User:Cleonis/Time dilation
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Relativistic physics has the following in common with Newtonian physics: in relativistic physics the structure of space and time is seen as a physical entity in itself, playing a part in the physics taking place. Metaphorically, one can speak of 'the fabric of space-time'. The physical properties of space-time play a part in the physics taking place.
[edit] Synchronisation procedure
The first animation represents a space-time diagram. The yellow lines represent the worldlines of pulses of light that are emitted at t=0. The consecutive frames of the animation combined represent a single diagram.
The three red circles represent a fleet of spaceships. The progressing segments inside the circles represent onboard clocks, counting time. The two orange circles represent miniclocks shuttling back and forth between the ships of the fleet, the miniclocks are used for a procedure to maintain synchronized fleet time. The ships of the red fleet are 4 units of distance apart. Here, 4 units of distance means that pulses of light take 4 units of time to propagate from one ship to another. In this article, distance is measured in terms of time: the amount of transit time of pulses of light. In this animation the miniclocks take 5 units of fleet time to travel from one ship to another, so their velocity relative to the fleet is 4/5 the speed of light. During the journey from one ship to another, the miniclocks count 3 units of proper time. This difference in lapse of proper time is called time dilation.
[edit] Minkowski space-time geometry
The line 
connects all the points in space-time that have in common that for an object moving with uniform velocity from point O in space-time to any point on that line, 3 units of proper time elapse. In Minkowski space-time, the proper time is a measure of the amount of separation between points in space-time. The amount of separation between point in space-time is called Spacetime interval. The line is a collection of all the points that are 3 units of proper time away from point O.
The yellow lines that represent pulses of light (which can also be used for a sychronisation procedure) are the most extreme case of time dilation. For light no time elapses in propagating from one point in space to another point. As far as light is concerned, all points in space-time are separated by null-intervals.
An observer situated at the origin can define a coordinate system for himself and any objects that are co-moving with himself, such as the ships of a fleet of space-ships that are co-moving. In the animations of this article the fleet consists of three ships, but it can be any number of ships, and those ships can be arranged to form a grid. That grid provides a coordinate system to assign a coordinate distance and a coordinate time-lapse between two events.
The concept of space-time interval in Minkowski space-time is somewhat analogous to the concept of radial distance in Euclidean space. In Euclidean space with 2 dimensions of space there is the relation:
- r2 = x2 + y2
The radial distance between two points is an invariant, in the sense that it is independent of the particular choice of mapping a space with a Euclidean coordinate system. Radial distance between two points is invariant under a coordinate transformation that corresponds to a spatial rotation.
The invariant space-time interval of Minkowski space-time geometry is as follows (Here, spatial distance is measured in units of time, the time it takes light to cover the coordinate distance):
- (space-time interval)2 = (coordinate time)2 - (coordinate distance)2
The radical difference is the minus sign. In the case of 3-dimensional Euclidean space, then a sphere around a chosen point of origin constitutes a finite surface with the property that all points on that surface are equally far away from the origin. In the case of Minkowski space-time continuum there is also a "surface" with the property that all points on that surface are an equal space-time interval away from the origin, but in Minkowski space-time this surface is infinitely extended, and it is more a hyperbola than a sphere.
Coordinate translation that involves the dimension of time and under which the space-time interval is invariant corresponds to a change of velocity.
In this article the word 'space' is used in a very abstract sense, in a meaning that is quite different from the everyday meaning of the word. In this article, everything is described in terms of time. Time is counted in units of time, and spatial distance is counted in units of time too! Proper time can be measured, measurement of spatial distance is inherently ambiguous.
[edit] Metric of Minkowski space-time
In the context of Euclidean space there is a natural concept of distance between two spatial points: the radial distance. This concept of radial distance is referred to as 'the metric of Euclidean space', as it provides a measure of spatial distance.
In the case of Minkowski space-time it is common to refer to its properties as 'geometry of Minkowski space-time'. (A more accurate expression would be 'chrono-geometry of Minkowski space-time', but that expression is rarely used.) By analogy with the concept of a metric in Euclidean context the formula for the invariant space-time interval is referred to as 'the metric of Minkowski space-time'. The expression 'metric of Minkowski space-time' is common usage, but because of the difference with the general concept of a metric it is also referred to as a 'pseudo-metric'. This signals that while in mathematical expressions the pseudo-metric performs exactly the same function as a metric, it is fundamentally different from a metric.
The concept of a metric can be applied in many different geometric contexts; A simple example of a metric is the metric of the way that in the game of chess the King moves around. To go from one corner to another along a column or a row takes 7 steps, and to go diagonally also takes 7 steps. That metric is an example of a non-euclidean metric, for Pythagoras' theorem does not apply.
The metric of Minkowski space-time, with the square of one dimension being subtracted from the square of another dimension, is unexplained. There is no theory to address the question of how the structure of space and time can be like that. At present, the Minkowski space-time geometry must be assumed in order to be able to formulate a theory at all.
[edit] Equivalence of different coordinate mappings
The three dark green circles represent a fleet of spaceships. As in the first animations miniclocks are shuttling back and forth between the ships of the fleet, as part of a procedure to maintain synchronized fleet time.
This space-time diagram represents how the motion of the green fleet in space-time is mapped in a coordinate system that has a velocity of 2/5 the speed of light with respect to the green fleet.
The procedure as it works for the green fleet is identical to what the red fleet does. The central ship sends the miniclocks in opposite directions and each miniclock has a relative velocity of 4/5 of the speed of light with respect to the fleet. For each leg of the procedere, the miniclocks count 3 units of proper time, and the fleet clocks count 5 units of proper time for each leg of the procedure.
This image shows space-time diagrams that map both the procedure of the red fleet and the procedure of the green fleet. The diagram on the left shows a mapping of events in space-time in a coordinate system that is co-moving with the green fleet, the diagram on the right shows a mapping of events in space-time in a coordinate system that is co-moving with the red fleet.
[edit] Connection with Principle of relativity of inertial motion
The concept of Minkowski space-time continuum is deeply interlinked with the principle of relativity of inertial motion. If you would assume independent 3-dimensional space and flow of time, then you would expect that this synchronization procedure can work fine only for one fleet, and that it must fail for any fleet that has a velocity relative to the fleet for which the procedure works fine. But in Minkowski space-time the procedure works fine for all relative velocities.
As stated in the introduction, relativistic physics has in common with ether theories that it is assumed that space (space-time) has physical properties. It is a feature of relativistic physics that space-time is assumed to be a physical entity, participating in the physics taking place. The nature of this participation in the physics taking place is that the amount of proper time that elapses in travelling from a certain point A to a certain point B is path dependent. In travelling from point A to point B along a trajectory that is not the spatially shortest trajectory, a smaller amount of proper time elapses than when travelling along the spatially shortest path. The major difference between ether theories and relativistic physics (arguably the only difference), is that in the case of relativistic physics velocity with respect the assumed background structure (space-time), does not enter the theory.
[edit] Equivalence class of coordinate systems
In this animation, a series of Minkowski coordinate systems is shown. All individual frames of the animation represent the physics taking place equally well. Together, the set of all frames in which the physics taking place is represented equally well constitutes an equivalence class of coordinate systems. The physical properties that are same in all representations (such as the space-time interval between two events), are considered to be the fundamental properties.
Because proper time is invariant under the coordinate transformations, proper time is regarded as true time.
The animation also suggests what happens during physical acceleration. During physical acceleration, an object's relation to its surroundings changes.
[edit] Symmetric velocity time dilation
The situation is symmetrical. The red fleet and the green fleet have a velocity relative to each other, so for each unit of red time less than one unit of green time elapses, and for each unit of green time, less than one unit of red time elapses.
At time t=0 the two central ships of both fleets pass each other. At t=0, let the red ship emit a signal with a particular frequency, as measured in red fleet time. The green ship receives that signal, and that signal will be shifted to a lower frequency, as measured by green fleet time.
Conversely: at t=0 let the green ship emit a signal with a particular frequency, as measured in green fleet time. The red ship receives that signal, and that signal will be shifted to a lower frequency, as measured by red fleet time.
This type of time dilation is called symmetric velocity time dilation. An example of that is the trajectories of the time-disseminating shuttles in the animations. At all times the shuttles have a velocity relative to each other, so there is a corresponding symmetric velocity time dilation. When the shuttles rejoin it is seen that there no difference in the amount of elapsed proper time.
[edit] Asymmetric velocity time dilation
Schematic representation of asymmetric velocity time dilation. The animation represents motion as mapped in a Minkowski space-time diagram, with two dimensions of space, (the horizontal plane) and position in time vertically. The circles represent clocks, counting lapse of proper time. The Minkowski coordinate system is co-moving with the non-accelerating clock.
The clock in circular motion counts less lapse of proper time than the non-accelerating clock. Here, the difference in the amount of lapse of proper time is in a ratio of 1:2, which corresponds to a transversal velocity of 0.866 times the speed of light.
Any light emitted by the non-accelerating clock and received by the circling clock is received as a blue-shifted signal, in a ratio of 1:2. Any light emitted by the circling clock and received by the non-accelerating clock is received as a red-shifted signal, in a ratio of 2:1 .
In this situation, symmetry is broken, and there is a difference in amount of proper time that elapses.
