Can An Effect Come BEFORE Its Cause?

Tuesday, September 18, 2018

Can An Effect Come BEFORE Its Cause?

Is Causality Invariant

In the previous post, we found out that the space-time interval is the same for all observers (it is an invariant).
𐤃s² = 𐤃s'²
But what about causality? Is this also an invariant?

We want to know how crazy the relativity of simultaneity really is. We want to know if observers can disagree on causality. Observers can claim events A, B and C happened in different orders. What if A caused B which caused C; can observers still claim that they happened in different orders?
Can I claim that Bob shot Joe while another observer claims that Joe was shot before Bob even pulled the trigger? Can another observer claim that Joe was actually dead long before Bob pulled the trigger?

What Does Causally Connected Mean

For two events to be causally connected; a signal slower than or the same as the speed of light must have been able to travel between them. In the unfortunate case of Joe, a bullet is slower than the speed of light and so it is possible that Bob pulling the trigger was the cause of Joe's death. This is because it is possible that a signal (a bullet in this case) could have travelled between Bob and Joe. If Joe and Bob were on separate sides of the universe and Joe died at the same time as Bob pulling the trigger; no signal could've travelled between them instantaneously and so it is impossible that the events of Bob shooting and Joe dying are causally connected.

Does Everyone Agree?

We need to find out if observers agree on the possibility of a signal travelling between. If all observers agree that a signal could have travelled between two points of space, then they will all agree that they may have been causally connected. Lets think about a person catching a ball. The ball in this case is the signal and it has speed v.
The ball travels a distance Δx at speed v for a time Δt.
Δx = vΔt

This distance is less than the distance light will travel in that time and so we get the inequality
vΔt < cΔt
Δx < cΔt

So for all observers is the distance the ball travels (Δx) less than the distance light would have travelling in that time? Phrasing the problem like this makes the answer seem obvious. Nothing can travel as fast as light and so light definitely travels further than the ball in all frames. We can prove this mathematically to confirm this logic.

Δx < cΔt
Δx² < c²Δt²
Δx² - c²Δt² < 0
Δs² < 0

We know the value for the space-time interval (Δs²) never changes between reference frames. If all observers agree on the space-time interval, they all agree on
Δs² < 0

From the mathematics above, it is clear that they must also agree that Δx < cΔt
We established earlier that Δx < cΔt
means the two events may have been causally connected because a signal may have been able to travel between the two events.
If they all agree Δs² < 0
they agree that Δx < cΔt t
and so all observers will agree on whether two events could have been causally connected.

Causally Connected Events

Δx² - c²Δt² < 0
A negative space-time interval means the events may have been causally connected as a signal could have travelled between the events.
We call this timelike seperation (there is a lot of time between the events but not a lot of space and so signals can travel between them).

Causally Disconnected Events

If a signal faster than light was needed to connect two events, they could not have been causally connected.
v > c
vΔt > cΔt

Δx = vΔt

Δx > cΔt (The distance between the events is greater than the distance light could have travelled in a given time)

Δx² > c²Δt²
Δx² - c²Δt² > 0
A positive space-time interval means the events could no have been causally connected since no signal could have travelled between the events in the given time.
We call this spacelike separation (there is a lot of space between the events but not a lot of time and so signals can't travel between events).

The Causal Boundary

v = c
vΔt = cΔt

Δx = vΔt

Δx = cΔt
Δx² = c²Δt²

Δx² - c²Δt² = 0
A space-time interval of 0 means that the events could have only been causally connected if the signal connecting the two events was light.
We call this lightlike separtation (the only signal that could travel between the two events is light).

Space-time Diagrams And Causality

This can all be represented nicely on a space-time diagram. This diagram shows what points in space-time may have been causally connected to the origin.

Areas separated by a lot of space (a lot of x) and not a lot of time (not a lot of t) could not have been connected. No signal could travel between the events.

Areas separated by a lot of time and not a lot of space could easily be connected. Even slow moving signals could have travelled between the events

Areas on the red line could only be connected if a beam of light was the signal used as this is the only signal that could have travelled between the events.

Notice that we extend the space-time diagram to include the past (negative t) because events in the past may be causally connected to the origin.

Same Place, Same Time

If you think back to how we draw space-time diagrams for moving observers there is something you may have already noticed. Due to the way the t' axis (x' = 0) rotates, if two events could be causally connected, there is a frame of reference where the two events happen at the same place.

This diagram shows two events occurring at the same place on the t' axis (x' = 0) in a moving reference frame. The t' axis represents a single point in space over time. They therefore occur at the same point in space.

This makes perfect sense. If you take a frame moving at the same speed as the signal, then the events happen at the same place. In the example of the ball, if you ran at the same speed of the ball, the throwing of the ball and catching of the ball would happen at the same place relative to you. They would both occur right in front of you.
We can also show this algebraically.
For there to be a frame where the events happen at the same place, their locations are both x' = 0
Δx' = 0
Δx' = γ(Δx - vΔt)

γ(Δx - vΔt) = 0

Δx - vΔt = 0
-vΔt = -Δx
v = Δx/Δt

And tells us the velocity of the frame where the events happen at the same place is Δx/Δt.
The frame covers the distance between the two events in the time between the events i.e. it moves at the same speed as the signal.
To draw a line representing this signal on a space-time diagram we draw the line t = x/v.

Similarly if two events are causally disconnected, there exists a frame where the events happen at the same time.

This diagram shows two events occurring at the same place on the x' axis (t' = 0) in a moving reference frame. The x' axis represents all of space at a moment in time and so the two events appear to occur at the time.

We can use algebra to find out what speed an observer would have to move at for this to occur
Δt' = 0
Δt' = γ(Δt - vΔx/c²)
γ(Δt - vΔx/c²) = 0
Δt - vΔx/c²= 0
- vΔx/c² = -Δt
v = Δtc²/Δx

The Sad Life Of A Photon Part 2

If two events are lightlike separated, they lie on a causal boundary. This means exists a frame where the events happen at the same place and at the same time. This is in a frame moving at the speed of light. When we considered the sad life of a photon, we found that photon's do not experience time. Their whole life happens at one moment in time. Well photons also do not see lengths either. Length contraction is so extreme that everything is at the same place for a photon. This means for photons, an event may happen at the same place and time while for a stationary observer, these events lie on the causal boundary.

What Does The Space-time Interval Physically Mean

The same place same time idea gives us a physical meaning of space-time distance.

If two events may be causally connected (timelike separated)
Δx² - c²Δt² < 0
Then we can consider the frame moving with the signal (the events happen at the same point in space)
Δx' = 0
and in this frame the space-time interval is - c²Δt'² = 0
This means the space-time interval is -c²(proper time between two events)²
(The proper time is the time elapsed in a frame where the events happen at the same location).

So the space-time interval between the events of you falling asleep tonight and you waking up tomorrow is -c² x the time you measure that you slept for ².
For the proper time we usually use the symbol τ (tau).

If two events are causally disconnected (spacelike separated)
Δx² - c²Δt² > 0
Then we there is a frame where the events are simultaneous
Δt' = 0
and in this frame the space-time interval is Δx'² = 0
The space-time interval is the proper distance between two events².
(The proper distance is the distance between two events in the frame in which they happen at the same time).
So the space-time interval between you and mars at this moment in time is the distance between Earth and Mars².

Now we have a physical understanding of the space-time interval and we know that causality is an invariant. We know that if two events may have been causally connected, then everyone agrees on this fact and the same applies for causally disconnected events. For observers to disagree on the order of events, they must be spacelike separated. If they are timelike separate, all observers will agree on the order. In the case of Bob and Joe, everyone agrees that Bob shot Joe.

With our new knowledge on the space-time interval, we will be able to consider what it means to think of yourself as moving through space-time at the speed of light and time dilation occurring because motion through time is being reduced and diverted into motion through space. This will come in the next post on special relativity.

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