# Outrunning Light by >(1/2)c

1. Jan 24, 2012

### spacebear2000

This may be old territory, but I am relatively (no pun intended) new to this stuff, so I apologize if I'm bringing up things that maybe I should have learned already or found among older posts.

I have a little thought experiment that I would like your opinions on. Let's suppose we have a square platform in space and two rockets on either side of it--to visualize, one is pointing "up" and the other is pointing "down." The two rockets launch and head in 180-degree opposite directions, each accelerating to just over 1/2 the speed of light. Can one rocket send a transmission that the other is capable of receiving, or will the each rocket exceed the light cone of the other? If they can each receive signals from the other, how can the signal traverse the distance without exceeding the speed of light? If they can't receive signals from each other, has one traveled faster than the speed of light when using the other rocket's cockpit as the inertial frame of reference?

2. Jan 24, 2012

### Staff: Mentor

3. Jan 24, 2012

### elfmotat

Try this: http://pdfcast.org/pdf/special-relativity-2/ [Broken] . Go to the section on velocity addition.

Last edited by a moderator: May 5, 2017
4. Jan 24, 2012

### Ryan_m_b

Staff Emeritus
Light always travels at C, it's not affected by the velocity of the transmitter. When a signal is sent from a ship it propagates at the speed of light. Eventually it will catch up to the other ship.

5. Jan 24, 2012

### spacebear2000

Thank you for the pointers.

I suppose another way to phrase my inquiry (or to add a question to it) would be to ask if there might be a scenario in which a body A traveling away from a body B could do so such that light emitted from body B in the direction of body A would not eventually arrive at or overtake body A.

While I can appreciate the Einstein velocity addition idea that says light can't be accelerated by the acceleration of the body that emits it, I have yet to come across an understanding that entails bodies moving away from each other in the manner I initially described as somehow unable to do so.

If the distance between two bodies increases by more than a light year each year (which could happen if they each travel at just over (1/2)c in opposite directions from a frame of reference between them, no?), how could light get from one to the other? And, in this scenario, wouldn't one of the bodies be traveling at faster than the speed of light from the frame of reference of the other?

6. Jan 24, 2012

### Ryan_m_b

Staff Emeritus
No because in the reference frame of either entity the other is moving at fractionally slower than the speed of light. Look at those relativistic velocity addition formulas, as far as I understand it it is because of effects like time dilation.

Perhaps this can clear up your confusion; if A and B head away from each other at .5c and A sends a transmission to B at 1C then the transmission is going at twice the velocity of B. Once the transmission is sent it is independent of A, we can forget about how fast A and B are receding from each other and focus on the relative speed of the transmission and B.

EDIT: here's a non relativistic analogy: Trucks A and B move away from each other at 30mph. The driver of truck A hits a button and from the back a motorbike pops out heading in the direction of B at 60mph. A and B may be moving away from each other at 60mph but the motorbike is heading towards B at 30mph. (To work that out with light and rockets you need the relativistic equations supplied above).

7. Jan 24, 2012

### spacebear2000

Ryan_m_b, thank you for using the terms of my example. I'm still pretty concrete about this stuff, so having an example helps.

I think that part of what I may be getting hung up on (or part of what I'm not making clear) is the concept of speed as distance/time. For example, if I go .5 a light year in a year, I must have gone at .5c. (Right?)

So if A and B are a light year apart, and each move away from each other at .5c for a year, they will be two light years apart after a year. (Is this incorrect as per velocity addition, or is this correct?)

If A and B are a light year apart and move away from each other at >.5c for a year, they will be more than two light years apart after a year. (Is this incorrect as per velocity addition, or is this correct?) If the difference in the distance between A and B exceeds a light year when only one year has elapsed, doesn't that mean one has traveled away from the other faster than c from the reference frame of A or B?

8. Jan 24, 2012

### Staff: Mentor

You are right that speed is distance/time, but.... A, B, and the hypothetical observer at the midpoint are in different inertial frames, so they are using different clocks and yardsticks. I was about to start explaining more... but then I realized that essentially the same question has been discussed in the last few weeks:

and
https://www.physicsforums.com/showthread.php?t=568571 (skip the first few posts about non-inertial rotating frames)

9. Jan 24, 2012

### spacebear2000

Thank you, Ryan_m_b and Nugatory, for bearing with me on this.
Having read the links and your offerings, let me see if I got it in these terms:

Let's say I'm a launch pad technician stationed in the launch pad from which my two rockets head off in opposing directions. I am able to measure their distance because they have a long, long measuring tape each attached to them that unspools as they go, revealing the maximum distance from the launchpad they've each achieved. They both travel at the same rate; when one year passes on my clock in the launch pad, I check the unspooled measuring tapes and find that the rockets have each traveled .51 light-years. This means that 1.02 light years separate them and that it took one year for the pair to generate this distance between them. This would seem to be okay, because they are each traveling well below the speed of light. As I take the measurement that they are 1.02 light-years apart, one of the rockets attempts to signal the other through radio waves. After one year, the signal is still .53 light-years away from the target rocket. However, the signal catches up to and overtakes the target rocket within the next year, because it is traveling at c and the rocket <c. Right?

10. Jan 24, 2012

### Passionflower

An observer can only potentially outrun a light signal if he keeps accelerating.
Lookup: Rindler horizon

11. Jan 24, 2012

### Staff: Mentor

Right. (Well, there's a pitfall in the words "as I take the measurement... One of the rockets attempts...". You seem to be trying to say that the signal is sent "at the same time" as you're taking the measurement and it will only be simultaneous in the launch pad frame. And .53 lightyears isn't exactly right either. But those are nits, and you're right that the signal will catch up with the target rocket, just as you say).

The observer in the rocket that receives the signal will tell the story different - he'll say that he was at rest while the launch pad and other rocket went zooming away and at some point the other rocket sent a light signal. While our rocket observer remained at rest, the light signal approached at the speed of light and was received. Different story, but it ends the same way with light signal received and nothing moving faster than light relative to anything else.

A surprisingly fun exercise is to try writing down the x and t coordinates of three events (rockets leave launch pad in opposite directions; sending rocket emits flash of light; receiving rocket sees flash of light) in the launch pad frame, then use the Lorentz transformations to see where and when these events happen in the two rocket frames.

12. Jan 25, 2012

### ghwellsjr

You might not think it's such a good idea to use a long, long measuring tape trailing behind your rocket to determine how far it has gone in one year. That tape will be length contracted to a factor of 86% which means the markings on it will progress at a faster rate than you had bargained for.