###

Sciences

### 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.

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.

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

** dS**^{2} = c^{2}
dT^{2}
- dX^{2}
The value of

**dS**^{2}
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

**dS**^{2}.

If two spacetime events
are separated by the spacetime interval

**dS**^{2}
= 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

**dS**^{2}
> 0.
Relativists call that a

**timelike path**. When two
spacetime events are
separated by the interval

**dS**^{2} >
0, we say those points have
a

**timelike separation**.

When two spacetime events
are separated by the interval

**dS**^{2}
< 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

**dS**^{2}
=
-1. Then we can find a Lorentz transformed coordinate system

**T',X'**
so
that

**dT' = 0, dX' = 1**. This is allowable because it
leaves

**dS**^{2}
= c^{2} dT^{2} - dX^{2}
= -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

**dS**^{2} = - dX^{2} <
0 .
But if the interval we're starting with has

**dS**^{2}
> 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!