Hello everyone,
I was thinking about Special Relativity and thought of this. Let's say that A is 1 light-hour to the left of B and B is at the mid-point of two light bulbs each 100 light-hours away so we have this:

Bulb<-------99 light-hours---->A<-----1 light-hour--->B<---100 light-hours---->Bulb.

Now, the bulbs each emit a flash of light at the same time from the reference frame of B such that they will reach B at the same time. As soon as the flashes are emitted, A starts moving towards B at .5c so that it takes two hours from A's reference frame to reach B. A stops as soon as he reaches B. Since the speed of light is always measured as c then the flash from the bulb on the left will reach A after 99 hours and the flash from the bulb on the right will reach A after 101 hours from A's point of view regardless of how A moves once the flashes are emitted. So we have the following possibilities:

1) Let's assume A's clock runs at the same rate as B's clock, in which case A would see flashes when B would not which seems impossible since they are both standing at the same physical location. For example: 4 p.m. A sees a flash and B doesn't. 5 p.m. B sees both flashes and A doesn't see anything. 6 p.m. A sees a flash and B doesn't.

2) Let's assume that A's clock runs at a different rate than B's clock. Well in order for the flash on the left to reach A in 99 hours and B in 100 hours then A's clock would have to run slower than B's. But, in order for the flash on the right to reach A in 101 hours and B in 100 hours then A's clock would have to run faster than B's. So, then how can A's watch run both faster and slower that B's watch at the same time?

What's the explanation from Special Relativity?

As soon as the flashes are emitted, A starts moving towards B at .5c so that it takes two hours from A's reference frame to reach B.

You forgot length contraction. Also, unless you want to bring in General Relativity, rather have A moving at a constant speed than 'starting' to move.

Nice mention of length contraction and GR. Now, does anyone care to resolve the paradox?

Since the speed of light is always measured as c then the flash from the bulb on the left will reach A after 99 hours and the flash from the bulb on the right will reach A after 101 hours from A's point of view regardless of how A moves once the flashes are emitted.

False. Your paradox is flawed.

Since the speed of light is always measured as c then the flash from the bulb on the left will reach A after 99 hours and the flash from the bulb on the right will reach A after 101 hours from A's point of view regardless of how A moves once the flashes are emitted.

Flawed? How so?

How about a simple example:
I'm 100 miles away from somebody driving towards me at 50 mi/hr. Relative to me this person is always driving towards me at 50 mi/hr. This person will reach me in two hours no matter what. If I don't move for the first hour the person is then 50 miles away after 1 hour. Let's say at the beginning of the second hour I start driving away from this person at 100 mi/hr(relative to the ground). The person will then be driving at 150 mi/hr(relative to the ground) and 50 mi/hr relative to me. In the second hour, the person will cover the 50 miles between us and the 100 miles I've covered and *voila* will reach me in two hours.

So, please explain the flaw.

Nice mention of length contraction and GR. Now, does anyone care to resolve the paradox?

*shrugs* I mention length contraction because it's the first error in your description. From A's reference frame it takes less than 2 hours to reach B because of length contraction, and B will agree with this because of time dilation - A's clock will run slow in B's reference frame.

The main problem, though, is the assumption that since the flashes are simultaneous in B's reference frame, they're also simultaneous in A's. Not so :D If you look at the Lorentz transforms (http://en.wikipedia.org/wiki/Lorentz_transformation#Lorentz_transformation_for_ frames_in_standard_configuration) you'll see that the time in one reference frame (t') depends not only on the time in the other reference frame (t) but also on position (x)! Thus events that are simultaneous but space-separated in one frame will be time-separated in other frames. Only events that occur at the same place and time will be agreed upon as simultaneous in all frames.

Hence you are right that the flashes should reach A at the same time. But by calculating, A will see that in his reference frame they weren't emitted at the same time.

See http://en.wikipedia.org/wiki/Relativity_of_simultaneity for a better explanation.

It's easy to forget one or another factor when working from first principles, but the Lorentz transforms work consistently (I think it was mentioned in another thread they describe a consistent mathematical Minkowski space or something) so they're a good place to start debugging.

Flawed? How so?

While the idea that the speed of light measures to be invariant is correct, it doesn't have the affect you propose.

Your scenario suggests that light takes the same time to reach A regardless of it having relocated. Light reaches point B after 100 hours regardless of what A does. Since A has moved to B, A will see the flash after 100 hours.

The invariance simply means that "if" A were in motion upon arrival of the flash the velocity of the flash will always be the same "c" regardless of the motion.

While the idea that the speed of light measures to be invariant is correct, it doesn't have the affect you propose.

Your scenario suggests that light takes the same time to reach A regardless of it having relocated. Light reaches point B after 100 hours regardless of what A does. Since A has moved to B, A will see the flash after 100 hours.

The invariance simply means that "if" A were in motion upon arrival of the flash the velocity of the flash will always be the same "c" regardless of the motion.

MacM

Let us take the same scale without motion.

Black Hole 0Bulb<-------99 light-hours---->A<-----1 light-hour--->B<---100 light-hours---->Bulb.


Now to the left of A there is a black hole.

Will the light from A ever reach B?

I assume you mean 'will the light from the bulb on the left reach A'? That would depend on the Schwarzchild radius of a moving black hole...

MacM

Let us take the same scale without motion.

Black Hole 0Bulb<-------99 light-hours---->A<-----1 light-hour--->B<---100 light-hours---->Bulb.


Now to the left of A there is a black hole.

Will the light from A ever reach B?

Your case is not stated very clear but if the BH is located in line between the left bulb and A then no I would think the flash would get absorbed into the event horizon of the BH and never come out.

If the BH is to the left of the left bulb but beyond the event horizon then it would have no affect.

The main problem, though, is the assumption that since the flashes are simultaneous in B's reference frame, they're also simultaneous in A's. Not so If you look at the Lorentz transforms you'll see that the time in one reference frame (t') depends not only on the time in the other reference frame (t) but also on position (x)! Thus events that are simultaneous but space-separated in one frame will be time-separated in other frames. Only events that occur at the same place and time will be agreed upon as simultaneous in all frames.
Since A is at rest with respect to B at the time of light flashes, aren't they in the same reference frame?

Since A is at rest with respect to B at the time of light flashes, aren't they in the same reference frame?

Yep and so again when the flash arrives at B after A has gotten there and become inertial rest to B again after having accelerated from its original location.

Hi Zeno,
A does not remain at rest with respect to B, so they are not in the same reference frame.

A is initially in the same non-accelerating frame as B, but that isn't enough.

There are a couple of ways to analyse the situation in SR that I know a little about.

The easiest (for me) is to consider only non-accelerating reference frames. In that case, we're not considering A's rest frame at all - we're considering several frames in which A happens to be at rest for a period of time.

The other way is to consider A's actual rest frame. This is harder, because this is an accelerating frame (ie it accelerates at least some of the time). I don't know precisely how to go about analysing accelerating frames in SR... for starters, SR says that in an accelerating frame the speed of light is not necessarily constant.

Your case is not stated very clear but if the BH is located in line between the left bulb and A then no I would think the flash would get absorbed into the event horizon of the BH and never come out.

If the BH is to the left of the left bulb but beyond the event horizon then it would have no affect.

All I was trying to point out is that Gravity has an effect on the speed of light.

So is the speed of light not a variable due to any Gravitational influences?

Is it proper to leave Gravity or Space curvature out of the equation?