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I read some book now about Einsteins thought experiments. And it seems to be a fallacy. They claim in the book that if you are running in the same direction of light and it's chasing you then you are supposed to see light moving slower than c. But because lights speed is constant it turns out that your clock would slow down so that , light wouldn't be measured by you as slower than c but rather exactly as c. My question is, and it looks like a stupid fallacy that's why I think I am maybe missing something. My question is, they only solved the problem in case that you are really going in the same direction of light therefore when your clock slows down it compensates over your own speed and makes you measure light correctly. But the thing that I don't understand is, how does it solve the problem if you were to run against lights direction. Suppose light goes from A to B and you are running from B to A. In such a case, even if your clock were to slow down when you are running, not only would light not be rated at speed C but you are even going to measure light to be much much faster. Cuz, 1) you are heading towards it. 2) your clock slows down.

It looks funny . It looks like a mistake. They only tackle the case of you running in same direction of light. And they explicitly say that for this we have 'time dilation ' so that your clock is ticking slowly so light would appear to you going faster at exact velocity of C. But they seem to forget that if I run towards light then in such a case I myself see light as going faster so in order for this to be corrected my clock shouldn't be going slowler but rather faster.

Am I missing something?

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You are missing a couple of things.

  1. I think it's a gross over-simplification to say that if you move in the same direction as a light beam, then it still travels at $c$ with respect to you, just because of time dilation.

  2. Space and time are relative to observer. To understand this, you need the Lorentz transformations. These are the correct transformation equations of space and time itself from one inertial frame to another, in your case from stationary guy seeing the light beam to a moving observer seeing it. The equations are:

$$x' = \dfrac{x - vt}{\sqrt{1 - \dfrac{v^2}{c^2}}}$$

$$t' = \dfrac{t - \dfrac{vx}{c^2}}{\sqrt{1 - \dfrac{v^2}{c^2}}}$$

Now suppose a stationary observer watches the light beam start at $x = 0$ and $t = 0$, or let me write that event as $(0, 0)$ and watches it reach a distance of $ct$ in time $t$ or $(ct, t)$. If you transform these to events to the moving guy's frame, we get that he watches it move from $(0, 0)$ to:

$$(c\sqrt{\frac{c - v}{c + v}}t, \sqrt{\frac{c - v}{c + v}}t)$$

So yes, the other guy sees light take less time between the events. But that factor is not entirely due to time dilation alone. Further, light also travels less distance compared to what the stationary guy measured, giving us the same speed $c$.

  1. Now I think the case of travelling in the direction opposite to that of light should be easier to understand. In that case, all the above equations hold provided you replace $v$ by $-v$. The thing is, in this case, even though time slows down for this observer as compared to the stationary observer, he detects light take more time to travel between these particular events. Light also travelled more distance. Thus the same speed $c$.
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  • $\begingroup$ "though time slows down for this observer as compared to the stationary observer, he detects light take more time to travel between these particular events." Can you please expand on it intuitively a little bit? According to what they write in the book. In that case if your time slows down you would measure light as going faster. Add to that that fact that you are running towards light. $\endgroup$ – bilanush Mar 1 '18 at 10:36
  • $\begingroup$ @bengy Measuring the time interval between two space-separated events is different from say a clock just sitting there in relativity. I don't know any intuitive way of putting it :( $\endgroup$ – PhyEnthusiast Mar 2 '18 at 13:29
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Apparently you speak about the one way speed of light. Measuring one way speed of light is quite well – known and very interesting problem. It is necessary to distinguish one – way speed of light and speed of light „back and forth“. What we know as c (299 792 458 m / s) is actually speed of light back and forth.

Speed of light „back and forth“ you can measure with a single clock. For example you send a beam, when this clock shows 0 and measure time, when the beam returns (Fizeau experiment). This speed of light has been measured many times with high precision and is c.

However, in your queston you probably mean one – way speed of light. That doesn‘t matter whether you catch up light or move towards it. To measure one – way speed of light you need two clocks – one at the starting point and another at endpoint. Then, you have to synchronize these clocks, so as they would show the same time.

The problem is that it is absolutely impossible to synchronize distant clocks without certain assumptions. To synchronize these clocks, you must send a signal so as it would reach another clock immediatelly. However, there is nothing that can move faster than light. This way, to synchronize clocks you must know one – way speed of light, and you measurement will depend on convention how to synchronize these clocks.

There is a heap of books in regard of this problem, but no one succeded to measure one way speed of light yet and will never probably will.

There are two empirically equivalent theories – Lorentz Ether theory and Special Relativity by A. Einstein.

Lorentz Ether theory considers measured two - way speed of light isotropic, while one- way speed of light is anisotropic. Reichenbah‘s synchrony convention allows anisotropic one - way speed of light, but isotropic two – way speed of light. In this theory dilation of moving clock and contraction of material bodies completely hides any anisotropy (look for Kennedy – Thorndike experiment in Wikipedia).

According to Reichenbach synchronization one way speed of light can be infinitely large in one direction and c/2 in another, so avearge speed will still be c.

Einstein‘s special relativity and Einstein synchronization admits, that one -way speed of light is always c.

https://en.wikipedia.org/wiki/One-way_speed_of_light

https://en.wikipedia.org/wiki/Einstein_synchronisation

Good book by Max Jammer: Concepts of Simultaneity– From Antiquity to Einstein and Beyond

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