","live":null,"does":[[344.51629166666663,"length_work is shown on the screen, written out."],[349.63629166666664,"length_work is hidden from the screen — left the board."],[349.63629166666664,"long_clock is hidden from the screen — left the board."],[349.63629166666664,"rear_mirror is hidden from the screen — long_clock left the board."],[349.63629166666664,"front_mirror is hidden from the screen — long_clock left the board."],[349.63629166666664,"clock_length is hidden from the screen — long_clock left the board."],[349.63629166666664,"outbound is hidden from the screen — long_clock left the board."],[349.63629166666664,"inbound is hidden from the screen — long_clock left the board."]]},{"start":350.8362916666666,"say":"Carry that platform result forward. In the clock's rest frame, the mirror separation is its proper length L zero, so the round trip is two L zero over c.","live":["length_heading"],"does":[[350.8362916666666,"roundtrip is shown on the screen, written out."],[357.32629166666663,"proper_roundtrip is shown on the screen, written out."]]},{"start":363.1857916666666,"say":"Time dilation says the platform round trip equals gamma times the proper round trip. Equate those two expressions.","live":["length_heading","roundtrip","proper_roundtrip"],"does":[[369.8032916666666,"dilation_relation is shown on the screen, written out."]]},{"start":372.7367916666666,"say":"Cancel the common factors. The moving length L equals the proper length divided by gamma. A moving object is shorter along its direction of motion.","live":["length_heading","roundtrip","proper_roundtrip","dilation_relation"],"does":[[378.3442916666666,"contraction is shown on the screen, written out."],[383.90579166666663,"A box is drawn around contraction."]]},{"start":384.5057916666666,"say":"Length means the distance between two endpoints measured at the same time. That last phrase is essential. If the endpoints are moving, recording one endpoint now and the other endpoint later does not measure one length.","live":["length_heading","roundtrip","proper_roundtrip","dilation_relation","contraction"],"does":[[384.85429166666665,"contraction (the \"L\" part) is emphasized."],[388.10529166666663,"contraction (the \"L\" part) is no longer emphasized."],[399.0292916666666,"contraction is hidden from the screen — left the board."],[399.0292916666666,"dilation_relation is hidden from the screen — left the board."],[399.0292916666666,"length_heading is hidden from the screen — left the board."],[399.0292916666666,"proper_roundtrip is hidden from the screen — left the board."],[399.0292916666666,"roundtrip is hidden from the screen — left the board."]]},{"start":400.22929166666665,"say":"To see why same time becomes delicate, put a lamp exactly at the centre of a train. The passengers know the rear and front walls are equally far from the lamp.","live":[],"does":[[400.22929166666665,"train_heading is shown on the screen, written out."],[400.22929166666665,"passenger_label is shown on the screen, written out."],[400.22929166666665,"train_frame is shown on the screen, written out."],[403.7122916666666,"lamp_rest is shown on the screen, written out."],[405.1992916666666,"train_body_rest is shown on the screen, written out."]]},{"start":410.9187916666666,"say":"The lamp flashes. Light travels left and right at the same speed c, through equal distances, so it strikes the rear and front simultaneously in the train frame.","live":["passenger_label","train_frame","train_heading","train_body_rest","lamp_rest"],"does":[[413.80929166666664,"left_flash_rest is shown on the screen, written out."],[414.18129166666665,"right_flash_rest is shown on the screen, written out."],[418.72029166666664,"rear_event_rest is shown on the screen, written out."],[419.0922916666666,"front_event_rest is shown on the screen, written out."]]},{"start":422.2112916666666,"say":"Now describe the same experiment from the platform. At emission, the lamp is between the moving rear and front walls. Both light pulses still travel at speed c in this frame.","live":["passenger_label","train_frame","train_heading","train_body_rest","lamp_rest","left_flash_rest","right_flash_rest","rear_event_rest","front_event_rest"],"does":[[424.5912916666666,"platform_label is shown on the screen, written out."],[424.5912916666666,"platform is shown on the screen, written out."],[427.2502916666666,"moving_lamp is shown on the screen, written out."],[428.1902916666666,"moving_train is shown on the screen, written out."],[428.5622916666666,"moving_rear is shown on the screen, written out."],[429.2122916666666,"moving_front is shown on the screen, written out."],[431.32529166666666,"left_pulse is shown on the screen, written out."],[431.52529166666665,"right_pulse is shown on the screen, written out."]]},{"start":435.84929166666666,"say":"The rear wall moves toward the left-going pulse. The front wall moves away from the right-going pulse. Let the experiment run until the left pulse meets the rear.","live":["passenger_label","train_frame","platform_label","platform","train_heading","train_body_rest","lamp_rest","left_flash_rest","right_flash_rest","rear_event_rest","front_event_rest","moving_train","moving_lamp","moving_rear","moving_front","left_pulse","right_pulse"],"does":[[443.53529166666664,"moving_train is redrawn as the numbers it depends on change."],[443.53529166666664,"moving_lamp is redrawn as the numbers it depends on change."],[443.53529166666664,"moving_rear is redrawn as the numbers it depends on change."],[443.53529166666664,"moving_front is redrawn as the numbers it depends on change."],[443.53529166666664,"left_pulse is redrawn as the numbers it depends on change."],[443.53529166666664,"right_pulse is redrawn as the numbers it depends on change."],[443.53529166666664,"platform_time ticks to 2.0."]]},{"start":446.9327916666666,"say":"The rear has already been struck, but the right pulse still has farther to go. Remove the completed pulse and continue.","live":null,"does":[[453.2952916666666,"left_pulse is hidden from the screen."]]},{"start":455.9152916666666,"say":"Only later does the right pulse catch the front wall. The platform observer therefore says rear first, front second.","live":["passenger_label","train_frame","platform_label","platform","train_heading","train_body_rest","lamp_rest","left_flash_rest","right_flash_rest","rear_event_rest","front_event_rest","moving_train","moving_lamp","moving_rear","moving_front","right_pulse"],"does":[[456.5882916666666,"moving_train is redrawn as the numbers it depends on change."],[456.5882916666666,"moving_lamp is redrawn as the numbers it depends on change."],[456.5882916666666,"moving_rear is redrawn as the numbers it depends on change."],[456.5882916666666,"moving_front is redrawn as the numbers it depends on change."],[456.5882916666666,"right_pulse is redrawn as the numbers it depends on change."],[456.5882916666666,"platform_time ticks to 6.0."],[458.2372916666666,"right_pulse is indicated — a transient flash."]]},{"start":464.4907916666666,"say":"This is not a disagreement about light taking time to reach someone's eyes. Each observer can correct for signal travel. They disagree about whether the two distant contact events share one time coordinate.","live":null,"does":[[478.3187916666666,"passenger_label is hidden from the screen — left the board."],[478.3187916666666,"platform is hidden from the screen — left the board."],[478.3187916666666,"moving_train is hidden from the screen — platform left the board."],[478.3187916666666,"moving_lamp is hidden from the screen — platform left the board."],[478.3187916666666,"moving_rear is hidden from the screen — platform left the board."],[478.3187916666666,"moving_front is hidden from the screen — platform left the board."],[478.3187916666666,"right_pulse is hidden from the screen — platform left the board."],[478.3187916666666,"platform_label is hidden from the screen — left the board."],[478.3187916666666,"train_frame is hidden from the screen — left the board."],[478.3187916666666,"train_body_rest is hidden from the screen — train_frame left the board."],[478.3187916666666,"lamp_rest is hidden from the screen — train_frame left the board."],[478.3187916666666,"left_flash_rest is hidden from the screen — train_frame left the board."],[478.3187916666666,"right_flash_rest is hidden from the screen — train_frame left the board."],[478.3187916666666,"rear_event_rest is hidden from the screen — train_frame left the board."],[478.3187916666666,"front_event_rest is hidden from the screen — train_frame left the board."],[478.3187916666666,"train_heading is hidden from the screen — left the board."]]},{"start":479.51879166666663,"say":"The algebraic statement is this. The time separation in the moving frame equals gamma times delta t minus v delta x over c squared.","live":[],"does":[[479.51879166666663,"sim_heading is shown on the screen, written out."],[479.9712916666666,"sim_formula is shown on the screen, written out."]]},{"start":490.80029166666657,"say":"If two events are simultaneous in one frame, delta t is zero. If they are separated in space, delta x is not zero, so another moving frame generally assigns a nonzero time separation.","live":["sim_formula","sim_heading"],"does":[[494.4222916666666,"sim_formula (the \"Delta t\" part) is emphasized."],[498.5442916666666,"sim_formula (the \"Delta t\" part) is no longer emphasized."],[498.5442916666666,"sim_formula (the \"Delta x\" part) is emphasized."],[505.1737916666666,"sim_formula (the \"Delta x\" part) is no longer emphasized."]]},{"start":505.7737916666666,"say":"This is difficult because everyday objects move far too slowly for the disagreement to notice. It is also difficult because simultaneity feels like a fact about the events themselves. Relativity says it is a relation between the events and a chosen inertial frame.","live":null,"does":[[513.2152916666665,"sim_note is shown on the screen, written out."]]},{"start":523.2772916666665,"say":"The train passengers and the platform observer agree on every event: the emission, the rear strike, and the front strike. What differs is the time coordinate assigned to separated events. A spacetime diagram makes that distinction visible.","live":["sim_formula","sim_note","sim_heading"],"does":[[533.1222916666666,"A box is drawn around sim_formula."],[539.6341458333333,"sim_formula is hidden from the screen — left the board."],[539.6341458333333,"sim_heading is hidden from the screen — left the board."],[539.6341458333333,"sim_note is hidden from the screen — left the board."]]}]},{"title":"Tilted Spacetime Axes","start":540.6758124999999,"end":720.1887083333332,"objects":{"diagram":"an Axes (x_range=(-5.0, 5.0), y_range=(-5.0, 5.0), aspect=(1.0, 1.0))","event_a":"a Point [green] labelled \"E_1\" drawn in diagram (location=(-2.0, 1.0))","event_b":"a Point [green] labelled \"E_2\" drawn in diagram (location=(2.0, 1.0))","event_c":"a Point [red] labelled \"E_3\" drawn in diagram (location=(2.0, 3.4))","heading":"a Heading that says \"One Picture of Space and Time\"","interval_law":"a Math [text] that says \"$(c Delta t)^2-Delta x^2=(c Delta tau)^2$\"","lab_simultaneous":"a Line [green] labelled \"Delta t=0\" drawn in diagram (start=(-2.0, 1.0), end=(2.0, 1.0))","light_left":"a Line [yellow] labelled \"upright(\"light\")\" drawn in diagram (start=(-5.0, 5.0), end=(5.0, -5.0))","light_right":"a Line [yellow] labelled \"upright(\"light\")\" drawn in diagram (start=(-5.0, -5.0), end=(5.0, 5.0))","moving_simultaneous":"a Line [red] labelled \"Delta t'=0\" drawn in diagram (start=(-2.0, 1.0), end=(2.0, 3.4))","moving_space_axis":"a Line [red] labelled \"x'\" drawn in diagram (start=(-5.0, -3.0), end=(5.0, 3.0))","moving_time_axis":"a Line [red] labelled \"c t'\" drawn in diagram (start=(-3.0, -5.0), end=(3.0, 5.0))","stationary_worldline":"a Line [blue] labelled \"upright(\"stationary observer\")\" drawn in diagram (start=(0.0, -5.0), end=(0.0, 5.0))","transforms":"a Derivation [text] that says \"$x' &= gamma (x-v t) \\ c t' &= gamma (c t-frac(v, c)x) \\ x'=0 &arrow.r x=frac(v, c)c t \\ t'=0 &arrow.r c t=frac(v, c)x$\""},"beats":[{"start":540.6758124999999,"say":"A spacetime diagram puts position on the horizontal axis and c times time on the vertical axis. Multiplying time by c gives both axes the same units.","live":[],"does":[[540.6758124999999,"heading is shown on the screen, written out."],[540.6758124999999,"diagram is shown on the screen, written out."],[546.0168124999999,"stationary_worldline is shown on the screen, written out."]]},{"start":552.6418124999999,"say":"An object at rest stays at one x position, so its history is a vertical worldline. A moving object changes x as time passes, so its worldline tilts.","live":["diagram","heading","stationary_worldline"],"does":[[557.2048124999999,"stationary_worldline is indicated — a transient flash."]]},{"start":564.9913124999999,"say":"Light is special. Since its speed is c, one unit of horizontal travel takes one unit of c times time. Light rays therefore follow these two forty-five-degree lines.","live":null,"does":[[565.5488124999999,"light_right is shown on the screen, written out."],[575.2428124999999,"light_right is indicated — a transient flash."],[576.0668124999999,"light_left is shown on the screen, written out."]]},{"start":577.5378124999999,"say":"Every inertial observer must draw light on those same lines. That shared light cone constrains how another observer's coordinate axes can sit in the diagram.","live":["diagram","heading","stationary_worldline","light_right","light_left"],"does":[]},{"start":588.1808124999999,"say":"The red c t prime axis is the moving observer's own worldline, x prime equal to zero. It tilts toward the right-going light ray.","live":null,"does":[[588.8658125,"moving_time_axis is shown on the screen, written out."],[596.0758124999999,"moving_time_axis is indicated — a transient flash."]]},{"start":598.8813124999999,"say":"The red x prime axis is the set of events that the moving observer calls simultaneous with the origin. It tilts upward by the matching amount.","live":["diagram","heading","stationary_worldline","light_right","light_left","moving_time_axis"],"does":[[599.8918124999999,"moving_space_axis is shown on the screen, written out."],[606.5788124999999,"moving_space_axis is indicated — a transient flash."]]},{"start":609.3503124999999,"say":"So the moving observer's space and time axes tilt toward the light lines and toward each other. They are not a rotated pair of ordinary Euclidean axes. Their scaling is controlled by gamma.","live":["diagram","heading","stationary_worldline","light_right","light_left","moving_time_axis","moving_space_axis"],"does":[[611.4868124999999,"moving_time_axis is indicated — a transient flash."],[614.2028124999999,"moving_space_axis is indicated — a transient flash."]]},{"start":622.3963124999999,"say":"The coordinate equations are the Lorentz transformation. Position x prime equals gamma times x minus v t.","live":null,"does":[[624.2998124999999,"diagram moves to a new place on the board."],[624.2998124999999,"transforms is shown on the screen, written out."],[629.0258124999999,"transforms (the \"x-v t\" part) is emphasized."],[630.4883124999999,"transforms (the \"x-v t\" part) is no longer emphasized."]]},{"start":631.0883124999999,"say":"The time equation mixes time with position. That mixed term, v x over c squared in ordinary time units, is the algebraic source of relative simultaneity.","live":null,"does":[[631.6688124999998,"transforms is shown on the screen, written out."],[634.9548124999999,"transforms (the \"frac(v, c)x\" part) is emphasized."],[642.3503124999999,"transforms (the \"frac(v, c)x\" part) is no longer emphasized."]]},{"start":642.9503124999999,"say":"Set x prime equal to zero. The result is the tilted time axis, the moving observer's worldline.","live":null,"does":[[644.7848124999999,"transforms is shown on the screen, written out."],[649.4748124999999,"moving_time_axis is indicated — a transient flash."]]},{"start":651.1603124999999,"say":"Set t prime equal to zero. The result is the tilted space axis, the moving observer's line of simultaneous events.","live":null,"does":[[652.9198124999999,"transforms is shown on the screen, written out."],[658.0398124999999,"moving_space_axis is indicated — a transient flash."]]},{"start":660.3753124999998,"say":"Now choose two events E one and E two on one horizontal line. The stationary observer assigns them the same time.","live":null,"does":[[662.1218124999999,"event_a is shown on the screen, written out."],[663.0858124999999,"event_b is shown on the screen, written out."],[664.2938125,"lab_simultaneous is shown on the screen, written out."]]},{"start":669.5953124999999,"say":"The moving observer's simultaneous line through E one is tilted. It reaches the same distant position at E three, not at E two.","live":["diagram","heading","stationary_worldline","light_right","light_left","moving_time_axis","moving_space_axis","event_a","event_b","lab_simultaneous"],"does":[[673.1128124999999,"moving_simultaneous is shown on the screen, written out."],[676.2938124999998,"event_c is shown on the screen, written out."]]},{"start":679.1578124999999,"say":"No event moves when we change coordinates. E one, E two, and E three stay where they are. What changes is which distant events one observer groups into a single present.","live":["diagram","heading","stationary_worldline","light_right","light_left","moving_time_axis","moving_space_axis","event_a","event_b","lab_simultaneous","event_c","moving_simultaneous"],"does":[[689.6418124999999,"lab_simultaneous is indicated — a transient flash."],[690.5708124999999,"moving_simultaneous is indicated — a transient flash."]]},{"start":692.4593124999999,"say":"The invariant quantity is not Euclidean distance on this page. Write it first. The time part is c delta t squared. The space part is delta x squared, and their difference has the same value for every inertial observer.","live":null,"does":[[692.4593124999999,"interval_law is shown on the screen, written out."],[698.3228124999999,"interval_law (the \"(c Delta t)^2\" part) is emphasized."],[701.1668124999999,"interval_law (the \"(c Delta t)^2\" part) is no longer emphasized."],[701.1668124999999,"interval_law (the \"Delta x^2\" part) is emphasized."],[704.4758124999998,"interval_law (the \"Delta x^2\" part) is no longer emphasized."]]},{"start":707.7923124999999,"say":"For events on one clock's worldline, that invariant equals c delta tau squared. This is why different paths between the same meetings can contain different amounts of proper time.","live":["interval_law","diagram","heading","stationary_worldline","light_right","light_left","moving_time_axis","moving_space_axis","event_a","event_b","lab_simultaneous","event_c","moving_simultaneous"],"does":[[717.8348124999999,"A box is drawn around interval_law."],[719.1470416666666,"diagram is hidden from the screen — left the board."],[719.1470416666666,"stationary_worldline is hidden from the screen — diagram left the board."],[719.1470416666666,"light_right is hidden from the screen — diagram left the board."],[719.1470416666666,"light_left is hidden from the screen — diagram left the board."],[719.1470416666666,"moving_time_axis is hidden from the screen — diagram left the board."],[719.1470416666666,"moving_space_axis is hidden from the screen — diagram left the board."],[719.1470416666666,"event_a is hidden from the screen — diagram left the board."],[719.1470416666666,"event_b is hidden from the screen — diagram left the board."],[719.1470416666666,"lab_simultaneous is hidden from the screen — diagram left the board."],[719.1470416666666,"event_c is hidden from the screen — diagram left the board."],[719.1470416666666,"moving_simultaneous is hidden from the screen — diagram left the board."],[719.1470416666666,"heading is hidden from the screen — left the board."],[719.1470416666666,"interval_law is hidden from the screen — left the board."],[719.1470416666666,"transforms is hidden from the screen — left the board."]]}]},{"title":"The Twin Paradox","start":720.1887083333332,"end":935.8256874999998,"objects":{"assignment_change":"a Math [text] that says \"$Delta t_(upright(\"assignment\"))=8.2-1.8=6.4 thin upright(\"y\")$\"","earth_age":"a Math [text] that says \"$Delta tau_E=10 thin upright(\"y\")$\"","earth_in_reading":"a Point [green] labelled \"8.2 thin upright(\"y\")\" drawn in twins (location=(0.0, 8.2))","earth_line":"a Line [blue] labelled \"upright(\"Earth twin\")\" drawn in twins (end=(0.0, 10.0))","earth_out_reading":"a Point [yellow] labelled \"1.8 thin upright(\"y\")\" drawn in twins (location=(0.0, 1.8))","gamma_value":"a Math [text] that says \"$gamma=frac(1, sqrt(1-0.8^2))=frac(5, 3)$\"","heading":"a Heading that says \"One Twin Stays, One Twin Turns Around\"","in_assignment":"a Math [text] that says \"$t_(E, upright(\"in\"))=5+0.8(4)=8.2 thin upright(\"y\")$\"","incoming_slice":"a Line [green] labelled \"upright(\"inbound now\")\" drawn in twins (start=(0.0, 8.2), end=(5.0, 4.2))","out_assignment":"a Math [text] that says \"$t_(E, upright(\"out\"))=5-0.8(4)=1.8 thin upright(\"y\")$\"","outgoing_slice":"a Line [yellow] labelled \"upright(\"outbound now\")\" drawn in twins (start=(0.0, 1.8), end=(5.0, 5.8))","outward_line":"a Line [red] labelled \"upright(\"outbound\")\" drawn in twins (end=(4.0, 5.0))","return_line":"a Line [red] labelled \"upright(\"inbound\")\" drawn in twins (start=(4.0, 5.0), end=(0.0, 10.0))","reunion":"a Point [green] labelled \"upright(\"reunion\")\" drawn in twins (location=(0.0, 10.0))","setup":"a Math [text] that says \"$v=0.8c, quad Delta t_(upright(\"leg\"))=5 thin upright(\"y\")$\"","traveler_age":"a Math [text] that says \"$Delta tau_T=2 frac(5 thin upright(\"y\"), 5/3)=6 thin upright(\"y\")$\"","turn_event":"a Point [yellow] labelled \"upright(\"turnaround\")\" drawn in twins (location=(4.0, 5.0))","twins":"an Axes (x_range=(-0.2, 5.0), y_range=(0.0, 11.0), aspect=(5.2, 11.0))"},"beats":[{"start":720.1887083333332,"say":"Two twins meet on Earth, synchronize clocks, and separate. One remains on Earth. The other travels outward at eighty percent of light speed, turns around, and returns at the same speed.","live":[],"does":[[720.1887083333332,"heading is shown on the screen, written out."],[720.1887083333332,"twins is shown on the screen, written out."],[725.2277083333332,"earth_line is shown on the screen, written out."],[727.9327083333333,"outward_line is shown on the screen, written out."],[731.3227083333333,"return_line is shown on the screen, written out."]]},{"start":733.7337083333332,"say":"The spacetime diagram makes the histories different immediately. The Earth twin follows one straight inertial worldline. The traveller follows one straight segment out and a different straight segment home.","live":["twins","heading","earth_line","outward_line","return_line"],"does":[[739.4927083333332,"earth_line is indicated — a transient flash."],[743.5677083333333,"outward_line is indicated — a transient flash."],[745.6107083333333,"return_line is indicated — a transient flash."]]},{"start":747.4067083333332,"say":"Let each leg last five Earth years. At speed zero point eight c, the turnaround is four light years away, and the reunion occurs after ten Earth years.","live":null,"does":[[748.8227083333333,"twins moves to a new place on the board."],[748.8227083333333,"setup is shown on the screen, written out."],[753.1187083333332,"turn_event is shown on the screen, written out."],[755.6617083333332,"reunion is shown on the screen, written out."]]},{"start":758.8162083333332,"say":"For zero point eight c, gamma is five thirds.","live":["setup","twins","heading","earth_line","outward_line","return_line","turn_event","reunion"],"does":[[761.6257083333333,"gamma_value is shown on the screen, written out."]]},{"start":763.4677083333332,"say":"The Earth clock is present at departure and reunion, and it records ten years of proper time.","live":["setup","gamma_value","twins","heading","earth_line","outward_line","return_line","turn_event","reunion"],"does":[[767.4737083333332,"earth_age is shown on the screen, written out."],[768.2277083333332,"earth_line is indicated — a transient flash."]]},{"start":770.0582083333333,"say":"On either travelling leg, five Earth-frame years correspond to five divided by gamma, which is three traveller years. Two legs therefore give six traveller years.","live":["setup","gamma_value","earth_age","twins","heading","earth_line","outward_line","return_line","turn_event","reunion"],"does":[[776.0137083333333,"traveler_age is shown on the screen, written out."],[777.8137083333332,"outward_line is indicated — a transient flash."],[779.0557083333332,"return_line is indicated — a transient flash."]]},{"start":781.1882083333333,"say":"At reunion the clocks can be compared at one place. The Earth twin has aged ten years and the travelling twin six. There is no dispute about that local comparison.","live":["setup","gamma_value","earth_age","traveler_age","twins","heading","earth_line","outward_line","return_line","turn_event","reunion"],"does":[[786.0297083333332,"earth_age is indicated — a transient flash."],[787.8867083333332,"traveler_age is indicated — a transient flash."],[792.0317083333332,"A box is drawn around traveler_age."]]},{"start":792.6317083333332,"say":"The apparent paradox comes from saying that each inertial observer sees the other's clock run slowly. If the roles were perfectly symmetric, why would one twin be younger?","live":null,"does":[[802.7557083333332,"traveler_age moves to a new place on the board."],[802.7557083333332,"earth_age is hidden from the screen — left the board."],[802.7557083333332,"gamma_value is hidden from the screen — left the board."],[802.7557083333332,"setup is hidden from the screen — left the board."],[802.7557083333332,"The box around traveler_age is lifted."]]},{"start":803.9557083333332,"say":"The roles are symmetric only during any one inertial leg. Earth remains in one inertial frame throughout the experiment. The traveller changes inertial frames at the turnaround.","live":["traveler_age","twins","heading","earth_line","outward_line","return_line","turn_event","reunion"],"does":[[812.8487083333332,"turn_event is indicated — a transient flash."]]},{"start":816.1887083333332,"say":"Before turning, the traveller uses an outbound line of simultaneity. On these axes, one year of c times time has the same scale as one light year. Draw that line through the turnaround event.","live":null,"does":[[818.7317083333332,"outgoing_slice is shown on the screen, written out."],[819.2417083333332,"earth_out_reading is shown on the screen, written out."]]},{"start":829.7807083333332,"say":"Where that line meets Earth's worldline, the outbound coordinates assign Earth a time of one point eight years at the distant turnaround moment.","live":["traveler_age","twins","heading","earth_line","outward_line","return_line","turn_event","reunion","outgoing_slice","earth_out_reading"],"does":[[835.0047083333333,"out_assignment is shown on the screen, written out."],[835.0047083333333,"earth_out_reading is indicated — a transient flash."]]},{"start":838.6117083333332,"say":"After turning, the traveller belongs to the inbound frame. Its line of simultaneity slopes the other way.","live":["traveler_age","twins","heading","earth_line","outward_line","return_line","turn_event","reunion","out_assignment","outgoing_slice","earth_out_reading"],"does":[[841.1427083333333,"incoming_slice is shown on the screen, written out."],[844.6377083333332,"earth_in_reading is shown on the screen, written out."]]},{"start":846.1892083333332,"say":"That new line meets Earth's worldline at eight point two years. The inbound coordinates assign a much later Earth time to events simultaneous with the same turnaround.","live":["traveler_age","twins","heading","earth_line","outward_line","return_line","turn_event","reunion","out_assignment","outgoing_slice","earth_out_reading","incoming_slice","earth_in_reading"],"does":[[848.6157083333333,"in_assignment is shown on the screen, written out."],[848.6157083333333,"earth_in_reading is indicated — a transient flash."]]},{"start":856.7737083333333,"say":"The difference between those coordinate assignments is six point four years. This is where the asymmetry lives: the traveller changes which distant Earth event belongs to the traveller's present.","live":["traveler_age","twins","heading","earth_line","outward_line","return_line","turn_event","reunion","out_assignment","in_assignment","outgoing_slice","earth_out_reading","incoming_slice","earth_in_reading"],"does":[[859.4087083333333,"assignment_change is shown on the screen, written out."],[864.5287083333333,"outgoing_slice is indicated — a transient flash."],[867.3737083333333,"incoming_slice is indicated — a transient flash."]]},{"start":868.8907083333332,"say":"Earth does not physically age six point four years in an instant. Nothing discontinuous happens to Earth's clock. What changes abruptly, in an idealized instantaneous turnaround, is the traveller's coordinate rule for distant simultaneity.","live":["traveler_age","twins","heading","earth_line","outward_line","return_line","turn_event","reunion","out_assignment","in_assignment","assignment_change","outgoing_slice","earth_out_reading","incoming_slice","earth_in_reading"],"does":[[881.8707083333333,"assignment_change (the \"upright(\"assignment\")\" part) is emphasized."],[884.3437083333332,"assignment_change (the \"upright(\"assignment\")\" part) is no longer emphasized."]]},{"start":884.9437083333332,"say":"With a realistic gradual turn, the reassignment is gradual too. The final proper times remain essentially the same if the acceleration interval is short compared with the journey.","live":null,"does":[]},{"start":896.4917083333332,"say":"Acceleration identifies which twin changes frames, but acceleration is not an extra aging penalty added by hand. The elapsed proper time comes from the complete spacetime path.","live":null,"does":[[906.3017083333332,"outward_line is indicated — a transient flash."],[907.4277083333333,"return_line is indicated — a transient flash."]]},{"start":908.8287083333332,"say":"Between the same departure and reunion events, the straight inertial worldline contains more proper time than the broken out-and-back path. Earth records ten years. The traveller records six.","live":null,"does":[[912.1147083333333,"earth_line is indicated — a transient flash."],[920.3807083333332,"traveler_age is indicated — a transient flash."]]},{"start":921.8047083333332,"say":"So the twin paradox is not a contradiction in time dilation. It is a warning that time dilation applies within one inertial comparison, while the traveller's full journey requires two different inertial frames.","live":null,"does":[[932.6827083333333,"A box is drawn around traveler_age."],[934.7840208333332,"assignment_change is hidden from the screen — left the board."],[934.7840208333332,"heading is hidden from the screen — left the board."],[934.7840208333332,"in_assignment is hidden from the screen — left the board."],[934.7840208333332,"out_assignment is hidden from the screen — left the board."],[934.7840208333332,"traveler_age is hidden from the screen — left the board."],[934.7840208333332,"twins is hidden from the screen — left the board."],[934.7840208333332,"earth_line is hidden from the screen — twins left the board."],[934.7840208333332,"outward_line is hidden from the screen — twins left the board."],[934.7840208333332,"return_line is hidden from the screen — twins left the board."],[934.7840208333332,"turn_event is hidden from the screen — twins left the board."],[934.7840208333332,"reunion is hidden from the screen — twins left the board."],[934.7840208333332,"outgoing_slice is hidden from the screen — twins left the board."],[934.7840208333332,"earth_out_reading is hidden from the screen — twins left the board."],[934.7840208333332,"incoming_slice is hidden from the screen — twins left the board."],[934.7840208333332,"earth_in_reading is hidden from the screen — twins left the board."]]}]},{"title":"Muons That Reach the Ground","start":935.8256874999998,"end":1154.4642499999998,"objects":{"approaching_ground":"a Point [green] labelled \"upright(\"ground\")\" drawn in muon_track","contracted":"a Math [text] that says \"$L'=frac(10 thin upright(\"km\"), 15.8) approx 0.63 thin upright(\"km\")$\"","contracted_distance":"a Brace [text] labelled \"0.63 thin upright(\"km\")\" drawn in muon_track (x_start=0.0, x_end=0.63)","earth_description":"a Tex [text] that says \"Earth description\"","earth_distance":"a Brace [text] labelled \"10 thin upright(\"km\")\" drawn in earth_track (x_start=0.0, x_end=10.0)","earth_muon":"a Point [yellow] labelled \"upright(\"muon\")\" drawn in earth_track (location=(<VariableNumber muon_position = 10.0>, 0.0))","earth_track":"a NumberLine labelled \"d thin upright(\"(km)\")\" (x_range=(0.0, 11.0), include_numbers=True)","ground_position":"a VariableNumber (initial_value=0.63)","m0":"a Math [text] that says \"$tau_0=2.2 thin upright(\"microseconds\")$\"","m1":"a Math [text] that says \"$c tau_0 approx 0.66 thin upright(\"km\")$\"","m2":"a Math [text] that says \"$v=0.998c &arrow.r gamma approx 15.8$\"","m3":"a Math [text] that says \"$tau_(upright(\"Earth\"))=gamma tau_0 approx 34.8 thin upright(\"microseconds\")$\"","m4":"a Math [text] that says \"$d=v tau_(upright(\"Earth\")) approx 10.4 thin upright(\"km\")$\"","muon_at_rest":"a Point [yellow] labelled \"upright(\"muon\")\" drawn in muon_track","muon_description":"a Tex [text] that says \"Muon description\"","muon_heading":"a Heading that says \"Two Frames, One Arrival\"","muon_position":"a VariableNumber","muon_time":"a Math [text] that says \"$Delta tau=frac(0.63 thin upright(\"km\"), 0.998c) approx 2.1 thin upright(\"microseconds\")$\"","muon_track":"a NumberLine labelled \"L' thin upright(\"(km)\")\" (x_range=(0.0, 0.7), include_numbers=True, ticks_every=0.1)","point":"a Point [yellow] drawn in earth_track (location=(0.66, 0.0))","question":"a Text [text] that says \"Muons are created about 10 km above Earth and have a proper mean lifetime of only 2.2 microseconds. Why are many detected at the ground?\"","recap_five":"a Text [text] that says \"Proper time belongs to a complete spacetime path.\"","recap_four":"a Text [text] that says \"Distant simultaneity depends on the inertial frame.\"","recap_heading":"a Heading that says \"What the Two Postulates Forced\"","recap_one":"a Text [text] that says \"Light speed is $c$ for every inertial observer.\"","recap_three":"a Text [text] that says \"Moving lengths contract along the motion.\"","recap_two":"a Text [text] that says \"Moving clocks accumulate less proper time.\""},"beats":[{"start":935.8256874999998,"say":"The last test comes from particles made high in the atmosphere. Muons are created roughly ten kilometres above the ground, but their proper mean lifetime is only two point two microseconds. Why do many reach detectors at sea level?","live":[],"does":[[935.8256874999998,"question is shown on the screen, written out."],[949.9316874999998,"question moves to a new place on the board."]]},{"start":950.5316874999999,"say":"Begin with the nonrelativistic expectation. Here is the ten-kilometre atmospheric depth. The muon's own lifetime is two point two microseconds.","live":["question"],"does":[[950.5316874999999,"earth_track is shown on the screen, written out."],[954.6646874999999,"earth_distance is shown on the screen, written out."],[957.4626874999999,"earth_muon is shown on the screen, written out."],[958.7746874999998,"m0 is shown on the screen, written out."]]},{"start":961.2091874999999,"say":"Even light travels only about zero point six six kilometres in that time. A slower particle should decay far above the ground.","live":["question","earth_track","m0","earth_distance","earth_muon"],"does":[[963.1946874999999,"m1 is shown on the screen, written out."],[964.3436874999999,"point is shown on the screen, grown."],[966.3436874999999,"point is hidden from the screen."]]},{"start":969.9946874999998,"say":"But atmospheric muons commonly travel near zero point nine nine eight c. At that speed gamma is about fifteen point eight.","live":["question","earth_track","m0","m1","earth_distance","earth_muon"],"does":[[976.8906874999998,"m2 is shown on the screen, written out."]]},{"start":979.0236874999998,"say":"In Earth's frame, the moving muon's lifetime is dilated. Two point two microseconds multiplied by gamma becomes about thirty-four point eight microseconds.","live":["question","earth_track","m0","m1","m2","earth_distance","earth_muon"],"does":[[981.9716874999998,"m3 is shown on the screen, written out."],[984.8626874999999,"m3 (the \"gamma tau_0\" part) is emphasized."],[989.2516874999999,"m3 (the \"gamma tau_0\" part) is no longer emphasized."]]},{"start":989.8516874999998,"say":"At almost light speed, that dilated lifetime corresponds to about ten point four kilometres. The muon can reach the ground.","live":["question","earth_track","m0","m1","m2","m3","earth_distance","earth_muon"],"does":[[994.0656874999999,"m4 is shown on the screen, written out."],[996.9106874999999,"earth_muon is redrawn as the numbers it depends on change."],[996.9106874999999,"muon_position ticks to 10.0."],[998.1291874999998,"A box is drawn around m4."]]},{"start":998.7291874999999,"say":"That is the Earth-frame explanation: the moving particle's internal clock accumulates less time while Earth clocks record the long flight.","live":["question","earth_track","m0","m1","m2","m3","m4","earth_distance","earth_muon"],"does":[[1007.7851874999999,"earth_track is hidden from the screen."],[1007.7851874999999,"m0 is hidden from the screen — left the board."],[1007.7851874999999,"m1 is hidden from the screen — left the board."],[1007.7851874999999,"m2 is hidden from the screen — left the board."],[1007.7851874999999,"m3 is hidden from the screen — left the board."],[1007.7851874999999,"m4 is hidden from the screen — left the board."],[1007.7851874999999,"question is hidden from the screen — left the board."]]},{"start":1008.9851874999998,"say":"Now use the muon's own frame. The muon is at rest, so its lifetime is still the proper value, about two point two microseconds. The atmosphere and ground move toward it.","live":["earth_distance","earth_muon"],"does":[[1008.9851874999998,"muon_heading is shown on the screen, written out."],[1008.9851874999998,"earth_description is shown on the screen, written out."],[1008.9851874999998,"muon_description is shown on the screen, written out."],[1008.9851874999998,"earth_track is shown on the screen, written out."],[1012.1196874999998,"muon_track is shown on the screen, written out."],[1012.8746874999998,"muon_at_rest is shown on the screen, written out."],[1019.3766874999999,"approaching_ground is shown on the screen, written out."]]},{"start":1021.5206874999999,"say":"The ten-kilometre atmospheric depth is moving, so it is length contracted. Divide ten kilometres by gamma fifteen point eight.","live":["earth_track","earth_distance","earth_muon","earth_description","muon_description","muon_track","muon_heading","muon_at_rest","approaching_ground"],"does":[[1025.3636874999997,"contracted is shown on the screen, written out."]]},{"start":1030.7006874999997,"say":"The muon measures only about zero point six three kilometres between its creation event and the approaching ground.","live":["earth_track","earth_distance","earth_muon","earth_description","muon_description","muon_track","contracted","muon_heading","muon_at_rest","approaching_ground"],"does":[[1032.6746875,"contracted_distance is shown on the screen, written out."],[1037.0636874999998,"approaching_ground is indicated — a transient flash."]]},{"start":1038.4411874999998,"say":"At zero point nine nine eight c, crossing that contracted distance takes about two point one microseconds of muon proper time.","live":["earth_track","earth_distance","earth_muon","earth_description","muon_description","muon_track","contracted","muon_heading","muon_at_rest","approaching_ground","contracted_distance"],"does":[[1043.2006874999997,"muon_time is shown on the screen, written out."]]},{"start":1047.0051875,"say":"Let the ground approach. It covers the shortened distance before the muon's typical lifetime has elapsed.","live":["earth_track","earth_distance","earth_muon","earth_description","muon_description","muon_track","contracted","muon_time","muon_heading","muon_at_rest","approaching_ground","contracted_distance"],"does":[[1047.9226875,"approaching_ground is redrawn as the numbers it depends on change."],[1047.9226875,"ground_position ticks to 0.0."]]},{"start":1053.7471874999999,"say":"Earth says the muon's clock runs slowly. The muon says the atmosphere is short. These are not competing explanations. They are the time and space parts of one Lorentz transformation.","live":null,"does":[[1058.0306874999999,"contracted is indicated — a transient flash."],[1062.6396874999998,"muon_time is indicated — a transient flash."]]},{"start":1066.9316875,"say":"Both descriptions predict the same local event: the muon and the detector meet. Experiments observe far more muons at the ground than a nonrelativistic lifetime calculation permits.","live":null,"does":[[1080.0051875,"contracted is hidden from the screen — left the board."],[1080.0051875,"earth_description is hidden from the screen — left the board."],[1080.0051875,"earth_track is hidden from the screen — left the board."],[1080.0051875,"earth_distance is hidden from the screen — earth_track left the board."],[1080.0051875,"earth_muon is hidden from the screen — earth_track left the board."],[1080.0051875,"muon_description is hidden from the screen — left the board."],[1080.0051875,"muon_heading is hidden from the screen — left the board."],[1080.0051875,"muon_time is hidden from the screen — left the board."],[1080.0051875,"muon_track is hidden from the screen — left the board."],[1080.0051875,"muon_at_rest is hidden from the screen — muon_track left the board."],[1080.0051875,"approaching_ground is hidden from the screen — muon_track left the board."],[1080.0051875,"contracted_distance is hidden from the screen — muon_track left the board."]]},{"start":1081.2051874999997,"say":"Now collect the argument. First, every inertial observer measures the same light speed c.","live":[],"does":[[1081.2051874999997,"recap_heading is shown on the screen, written out."],[1084.0266874999998,"recap_one is shown on the screen, written out."]]},{"start":1089.1426874999997,"say":"The moving light clock then forces moving clocks to accumulate less proper time between shared events.","live":["recap_one","recap_heading"],"does":[[1091.2556874999998,"recap_two is shown on the screen, written out."]]},{"start":1096.3371874999998,"say":"A longitudinal light clock and the same time-dilation law force moving lengths to contract along the motion.","live":["recap_one","recap_two","recap_heading"],"does":[[1101.2716874999999,"recap_three is shown on the screen, written out."]]},{"start":1104.0191874999998,"say":"The train lamp shows the price of combining those results with constant light speed: distant simultaneity depends on the inertial frame.","live":["recap_one","recap_two","recap_three","recap_heading"],"does":[[1109.7666874999998,"recap_four is shown on the screen, written out."]]},{"start":1113.0366875,"say":"The spacetime diagram gathers all of this into tilted coordinate axes and invariant proper time. It also reveals the twin asymmetry: only the travelling twin changes inertial frames.","live":["recap_one","recap_two","recap_three","recap_four","recap_heading"],"does":[[1118.4586874999998,"recap_five is shown on the screen, written out."]]},{"start":1126.5126874999999,"say":"Finally, atmospheric muons turn the geometry into an observation. Long lifetime in Earth's coordinates and short atmosphere in the muon's coordinates are two descriptions of the same arrival.","live":["recap_one","recap_two","recap_three","recap_four","recap_five","recap_heading"],"does":[[1132.4336875,"recap_two is indicated — a transient flash."],[1134.1056875,"recap_three is indicated — a transient flash."]]},{"start":1139.3956875,"say":"Special relativity is not a story in which appearances deceive us. It is a precise rule for how different observers divide one spacetime into space and time, while agreeing on light, meetings, and every measurable event.","live":null,"does":[[1147.0466874999997,"recap_five is indicated — a transient flash."],[1149.9726875,"recap_one is indicated — a transient flash."],[1153.4225833333333,"recap_five is hidden from the screen — left the board."],[1153.4225833333333,"recap_four is hidden from the screen — left the board."],[1153.4225833333333,"recap_heading is hidden from the screen — left the board."],[1153.4225833333333,"recap_one is hidden from the screen — left the board."],[1153.4225833333333,"recap_three is hidden from the screen — left the board."],[1153.4225833333333,"recap_two is hidden from the screen — left the board."]]}]}]},"durationSeconds":1154,"chapters":[{"title":"Two Postulates and One Clock","startSeconds":0,"narration":"Special relativity begins with two statements that sound modest and turn out to reorganize space and time. We will keep one physical clock beside us while we follow their consequences. First, the laws of physics have the same form in every inertial frame. An inertial observer is simply one who is not accelerating. Second, every inertial observer measures the same speed c for light in a vacuum. The source may move, and the observer may move, but the measured light speed is still c. That second statement is the unfamiliar one. Ordinary speeds add. If you throw a ball forward from a train, someone beside the track combines the ball's speed with the train's speed. Light refuses that rule. We will not begin by guessing how clocks and rulers must change. We will ask what these two postulates force one very simple clock to do. Here is the clock. Two parallel mirrors face one another, separated by a fixed distance h. A pulse of light bounces between them. Let the pulse begin at the lower mirror. It travels upward at speed c, reaches the upper mirror, and returns. One complete up and down journey is one tick. The mirrors are separated by h, so the one-way journey has distance h. Speed is distance divided by time, which means the one-way time is h divided by c. A complete tick travels that distance twice. The clock therefore records delta tau equal to two h over c. I have called this interval delta tau because both endpoint events happen at the same place in the clock's own frame. The clock is there when the pulse leaves, and the same clock is there when the pulse returns. That is proper time: the time measured by a clock travelling with both events. Nothing relativistic has happened yet. We have only built a clock whose operation follows directly from the constant speed of light. Now let this entire clock move sideways. The passengers who carry it still see the same vertical pulse and the same tick delta tau. The interesting question is what someone beside the track must see."},{"title":"The Moving Light Clock","startSeconds":135.0288125,"narration":"The passenger sees the clock at rest. The pulse goes straight up and straight down, and one tick lasts delta tau. The platform observer sees the whole apparatus move to the right. During the first half of the tick, the upper mirror moves sideways. The light still travels at speed c, so it follows a diagonal path to catch that mirror. During the second half, the clock keeps moving while the pulse returns to the lower mirror. The platform observer therefore sees two diagonal legs, not one vertical round trip. Both observers must use the same light speed c. The platform observer sees a longer light path, so that observer must assign more time to the tick. The geometry will tell us exactly how much more. Take only the upward half of the journey. The vertical separation of the mirrors is h. If the full platform time is delta t, this half lasts delta t over two. During that half tick, the clock moves horizontally by v times delta t over two. That is the bottom leg of a right triangle. Light covers the diagonal. Since its speed is c, the diagonal length is c times delta t over two. Now use only the Pythagorean theorem. The square of the diagonal equals the square of h plus the square of the horizontal leg. Multiply out the common factors and move the horizontal contribution to the left. Delta t squared over four multiplies c squared minus v squared. Solve for delta t. The answer is two h divided by the square root of c squared minus v squared. Carry that result forward. Factor c squared out of the square root. Two h over c is exactly the proper tick delta tau that the passenger measured. The remaining factor is one over the square root of one minus v squared over c squared. We call this factor gamma. So the platform time is gamma times the proper time. Since gamma is at least one, the moving clock accumulates less time between the same two meetings. Watch the triangle as the speed rises from forty percent to eighty percent of c. The mirror separation stays fixed, but the horizontal leg and the light path both grow. At low speed gamma is almost one, which is why ordinary experience hides relativity. At eighty percent of c, gamma is five thirds. Five platform seconds correspond to only three seconds on the moving clock. Time dilation is not a mechanical defect in the light clock. Every process carried with the moving observer follows the same spacetime geometry, including chemical reactions, heartbeats, and aging."},{"title":"Length and Simultaneity","startSeconds":308.4012916666666,"narration":"Time dilation also forces a change in measured length. Turn a light clock so its pulse travels along the direction of motion. In the platform frame, call the moving mirror separation L. On the outward trip the front mirror runs away from the light, so the closing speed is c minus v. On the return trip the rear mirror moves toward the light. The closing speed is c plus v. Add the two travel times. The common denominator is c squared minus v squared. After simplifying, the platform round trip is two L gamma squared over c. Carry that platform result forward. In the clock's rest frame, the mirror separation is its proper length L zero, so the round trip is two L zero over c. Time dilation says the platform round trip equals gamma times the proper round trip. Equate those two expressions. Cancel the common factors. The moving length L equals the proper length divided by gamma. A moving object is shorter along its direction of motion. Length means the distance between two endpoints measured at the same time. That last phrase is essential. If the endpoints are moving, recording one endpoint now and the other endpoint later does not measure one length. To see why same time becomes delicate, put a lamp exactly at the centre of a train. The passengers know the rear and front walls are equally far from the lamp. The lamp flashes. Light travels left and right at the same speed c, through equal distances, so it strikes the rear and front simultaneously in the train frame. Now describe the same experiment from the platform. At emission, the lamp is between the moving rear and front walls. Both light pulses still travel at speed c in this frame. The rear wall moves toward the left-going pulse. The front wall moves away from the right-going pulse. Let the experiment run until the left pulse meets the rear. The rear has already been struck, but the right pulse still has farther to go. Remove the completed pulse and continue. Only later does the right pulse catch the front wall. The platform observer therefore says rear first, front second. This is not a disagreement about light taking time to reach someone's eyes. Each observer can correct for signal travel. They disagree about whether the two distant contact events share one time coordinate. The algebraic statement is this. The time separation in the moving frame equals gamma times delta t minus v delta x over c squared. If two events are simultaneous in one frame, delta t is zero. If they are separated in space, delta x is not zero, so another moving frame generally assigns a nonzero time separation. This is difficult because everyday objects move far too slowly for the disagreement to notice. It is also difficult because simultaneity feels like a fact about the events themselves. Relativity says it is a relation between the events and a chosen inertial frame. The train passengers and the platform observer agree on every event: the emission, the rear strike, and the front strike. What differs is the time coordinate assigned to separated events. A spacetime diagram makes that distinction visible."},{"title":"Tilted Spacetime Axes","startSeconds":540.6758124999999,"narration":"A spacetime diagram puts position on the horizontal axis and c times time on the vertical axis. Multiplying time by c gives both axes the same units. An object at rest stays at one x position, so its history is a vertical worldline. A moving object changes x as time passes, so its worldline tilts. Light is special. Since its speed is c, one unit of horizontal travel takes one unit of c times time. Light rays therefore follow these two forty-five-degree lines. Every inertial observer must draw light on those same lines. That shared light cone constrains how another observer's coordinate axes can sit in the diagram. The red c t prime axis is the moving observer's own worldline, x prime equal to zero. It tilts toward the right-going light ray. The red x prime axis is the set of events that the moving observer calls simultaneous with the origin. It tilts upward by the matching amount. So the moving observer's space and time axes tilt toward the light lines and toward each other. They are not a rotated pair of ordinary Euclidean axes. Their scaling is controlled by gamma. The coordinate equations are the Lorentz transformation. Position x prime equals gamma times x minus v t. The time equation mixes time with position. That mixed term, v x over c squared in ordinary time units, is the algebraic source of relative simultaneity. Set x prime equal to zero. The result is the tilted time axis, the moving observer's worldline. Set t prime equal to zero. The result is the tilted space axis, the moving observer's line of simultaneous events. Now choose two events E one and E two on one horizontal line. The stationary observer assigns them the same time. The moving observer's simultaneous line through E one is tilted. It reaches the same distant position at E three, not at E two. No event moves when we change coordinates. E one, E two, and E three stay where they are. What changes is which distant events one observer groups into a single present. The invariant quantity is not Euclidean distance on this page. Write it first. The time part is c delta t squared. The space part is delta x squared, and their difference has the same value for every inertial observer. For events on one clock's worldline, that invariant equals c delta tau squared. This is why different paths between the same meetings can contain different amounts of proper time."},{"title":"The Twin Paradox","startSeconds":720.1887083333332,"narration":"Two twins meet on Earth, synchronize clocks, and separate. One remains on Earth. The other travels outward at eighty percent of light speed, turns around, and returns at the same speed. The spacetime diagram makes the histories different immediately. The Earth twin follows one straight inertial worldline. The traveller follows one straight segment out and a different straight segment home. Let each leg last five Earth years. At speed zero point eight c, the turnaround is four light years away, and the reunion occurs after ten Earth years. For zero point eight c, gamma is five thirds. The Earth clock is present at departure and reunion, and it records ten years of proper time. On either travelling leg, five Earth-frame years correspond to five divided by gamma, which is three traveller years. Two legs therefore give six traveller years. At reunion the clocks can be compared at one place. The Earth twin has aged ten years and the travelling twin six. There is no dispute about that local comparison. The apparent paradox comes from saying that each inertial observer sees the other's clock run slowly. If the roles were perfectly symmetric, why would one twin be younger? The roles are symmetric only during any one inertial leg. Earth remains in one inertial frame throughout the experiment. The traveller changes inertial frames at the turnaround. Before turning, the traveller uses an outbound line of simultaneity. On these axes, one year of c times time has the same scale as one light year. Draw that line through the turnaround event. Where that line meets Earth's worldline, the outbound coordinates assign Earth a time of one point eight years at the distant turnaround moment. After turning, the traveller belongs to the inbound frame. Its line of simultaneity slopes the other way. That new line meets Earth's worldline at eight point two years. The inbound coordinates assign a much later Earth time to events simultaneous with the same turnaround. The difference between those coordinate assignments is six point four years. This is where the asymmetry lives: the traveller changes which distant Earth event belongs to the traveller's present. Earth does not physically age six point four years in an instant. Nothing discontinuous happens to Earth's clock. What changes abruptly, in an idealized instantaneous turnaround, is the traveller's coordinate rule for distant simultaneity. With a realistic gradual turn, the reassignment is gradual too. The final proper times remain essentially the same if the acceleration interval is short compared with the journey. Acceleration identifies which twin changes frames, but acceleration is not an extra aging penalty added by hand. The elapsed proper time comes from the complete spacetime path. Between the same departure and reunion events, the straight inertial worldline contains more proper time than the broken out-and-back path. Earth records ten years. The traveller records six. So the twin paradox is not a contradiction in time dilation. It is a warning that time dilation applies within one inertial comparison, while the traveller's full journey requires two different inertial frames."},{"title":"Muons That Reach the Ground","startSeconds":935.8256874999998,"narration":"The last test comes from particles made high in the atmosphere. Muons are created roughly ten kilometres above the ground, but their proper mean lifetime is only two point two microseconds. Why do many reach detectors at sea level? Begin with the nonrelativistic expectation. Here is the ten-kilometre atmospheric depth. The muon's own lifetime is two point two microseconds. Even light travels only about zero point six six kilometres in that time. A slower particle should decay far above the ground. But atmospheric muons commonly travel near zero point nine nine eight c. At that speed gamma is about fifteen point eight. In Earth's frame, the moving muon's lifetime is dilated. Two point two microseconds multiplied by gamma becomes about thirty-four point eight microseconds. At almost light speed, that dilated lifetime corresponds to about ten point four kilometres. The muon can reach the ground. That is the Earth-frame explanation: the moving particle's internal clock accumulates less time while Earth clocks record the long flight. Now use the muon's own frame. The muon is at rest, so its lifetime is still the proper value, about two point two microseconds. The atmosphere and ground move toward it. The ten-kilometre atmospheric depth is moving, so it is length contracted. Divide ten kilometres by gamma fifteen point eight. The muon measures only about zero point six three kilometres between its creation event and the approaching ground. At zero point nine nine eight c, crossing that contracted distance takes about two point one microseconds of muon proper time. Let the ground approach. It covers the shortened distance before the muon's typical lifetime has elapsed. Earth says the muon's clock runs slowly. The muon says the atmosphere is short. These are not competing explanations. They are the time and space parts of one Lorentz transformation. Both descriptions predict the same local event: the muon and the detector meet. Experiments observe far more muons at the ground than a nonrelativistic lifetime calculation permits. Now collect the argument. First, every inertial observer measures the same light speed c. The moving light clock then forces moving clocks to accumulate less proper time between shared events. A longitudinal light clock and the same time-dilation law force moving lengths to contract along the motion. The train lamp shows the price of combining those results with constant light speed: distant simultaneity depends on the inertial frame. The spacetime diagram gathers all of this into tilted coordinate axes and invariant proper time. It also reveals the twin asymmetry: only the travelling twin changes inertial frames. Finally, atmospheric muons turn the geometry into an observation. Long lifetime in Earth's coordinates and short atmosphere in the muon's coordinates are two descriptions of the same arrival. Special relativity is not a story in which appearances deceive us. It is a precise rule for how different observers divide one spacetime into space and time, while agreeing on light, meetings, and every measurable event."}]}}