Virginia Tech > College of Science > Physics Dept > Tatsu Takeuchi > Special Relativity > Lecture Notes > Section 10
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10. Faster than Light Travel

If the concept of before and after depends on the observer, we must worry about what happens to the concept of causality. What I mean by causality is the concept of cause and effect.

For an event A to be the cause of another event B, then A must occur before B. But if some observers think A happened before B, while others think B happened before A, won't there be a contradiction?

For instance, consider the spacetime diagram we just studied in the previous section. Assume that

  1. an arrow is shot at A,
  2. it travels along the line connecting A and B,
  3. it reaches B where it hits an apple.
In the (x,t) frame, the apple is hit after the arrow was shot. In the (x',t') frame, the arrow being hit and the shooting of the arrow occur simultaneously. But in the (x'',t'') frame the apple is hit before arrow is shot!. How can this possibly happen? This contradiction occurs because we allow the arrow to travel along the line connecting A and B which requires the arrow to travel at a speed exceeding the speed of light.

What can happen if an arrow could fly faster than the speed of light

Let's consider another example. This thought experiment is discussed in Chapter 7 of the book "The Einstein Paradox and other Science Mysteries Solved by Sherlock Holmes" by Colin Bruce (Perseus Books, ISBN 0738200239). Assume that instantaneous communication were possible between remote points. In the spacetime diagram below, A and D are standing by the railroad tracks along which a speeding train passes by carrying B and C. Consider the following sequence of events:

  1. A sends a message to B as the leading car carrying B passes by.
  2. B relays the message to C riding the last car of the train via the instantaneous communication device.
  3. C relays the message to D who is standing by the railroad tracks.
  4. D relays the message back to A via the instantaneous communication device.
It is clear from the spacetime diagram that A will receive the message from D before he has sent out the original!

What can happen if instantaneous communication were possible

As these examples show, contradictions can occur if objects or signals could travel faster than the speed of light. You could have arrows hitting their targets before they are shot or messages received before they are sent. However, if nothing could travel faster than the speed of light, we would not have this difficulty. This is because for objects or signals traveling at speeds slower than the speed of light from point A to point B, all observers will agree that A happened before B.

Chronological order of timelike (AB) and spacelike (CD) separated events in different frames

So faster than light travel is impossible. This also means that Newton's second law must be modified since it says that you can accelerate anything to any speed you want. (Note that Newton's first law is still correct.) We won't be looking at HOW it has to be modified though, since it would be too mathematical. All you need to know is that the real theory says that you cannot accelerate things beyond the speed of light. (I'll add a section dealing with how Newton's second law must be modified later.)

Some science fiction writers think that some new invention (like the warp drive, whatever that is) may let us accelerate to speeds faster than c. What you need to understand is that faster than light travel breaks causality so that it is inconceivable that some new invention is going to let us do it. Otherwise, we can have faster than light speed travel before we invent it!

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