D = proper distance between mirrors
T = proper light clock period = D/c
SR: Lab Rocket
GR: Rain Shell Far
speed (v): -c c
v/c = 0    γ = 1   rc²/GM = ∞
Photons Annotation   Center Clock
waves: Retarded Advanced Full
space: xy x'y' x"y"
time: t t' t" mini light clocks
diagrams: S S' S"
↔ xt ↑ xt ↓ xt ↕ yt
zoom: - 0 + rotate: - 0 +
time direction: Forward Backward
Slower Faster Reset   /T =
Exit Pause single step: - +
Events:  A Set GoTo B Set GoTo

Using an animation of light and mirrors functioning as a Light Clock we explore the nature of space and time.
Spacetime kinematics are depicted, without dynamical effects (no massive particles, forces, or accelerations).
We adhere to Einstein's stipulation that the local speed of light is the same in all directions.
To keep the Light Clock ticking while it is moving at a constant speed,
the mirrors are positioned so the round trip times are the same in all directions.
This results in length contraction (in the direction of motion) and time dilation, consistent with Relativity.

Special Relativity General Relativity

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x' = γ(x-vt) y' = y z' = z t' = γ(t-vx/c²)

x = γ(x'+vt') y = y' z = z' t = γ(t'+vx'/c²)

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Given a non-rotating neutrally charged point mass M, the general relativistic effects at radial distance r
are the same as the special relativistic effects due to relative speed v = (2GM/r)^½ [see derivation below].
Thus, warped 4D spacetime is equivalent to 3D space moving at speed v toward M (relative to a stationary observer at distance r from M).

References:

To see the equivalence of gravity and moving space select Rain and 'Center Clock'.
Consider the Light Clock to be stationary in a gravitational field (constant r).

Select Shell to see Schwarzschild geometry from a Shell Frame Observer's point of view (at constant distance r from M).

Select Far to see Schwarzschild geometry from a far-away bookeeper's point of view (at large distance from M).

[the far-away frame is also the Schwarzschild frame which is usually represented by (t, x, y), but here (t", x", y") is used instead]

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Derivation of the "speed of space"

Modeling gravity as a flow of space means a test particle free-falling from infinity rides along with space.
The test particle motion reveals the motion of space (is comoving with space) at speed v.
We use Newton's laws to derive the motion. This is valid because,
in the special case of radial free fall toward a non-rotating uncharged point mass M,
General Relativity and Newtonian physics agree exactly.

Let mg be the gravitational mass of the test particle.
Let mi be the inertial mass of the test particle.
Newtonian force of gravity on the test particle is F = -GmgM/r²
Newtonian acceleration is directly proportional to the force of gravity: F = mia
Therefore, F = mia = -GmgM/r²
Using the Equivalence Principle (mg = mi) gives: a = -GM/r²

Prove this is consistent with v² = 2GM/r by
Differentiate wrt t:  2vv̇ = (-2GM/r²)ṙ = -2vGM/r²
Divide by 2v:  v̇ = a = -GM/r² ✓

At the Schwarzschild radius (event horizon of a non-rotating black hole) r = 2GM/c² and v = -c

What is a Light Clock?
Following Einstein we define time to be what is measured with a clock, and the ideal time measurement device is a Light Clock.
A Light Clock uses a light pulse sent from one location to a mirror at another location, which reflects the pulse back. The distance to the mirror is held constant.
The completion of each roundtrip increments the clock by one 'tick' and initiates the next cycle. Counting only completed roundtrips complies with the "all physics is local" rule.
The tick count of your clock is in the bottom right corner of the window; it shows your proper time progressing at a constant rate, unaffected by the Lorentz factor (γ).
The tick count of the Light Clock (S' clock) is centered between the mirrors; it shows the proper time of the S' clock.
No scale is assumed for the Lab or Rocket animations. For the Rain, Shell and Far animations the scale is assumed to be small enough for tidal effects to be insignificant.
In this animation the main Light Clock is stationary in the Rocket, Shell, and Far frames.
Tip: Click the t time button, then click the 'mini light clocks' button to see how light clocks look in the Lab or Rain frames.
Note that times are shown as decimal fractions of the clock period (e.g., t/T=0.750 has the S clock hand in the "9 o'clock" position).

What do the colors signify?
Colors are used to indicate different reference frames.
Lime represents the S inertial reference frame of a stationary inertial observer (laboratory frame) in Special Relativity.
S is also the frame of a Free-Falling observer in General Relativity (local rain frame).
Cyan represents the S' inertial reference frame of the moving unaccelerated Light Clock (rocket frame) in Special Relativity.
S' is also the frame of a stationary fiducial observer in General Relativity (local shell frame).
Orange represents the S" reference frame of a far-away bookkeeper in General Relativity (global Schwarzschild frame).
S" is an idealized global coordinate system which extends far-away flat spacetime to radius r as if there where no gravitating mass M.
You can select whether the animation shows Special or General Relativity, and which reference frame is yours:

S	Lab	Laboratory Frame	Special Relativity	You are a local observer with the Light Clock moving to the right (left) at speed v (-v) through motionless space.
S'	Rocket	Rocket Frame	Special Relativity	You are a local observer with the Light Clock stationary in your frame, space is motionless.
S	Rain	Rain Frame	General Relativity	You are a local observer free-falling past the Light Clock. The Light Clock is at a constant distance r from mass M. You are comoving with space at speed [2GM/r]1/2 toward mass M.
S'	Shell	Shell Frame	General Relativity	You are a local observer with the Light Clock stationary in your frame. The Light Clock is at a constant distance r from mass M. Space is moving past you at speed [2GM/r]1/2 toward mass M.
S"	Far	Far-Away Frame	General Relativity	You are a global bookkeeper far away from mass M (your location is at r = ∞). The Light Clock is at a constant distance r from mass M. You are unaware (or choose to ignore) that space is moving at speed γ-2[2GM/r]1/2, in your frame, past the stationary Light Clock toward mass M.

Local reference frames with the same color have the same relativistic effects (Lab = Rain, Rocket = Shell). In special relativity the effects are due to relative motion through space (motionless space, flat spacetime), while in general relativity the effects are due to relative motion of space (moving space, warped spacetime). This is how the Special and General Theories are linked.
Tip: Toggle between the Lab frame and the Rain frame to see that there is no difference in space or time effects. The same is true for toggling between the Rocket frame and the Shell frame.
The Far frame is not like any of the other frames; the extra space and time distortions (spacetime warping) you see are due to not accounting for the motion of space.

Why does the Light Clock stay contracted and its time dilated when 'Center Clock' and Lab are selected?
Centering the Light Clock automatically drags the scene to the left (right) at constant speed v (-v), with the Light Clock held in the center of the window.
It is an animation trick for the convenience of you the S observer. It is not the same as viewing the Light Clock in its own rest frame.
From the point of view of the Light Clock's S' frame it is your lengths and clocks that are distorted.
You are not moving at the same speed as the S' clock; it just looks that way. By definition, you and your space coordinate grid have no relative motion, despite how it appears when the clock is centered.
Tip: Set v to a high positive value, Select Lab, Select Center Clock, and check the xy space grid to see it being dragged to the left.
Pause the animation or check Uncenter Clock to see a correct depiction of the physics, without the illusion that your space grid is in motion.

How are the spatial coordinate axes oriented?
The x, x', and x" axes are horizontal, increasing to the right. The y, y', and y" axes are vertical, increasing upward.
Only the xy, x'y', and x"y" planes are shown in the animation (z = z' = z" = 0) so each circular wavefront is a 2D slice through the center of a 3D spherical wavefront.
Only the x' axis is Lorentz contracted; the y, y', and y" axes are identical, and the z, z', and z" axes are identical.
Tip: The positive z, z', and z" axes extend out of the screen toward you.

What is Time Dilation?
Time Dilation is the lengthening of time durations of a moving clock, as measured by a stationary clock.
A clock measures time differences between two events at the same position within a reference frame: S clock has Δx = 0, S' clock has Δx' = 0
The slower ticking S' moving clock measures less elapsed time Δt' than the elapsed time Δt of the stationary clock.
The ratio of measured durations between the two frames is the Lorentz factor: Δt = γΔt'
Tip: Select the Lab reference frame. Set γ to 2 and compare clock ticks. The S' clock runs two times slower than your S clock, or equivalently, your S clock runs twice as fast as the S' clock.

What is Lorentz Contraction?
Lorentz Contraction is the shortening of the length of a moving object in the direction of its motion as measured by a stationary observer,
which requires the distance between two positions be measured at the same time by the stationary observer: S length Δ x has Δt = 0.
The ratio of measured lengths between the two frames is the Lorentz factor: Δx' = γ(Δx-vΔt)=γΔx.
In this animation two pairs of mirrors are used to demonstrate that length contraction happens only in the direction of motion, not perpendicular to the motion.
Shortened separation between mirrors in the direction of motion is necessary to keep the clock ticking by ensuring all light pulses return to the source at the same time.
Tip: Pause the animation to see a snapshot with all events having the same time in your reference frame, e.g. in the S frame (Δt = 0 for all event pairs).

What is a Spacetime Interval?
The Spacetime Interval between two spacetime events is defined to be: spatial distance squared minus time duration (multipled by c) squared.
It has the same value in all inertial reference frames and global metrics:

Invariant Spacetime Interval ≡ (Δx)² + (Δy)² + (Δz)² - (cΔt)² = (Δx')² + (Δy')² + (Δz')² - (cΔt')² = (γΔx")² + (Δy")² + (Δz")² - (cΔt"/γ)²

   [The Lorentz factor γ in the Δx" and Δt" terms represent the spacetime warping due to gravitation]

Types of Spacetime Intervals: negative is "timelike", zero is "lightlike", positive is "spacelike".
Relative motion in only the x, x' or x" direction means the Interval is (Δx)² - (cΔt)² = (Δx')² - (cΔt')² = (γΔx")² - (cΔt"/γ)² because Δy = Δy' = Δy" and Δz = Δz' = Δz".
All events in each Lab or Rain animation frame have the same t, therefore the Interval is (Δx)² = (Δx')² - (cΔt')² = (γΔx")² - (cΔt"/γ)² for all event pairs in an animation frame.
All events in each Rocket or Shell animation frame have the same t', therefore the Interval is (Δx)² - (cΔt)² = (Δx')² = (γΔx")² - (cΔt"/γ)² for all event pairs in an animation frame.
All events in each Far animation frame have the same t", therefore the Interval is (Δx)² - (cΔt)² = (Δx')² - (cΔt')² = (Δx")² for all event pairs in an animation frame.
Tip: Check Retarded Waves. Pause and Reset the animation. Check Events and Set Event A at the center of the Light Clock. Step the animation forward a few steps to expand the wave without reflection.
Set Event B on the wave and notice the Interval is lightlike (= 0). Drag Event B inside the wave for timelike (< 0), or outside the wave for spacelike (> 0).

What is an Inertial Reference Frame?
An Inertial Reference Frame has a uniform spatial coordinate grid with synchronized identical clocks at each position in the grid, to locally measure the position and time of spacetime events.
Clocks are synchronized using Einstein's method of round-trip light signals that are assumed to travel at the same speed c in all directions (A-to-B light travel time same as B-to-A).
Check any of the xy x'y' x"y" space grids to see the spatial coordinates. Check any of the t t' t" times to see the time coordinates depicted as clocks.
Tip: Watch the four photons travel round-trip from S' clock tick to the next tick. Notice the mirror reflection events are all at half-tick t' times, even with the mirrors rotated.

What is Relativity of Simultaneity?
By definition an observer's simultaneous events happen at the same time, so they have the same time coordinate: Δt = 0 within S, Δt' = 0 within S', Δt" = 0 within S".
However, two reference frames that are moving relative to each another will generally assign different times to two events, hence simultaneity is relative to the reference frame.
The only case where both frames assign the same time is when the relative motion is perpendicular to the line connecting the locations of the two events.
Consider, for example, frames S and S' with two events that have times Δt = Δt' = 0, therefore Δx² = Δx'². With Δx = Δx'/γ, this means γ = 1 or Δx = Δx' = 0.
If γ is not 1 and two S' clocks have non-zero Δx' separation then they do not measure the same time as observed by S.
Note that all events depicted in an animation frame are simultaneous from the selected observer's perspective, all the observer's clocks within an animation frame are synchronized with each other.
Tip: Select the Lab frame. Set γ to 2 and show t' time. Notice the clocks are only synchronized where Δx = Δx' = 0. Check the t time to see that all the S clocks are synchronized with each other.

What is Relativistic Causality?
If event A is said to cause event B, then relativity requires B to be on or inside the future light cone of A (A must lie on or inside the past light cone of B).
Event A can cause event B (A and B are said to be causally connected) only if a signal can pass from A to B at a speed less than or equal to the speed of light.
Two causally connected (timelike or lightlike separated) events always appear in the same time order as viewed from any frame of reference.
Two causally disconnected (spacelike separated) events, however, can appear in any time order by choosing to view them from different frames of reference. [For three or more events see this reference].
Tip: Select the Lab frame. Set γ to 2. Check Retarded Waves. Pause and Reset the animation. Check Events and Set Event A at the center of the Light Clock. Event A has Lab frame time t = 0.
Step the animation forward a few steps to advance the time. Set Event B somewhere inside the circular wavefront. Event B has Lab frame time t > 0. Event A always precedes Event B in the Lab frame.
Drag Event B to other positions inside the wavefront (timelike separation) and notice how its time changes in the Rocket frame, with Event B always later than Event A in that frame.
Drag Event B to positions outside the wavefront (spacelike separation) and notice that Event B's spatial position determines whether its Rocket frame time is earlier than, simultaneous with, or later than Event A's Rocket frame time.

Why do waves and photons reflect off a moving mirror differently than off a stationary mirror?
The apparent angle of a rotated mirror is due to length contraction along the direction of motion.
The waves obey the Huygens Principle with each wavelet centered on a different spot in space due to the mirror's motion.
The photons obey the principle of least time. [see references]
Tip: Set gamma to 2, Pause, and Reset. Check Retarded Waves and Full Arcs. Check Events and Set Event A at center of Light Clock, Resume.
Pause when the center of the Full Arc of the reflected wave for the right mirror first appears. Set Event B there. Notice that Events A and B have the same t' time.

Why are the light waves elliptical when the Far option is selected?
Far depicts how a far-away bookkeeper represents the events in the vicinity of the light clock. It is not what the bookkeeper observes directly; he is not a local observer of those events.
The free-falling observer (Rain Frame) and the stationary observer (Shell Frame) are local observers, each with their own local inertial frame; therefore they see events consistent with special relativity.
The Far distortions are the result of extrapolating from far-away flat spacetime to the vicinity of the light clock, without compensationg for the effects of the gravitating mass.
These distortions (apparent spacetime warping) include contracted radial lengths (Δx" = Δx = Δx'/γ) and slowed clocks (Δt" = γ²Δt = γΔt'). This explains the Far view of Shell Frame length contractions (e.g. elliptical appearence of the light waves, contracted appearance of the Light Clock even though it is motionless in the Far frame), time dilations (e.g. slowed clocks), and the extra slow inward speed (v/γ²) of the free-falling Rain Frame.
Tip: Set the speed to zero to see the distortions disappear: same effects are seen by the far-away bookkeeper, free-falling observer, and stationary shell observer.
This is because zero relative speed between the stationary shell observer and the free-falling observer happens only at r → ∞ (or M=0), which means in this special case the far-away bookkeeper is a local observer.

What are Retarded and Advanced waves?
A Retarded wave propagates forward in time, producing effects which are retarded from (later than) the originating event at the wave source. An Advanced wave propagates backward in time, producing effects which are in advance of (earlier than) the originating event at the wave source. A Retarded wave from a source is the top half of the lightcone with the source event at the vertex; an Advanced wave from that event is the bottom half of the light cone.
When viewed forward-in-time a Retarded wave expands outward from its source, and an Advanced wave contracts inward toward its source. When viewed backward-in-time a Retarded wave contracts inward toward its source, and an Advanced wave expands outward from its source.
Notice that a Retarded wave from an emission event overlaps at only one point with the Advanced wave from the corresponding absorption event. This results in particle-like behavior from waves: point-like superposition moves at light speed in a straight line from emission to absorption, tracking exactly the photon path(s).
Add many iterations forward-and-backward in time, plus a π phase shift at the end points, and one arrives at the Transactional Interpretation of Quantum Mechanics (TIQM). [see references, especially John Cramer's book and video]
Tip: Consider whether combining TIQM with the flowing-space model of General Relativity (as depicted in the animation) might be a path toward Quantum Gravity.

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Description

FAQ

User Interface Help

	web browser	smart phone or tablet
show/hide control menu	left doubleclick	two touches
move grid and clock	left mouse drag	touch and drag
zoom grid/rotate clock	right mouse drag	two finger gesture

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