Simultaneous Events in Special Relativity

When particles travel at speeds that approach the speed of light, the laws of classical mechanics are no longer valid, and the laws of special relativity apply instead.

In classical mechanics, there is no issue with making a statement such as event X takes place before event Y; but in special relativity, this is no longer the case. In fact, the special theory of relativity forces us to reconsider the concept of what it means for two things to happen simultaneously.

This was first shown by Einstein in 1905: suppose that observer A is at rest, and perceives that event X and event Y take place at the same time. Now suppose that observer B is travelling close to the speed of light relative to observer A, and is travelling away from X and towards Y. Then since observer B is moving at almost the speed of light away from the light coming from X, and towards the light coming from Y, observer B will perceive that Y actually takes place before X.

Therefore, in special relativity it is required that when describing a physical system, one must also specify a frame of reference, because in this case, the statement that two events take place at the same time is well defined.

The special theory of relativity is founded on two principles: that the laws of physics are symmetrical with respect to the choice of frame of reference, and that the speed of light is constant for all observers.

The Twin Paradox

It turns out that the rate of the passage of time is not the same for all frames of reference, it depends on the relative velocity of the frames.

For example: imagine two twins, A and B; if twin A stays on earth while twin B travels in a ship at close to the speed of light, then time will pass at a slower rate for B than for A, and when B returns he will be younger than his twin. This means that according to the special theory of relativity, time travel into the future is theoretically possible.

The Space-Time Continuum

In classical mechanics, the position of a particle along its path is given by four numbers, the three spatial coordinates and the time. In special relativity the same thing is true. Therefore in classical mechanics, as well as in special relativity, time can be thought of as “the fourth dimension”.

The drastic change in our conception of the nature of time demanded by the theory of relativity comes from the way in which time and space are mixed together, and put on a more equal footing. There is a famous set of equations called the “Lorentz Contraction” which show clearly how when changing from one frame of reference to another, time and space actually become mixed together.

One consequence of this is that the spatial distance between two particles is not the same in all frames of reference, and actually depends on the relative velocity of the frames. Therefore in the theory of relativity, rather than measuring the spatial distance between two particles, it is customary to measure the space-time distance between two events. The function which defines the space-time distance must satisfy the condition that it does not depend in any way on the velocity of the frame of reference. In the special theory of relativity there is a particular choice made for the distance function; but in the general theory the gravitational field defines, and is defined by, the choice of distance function.

The Passage of Time in a Gravitational Field

According to the the general theory of relativity, a gravitational field is equivalent, in a sufficiently small region of space, to frame of reference that is accelerating at a constant rate.

This principle alone is sufficient to demonstrate that clocks which are located at different points in a gravitational field, will appear to run at different speeds.

To see this, let us consider two clocks A and B which are both in a gravitational field, with clock A accelerating towards clock B. Furthermore, let us assume that every second, clock B emits a pulse of light. Then since A is accelerating towards the pulse of light coming from B, it will appear to A that the pulses arrive in less time than one second; and thus it will appear to an observer at A that clock B is running fast. By a similar argument, an observer at B would perceive that clock A is running slow.

This means, for example, that a clock on a satellite that is orbiting the earth will appear to run slow compared to a clock on the surface. For the case of satellite GPS systems, taking this into account has been shown to improve the accuracy of the results, demonstrating that the effect is indeed very real.

The Principle of Maximum Proper Time

An important role is played in general relativity by the concept of proper time. If we consider a frame of reference which is moving and accelerating in space in an arbitrary manner, then the proper time for the motion is defined as the amount of time that has passed on a clock that moves with the frame.

As the velocity of the motion increases, the rate of the passage of proper time decreases; therefore the amount of proper time which is associated with the motion depends not only on it’s position in the gravitational field, but also on how fast it is moving.

The principle of maximum proper time defines the way that particles move in a gravitational field, and it states that free particles in a gravitational field will move along a path that maximizes the proper time for the motion.

The Passage of Time Inside of a Black Hole

A black hole is a region of space in which the gravitational field is so strong that light cannot escape if it travels past a certain critical event horizon. In these regions of space, according to general relativity, the strength of the gravitational field is infinite, and this would imply that inside a black hole the passage of time has ceased.

It has been predicted that quantum effects from inside of a black hole actually cause it to evaporate over time. In general, the presence of an infinity in a physical theory indicates a problem with the theory, and at the energy scales present inside of a black hole, the theory of general relativity is no longer valid.

The Beginning of Time

The theory of the creation of the universe is based on the assumption that at the beginning of time everything was pure energy, the same everywhere and in all directions, and furthermore that the strength of the gravitational field was infinite.

10^-36 seconds after the beginning of time, there is a period of rapid inflation which lasts about 10^-33 seconds, followed by a slower expansion at a rate similar to what we observe experimentally today. But whatever happened before the first 10^-36 second remains unknown.

tl;dr

The laws of physics describe the process of change, but these laws themselves are timeless and unchanging. If nothing ever changed, then the concept of time would have no meaning. Time is something that is used in physics to describe change, and the thing that changes is energy. The way energy moves around and changes form depends on the scales at which we are looking, and therefore the meaning of the word “time” depends on which part of physics we are talking about.