Explain Paradox of Light Moving at c in All Frames

[stever]

My friend is moving in my inertial frame and receives light in the two directions parallel to her movement: from behind and from ahead. The photons that reach her travel at c in my frame, so I presume they will approach my friend at different speeds (as I view it): faster from ahead and slower from behind. But we know from SR that my friend would measure all photons to be moving at speed c. Explain (away) the seeming contradiction so that the photons coming from behind and in front both measure at c on board her frame.

 

[D H]

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Why do you think there's a contradiction? What do you think the answer is?

 

[stever]

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Why do you think there's a contradiction? What do you think the answer is?

Are you saying you cannot see a contradiction?

To this puzzling question I have no answer. I have been looking for the answer for 25 years and have yet to see it?

I would like to know where the people are who can answer this question.

 

[D H]

It only seems contradictory because you have certain expectations based on living in a slow-moving world regarding how velocities add. You are assuming that the same rules pertain at very high velocities. They don't.

For example, suppose a police officer driving 80 mph detects someone via radar going 30 mph faster than the police car. We assume that this can only mean that the speedster was going 110 mph. That isn't exactly correct. The speeding car is going 109.99999999999941 mph relative to an observer who is standing still on the side of the road. No radar can measure that tiny, tiny difference of 5.9×10^-13 mph. As far as we are concerned, the speeding car is going 110. In the example above, if you replace the cars with spaceships and change the speeds from 80 mph and 30 mph to 0.8c and 0.3c, the combined speed 0.887c rather than 1.1c.

The rules that we thought were the rules of the universe are just an approximation of the true rules, and an approximation that is valid at very small speeds. What is a rule of the universe (as far as we know) is that the speed of light is the same to all inertial observers. This is a very well-observed fact.

 

[vela]

At everyday, low speeds, time and space seem to be independent. Time for me appears to be the same as time for you; similarly, we would agree on the spatial distance between two points. It turns out, though, that time and space aren't actually independent. Moving clocks run more slowly, and moving objects shrink in size. Those effects, however, only become apparent as speeds approach that of light.

Special relativity tells us that one observer's space and time dimensions are a mixture of another's space and time dimensions and that how they mix depends on their relative motion. This mixing occurs in exactly the right way so that both observers would see that a light ray moves at speed c.

 

[stever]

These responses don't really answer the question do they?

 

[Hurkyl]

To this puzzling question I have no answer.

Here is the problem:

I presume

 

[DaveC426913]

These responses don't really answer the question do they?

Well, these answers don't teach Special Relativity. They answer the question if you know SR already. You can;t learn that from a forum thread.

The simplest answer is that, at the very essence of relativity is the relativity of simultaneity. Simply put: two observers, traveling in different FoRs will have different ideas of simultaneous. One will see two events occurring simultaneously, the other will see them occurring separately.

Classic thought experiments about this involve a train moving at relativistic velocities with two lights at either end of a passenger car. Both lights are switched on "simultaneously". Depending on who you ask (guy on train versus guy on platform), the lights may or may not have turned on simultaneously.

There are literally countless examples of this. Google "relativity simultaneity train" or any other combo and you'll get more hits than you know what to do with.

 

[stever]

Dave, are you saying the phenomenon is just a measurement problem because of the time it takes light to reach different observers??

 

[stever]

Hurkyl, try to explain why light arrives at a moving object at speed c.

 

[DaveC426913]

Is that so Dave? Care to prove that?

Prove what? A postulate of relativity??

Hurkyl, try to explain why light arrives at a moving object at speed c.

Do you need an explanation more comprehensive than any book on relativity basics would deliver? Not sure if an answer needs to be recut from whole cloth when it's readily available.

 

[DaveC426913]

OK, before this gets much farther:

stever: do you think that relativity is wrong, or do you think perhaps you just don't understand it?

i.e. are you looking for enlightenment, or are you looking for an argument?

 

[stever]

Dave, are you saying the speed of light is constant in different frames because of the different times it takes light to reach different observers?

 

[Hurkyl]

Hurkyl, try to explain why light arrives at a moving object at speed c.

Why not first try to explain why light wouldn't "arrive at a moving object at speed c".

Actually, you should first clarify what you mean by that phrase. The most important aspect is what you mean by "arrives at speed c".

 

[DaveC426913]

Dave, are you saying the phenomenon is just a measurement problem because of the time it takes light to reach different observers??

No. The two spaceships could pass millimetres from each other (negating time lag) and it would not change the results.

I'm saying simultaneity is not absolute. There is no preferred frame in the universe in which one person is right and the other wrong.

 

[stever]

What I don't understand is the answer to the question I originally posed. I have not seen an answer to it anywhere, including SR101. If you have an idea about the answer I would like to hear it.

 

[stever]

"No. The two spaceships could pass millimetres from each other (negating time lag) and it would not change the results."

Yes. I agree. So what then is happening?

 

[Hurkyl]

I have not seen an answer to it anywhere, including SR101.

We cannot explain away the "seeming contradiction" without first understanding why it seems contradictory to you. That requires you to explain why you think it's contradictory.

There's also may be a Socratic thing going on here, depending on your responses. If you explain your own reasoning in greater detail, you might spot the gaps/errors yourself.

 

[Hurkyl]

That requires you to explain why you think it's contradictory.

To illustrate by example, the thing following your words "I presume" could mean a couple different things. The thing I originally thought you meant was in error. However, I've thought of another thing you might have meant which is correct and reasonable, but makes a later statement in error (and when explained it makes the later error easier to spot) (assuming I've correctly interpreted everything else you've said and have left unsaid).

 

[stever]

I don't see how the light arrives at c everywhere in space regardless of movement of the emitter or receiver. How does the light adjust its speed or how does the measurer make a measurement such that the speed is always c?

 

[Hurkyl]

How does the light adjust its speed

It doesn't. Its (coordinate) speed is always c in an inertial reference frame. No adjustments necessary.

How does the measurer make a measurement such that the speed is always c?

They don't do anything special; they just measure (coordinate) displacement over (coordinate) time, like you would expect them to.

 

[DaveC426913]

What I don't understand is the answer to the question I originally posed. I have not seen an answer to it anywhere, including SR101. If you have an idea about the answer I would like to hear it.

How much reading have you done?

I think we're getting the impression that you expect us to do all the work - essentially teach you SR.

 

[zzzoak]

Copy from http://en.wikipedia.org/wiki/Special_relativity"
If the observer in S sees an object moving along the x axis at velocity w, then the observer in the S' system, a frame of reference moving at velocity v in the x direction with respect to S, will see the object moving with velocity w' where
$$w'=\frac{w-v}{1-\frac{wv}{c^2}}$$

So you see w=+c. The light is going to +x direction and reaches your friend from behind.
Your friend sees this light with speed
$$w'=\frac{c-v}{1-\frac{cv}{c^2}}=\frac{c-v}{\frac{c-v}{c}}=c$$

You see another light w=-c is going to -x direction and reaches your friend from front.
Your friend sees this light with speed
$$w'=\frac{-c-v}{1+\frac{cv}{c^2}}=\frac{-c-v}{\frac{c+v}{c}}=-c$$

So you see the light going to your friend with speed +c and -c.
Your friend sees this light with the same speed +c and -c.
This doesn't depend on the speed v your friend is moving with.

 

[stever]

It doesn't. Its (coordinate) speed is always c in an inertial reference frame. No adjustments necessary.


They don't do anything special; they just measure (coordinate) displacement over (coordinate) time, like you would expect them to.

Yes I know. It is a given from the experimental data. What we are trying to explain, or at least I am, is why it is at c.

If two observers are moving relative to each other, or in relation to a light source, something is happening such that they both can measure light at c. Yes it happens. But it does not just happen. Even death and taxes can be explained to a certain degree, but light...?

 

[stever]

Copy from http://en.wikipedia.org/wiki/Special_relativity"
If the observer in S sees an object moving along the x axis at velocity w, then the observer in the S' system, a frame of reference moving at velocity v in the x direction with respect to S, will see the object moving with velocity w' where
$$w'=\frac{w-v}{1-\frac{wv}{c^2}}$$

So you see w=+c. The light is going to +x direction and reaches your friend from behind.
Your friend sees this light with speed
$$w'=\frac{c-v}{1-\frac{cv}{c^2}}=\frac{c-v}{\frac{c-v}{c}}=c$$

You see another light w=-c is going to -x direction and reaches your friend from front.
Your friend sees this light with speed
$$w'=\frac{-c-v}{1+\frac{cv}{c^2}}=\frac{-c-v}{\frac{c+v}{c}}=-c$$

So you see the light going to your friend with speed +c and -c.
Your friend sees this light with the same speed +c and -c.
This doesn't depend on the speed v your friend is moving with.

I am impressed by how neatly the math works out. Is there something hidden in the math that explains to you how the light can go at c, in space, for all observers?

 

[DaveC426913]

I am impressed by how neatly the math works out. Is there something hidden in the math that explains to you how the light can go at c, in space, for all observers?

The key is time dilation.

Have you read the spaceship with a flashlight thought experiment?

Bob's spaceship flies past Earth at .99c. As he does so, Bob points a flashlight out the front window of his ship. Bob sees the beam of light travel away from him at c. In one second, he sees the beam of light travel out from his ship a distance of one light-second.

On Earth, Alice sees the spaceship fly by, and sees the beam of light leave the ship. In one second, she sees the ship move .99 light-seconds and she sees the beam of light has traveled at c. It this case, the beam of light is closely followed by Bob's ship trsavelling a .99c, so Alice sees that the beam is only .01 light-second out from Bob's ship.

Let's take stock:

Each observer, before comparing notes, has seen exactly what they expect to see. Each observer knows that light always travels at c, and they have seen this in action.

The paradox is that, when they trry to reconcile their experiences, they see that in the same length of time (one second), they have had two very different realities.

The key is that their experience of the passage of time is not in sync.

Alice saw Bob's spaceship travel .99 light-second and the beam of light has traveled .01 light-second beyond that.

In that time, Bob has only experienced .01 second passing. Alice sees Bob moving very slowly.

It is not until the light beam is a full light-second beyond his ship that Bob stop his stopwatch. He counts one second. Alice, meanwhile, has been waiting 100 seconds for him to stop his stopwatch.

You can see how they have two completely different experiences of how long it took for the beam of light to get 1 light-second beyond Bob's spaceship.

 

[stever]

I don't recall this thought experiment but it makes sense to me.

I wonder what distance the light would be in front of Bob after one of his seconds? Is it about the same distance that light travels in one second in Alice's frame?

What if Bob points a flashlight out the back window of his ship. This light would move at c for Alice I presume. What would it move at for Bob? At the end of one second on Bob's ship, the light would be almost 200 of Alice's light-seconds of distance from Bob, wouldn't it. If true, this is a much further distance behind Bob than the front light has traveled in front of Bob. Is this asymetric distance really happening? And the light speed for Bob would be close to 2c, which is not allowed. It seems to me that the light should move from Bob the same distance in every direction in a given amount of time. Every frame should have the same behavior.

 

[Hurkyl]

I don't recall this thought experiment but it makes sense to me.

I wonder what distance the light would be in front of Bob after one of his seconds? Is it about the same distance that light travels in one second in Alice's frame?

By what measurement convention?

Is Bob choosing an inertial coordinate chart in which he is at rest, waiting one second (as defined by the coordinates), identifying the point (as defined by the coordinates) where the light is at that time, and then measuring the distance (as defined by the coordinates) from himself to that point?

And is Alice choosing an inertial coordinate chart in which she is at rest, waiting one second (as defined by the coordinates), identifying the point (as defined by the coordinates) where the light is at that time and the point (as defined by the coordinates) where Bob is at at that time, and then measuring the distance (as defined by the coordinates) between those points?

Then Alice and Bob are making two very different measurements. They both get c, of course; but the particular distances they are doing are quite different.

 

[stever]

By what measurement convention?

Is Bob choosing an inertial coordinate chart in which he is at rest, waiting one second (as defined by the coordinates), identifying the point (as defined by the coordinates) where the light is at that time, and then measuring the distance (as defined by the coordinates) from himself to that point?

And is Alice choosing an inertial coordinate chart in which she is at rest, waiting one second (as defined by the coordinates), identifying the point (as defined by the coordinates) where the light is at that time and the point (as defined by the coordinates) where Bob is at at that time, and then measuring the distance (as defined by the coordinates) between those points?

Then Alice and Bob are making two very different measurements. They both get c, of course; but the particular distances they are doing are quite different.

I think I was conceiving of the measurements using Alice's coordinate system for all distance measurements.

I see I should not have calculated the speed for the rear-going light for Bob because there was no specification of Bob's distance units (which are shortened to some degree).

What struck me about this thought experiment when I proposed a rear-going light was how asymmetric the distances would be for Bob, meaning that the back-going light gets much further from Bob than does the front-going light, (assuming that the lights are moving at c in Alice's frame). (I mis-stated the case earlier when I said "how far the front light traveled" and should have said how far it is from Bob.) This asymmetry isn't allowed in SR is it? The forward and rear going lights both must stay at equal (but increasing) distances from Bob, just as they do from Alice, so this thought experiment seems to fail when it is exrapolated to include rear-going light. Unless of course I got something wrong.

 

[Hurkyl]

The forward and rear going lights both must stay at equal (but increasing) distances from Bob,

If Bob measures time using "his" clocks, but measures position using the marks on the rulers "belonging" to Alice, Ihe measures:

• The "forward" beam of light has a higher velocity than the "rearward" beam
• Bob himself is not at the origin of these coordinates; he's moving.

I'm having trouble with the picture for whether the displacement vectors from him to the beams of light grow at different rates, though, and am too lazy to do the math to calculate it right now.


If Bob instead uses the clocks "belonging" to Alice, he will of course make exactly the same measurements as Alice.

If Bob instead uses "his" rulers, he will also see light traveling at the same speed in any direction (as well as the rate of growth of the displacement vectors), although he will measure both position and time differently than Alice does.