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Space Time Theory

What is "causality" and what does it have to do with time travel?


What do we mean by time travel?

Technically we're all travelling in time just by existing. But we can't seem to control our motion through time in the same way we can control our motion through space. What most people usually mean by time travel is the ability to drive around in time they same way they'd drive around in a city. Spacetime paths Let's look at a spacetime diagram that shows a few examples of different ways observers or objects can travel in time and space.
It's important to remember that any single point on this diagram represents an event - both a moment in time and a location in space. The origin of the axes in the figure represents the place X=0 at the time T=0. This is what we mean by a spacetime event.
The figure to the left is a spacetime diagram showing spacetime paths of various observers moving in 1+1 dimensional spacetime. Paths A, B, C and D represent the normal kind of time travel that we find in our world. Path E shows an example of a kind of time travel that is not allowed in Special Relativity.

The allowed time travellers

The light blue spacetime path A represents a flash of light from a laser coming from off of the diagram. The flash of light travels into the future to the spacetime event T=0, X=0 at which it intersects with the spacetime path B.
Path B is a vertical line. A vertical line on this diagram means an observer or object at rest in this coordinate system, staying at the same value of X for all time T. In this example, the green spacetime path B is the worldline of a mirror standing on a table. Let's say that this mirror is only half-silvered, so that when the laser pulse path A intersects with it, half of the pulse is reflected back to the source.
The reflected half of the laser pulse is shown by the light blue spacetime path C. The half of the pulse that is not reflected but transmitted is found by continuing path A.
The purple spacetime path D starts at rest, then accelerates and keeps accelerating until it approaches the speed of light.

The forbidden time travellers

Our abnormal time traveller is shown on the red spacetime path E. Notice this path is a circle. But it isn't like a circle in space -- anyone can walk around a circle in space. This path is a circle in spaceTIME -- this path keeps going back to the same TIME as well as back to the same space.
Notice also that spacetime path E is almost always in two places at the same time.
What's wrong with this spacetime path?
Over 50% of spacetime path E is travelling faster than the speed of light. At the top and bottom of the circle, the observer would have to be travelling infinitely fast.
We'll explore the difference between normal and abnormal time travellers in the next section.

What is a light cone?

Since the observed properties of light were so vastly important in reshaping the mathematical models of space and time in Special Relativity, the propagation of light occupies a very special place in this model. Light cones
Relativists like to speak of the behavior of light in terms of light cones. Light cones come in two kinds: past light cones and future light cones.
The future light cone of a spacetime event E consists of all the paths of light that begin at E and travel into the future. One could imagine a flash of light at event E. The future light cone of E would be everywhere the flash went in space and time after leaving E.
In one time and one space dimension, light cones are lines in spacetime. The future light cone of event E is shown in the figure as the blue diagonal lines to the future of E.
The past light cone of a spacetime event E consists of all the paths of light that begin at E and travel into the past. The past light cone of E would be similar to flashes of light all converging at the event E. The past light cone of event E is shown in the figure as the pink diagonal lines to the past of E.

How do light cones limit time travel?

Recall the Minkowski metric
        dS2 = c2 dT2 - dX2

    The value of dS2 is important in Special Relativity because it stays the same under a Lorentz transformation. That means all observers moving at constant velocity with respect to each other will agree on the value of dS2. Spacetime intervals
If two spacetime events are separated by the spacetime interval dS2 = 0, then we say those two points have a lightlike separation. Only a light ray, and nothing else, can connect those two spacetime events.
The path of an observer travelling less than the speed of light satisfies dS2 > 0. Relativists call that a timelike path. When two spacetime events are separated by the interval dS2 > 0, we say those points have a timelike separation.
When two spacetime events are separated by the interval dS2 < 0, we call that a spacelike separation. For events with a spacelike separation, there always exists some moving observer who will say those two events happened at the same time.
To prove that, look at the Minkoski metric. Suppose we have a spacelike separation with dS2 = -1. Then we can find a Lorentz transformed coordinate system T',X' so that dT' = 0, dX' = 1. This is allowable because it leaves dS2 = c2 dT2 - dX2 = -1 unchanged.
The same logic shows that if two spacetime events have a timelike separation, then there is no Lorentz transformation to coordinates in which they occur at the same time. This is because the definition of simultaneous in this context is dT' = 0. That would make dS2 = - dX2 < 0 . But if the interval we're starting with has dS2 > 0 , and any Lorentz transformation will preserve that condition.

What light cones do

The light cone of an event E forms the boundary between all spacetime events with a timelike separation from E and all events with a spacelike separation from E.
Observe in the figure above that the light cone disconnects the past and future timelike areas with areas of spacelike separation in a Lorentz-invariant manner, so that the gap between the past and future of an event is absolute and unbridgable - for all observers.
This is how Special Relativity implements causality and makes nontrivial time travel so mentally challenging to cook up!
The challenge is to find some way to make the past and future light cones of some event bend around or overlap. But we can't do that in Special Relativity, the light cones in Special Relativity are rigid and unchanging. We need to move up to General Relativity if we want to try messing with causality. Stay tuned!