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In water the speed of light is slower than it is in vacuum. By special relativity the speed of light is constant. Typically, for the basic explanation of special relativity a clock which has a mirror is used. Then the fact that the light travels a different path for a stationary observer and an observer who is traveling with the clock, is used to demonstrate the principle of special relativity. The following link should serve as an example: http://galileoandeinstein.physics.virginia.edu/lectures/srelwhat.html

Of course time doesn't pass slowly in water. But what would be a simple way (if there's one) to explain this to a student who has just learnt how special relativity works?

I will update this question in the future in case I haven't made it clear enough. I also know that when it is postulated that speed of light is constant it is quite explicitly clear that the speed of light is constant in vacuum. What I am interested in is, how does (or why doesn't) it affect the calculations and Lorentz transformations that come along with it.

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The two phenomena are very different.

One is due to the bulk properties of a polarizable or magnetizable solid (properties of avogadro's-number-of electrons bound to atoms) and so is very much an emergent effect and can be violated.

The other is a nice example of the properties of the Lorentz transformation.

If you boosted into a frame travelling with the light in the material - which is well possible, since the speed of light in the material is less than $c$ - things would be MORE complicated. You get drag exactly as in the Fizeau experiment. So now you have a bulk magnetizable polarizable material that's been boosted as well as Fizeau drag, and you haven't simplified the situation one bit.

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In water the speed of light is slower than it is in vacuum.

That's at best imprecise: The phase speed of electro-magnetic waves in water, for instance, is less than the corresponding signal front speed (a.k.a. "speed of light in vaccuum") in the frequency range of the visible spectrum; but indeed larger for some range of the X-ray spectrum.

(Similarly, the group speed of some electro-magnetic pulse in water is not necessarily smaller than the corresponding signal front speed.)

However: signal front speed is of course the same in all cases; plainly due to the (chrono-geometric) definition of "distance", $$\frac{c}{2}\text{ ping duration},$$ referring to signal fronts of the ping signals which are being considered.

By special relativity the speed of light is constant.

Right; or more specificly: by the notion of "distance" in the RT being defined as "chrono-geometric distance"; together with the subsequent definitions of "speed" etc. in terms of "distance" ratios.

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I have a problem with the 'light clock' operation... postulate 2 says that light is propagated in free space with a defined speed that is independent of the emitting source... velocity is a vector quantity, ie it is directional. so in order for the velocity to remain independent of the emitting source, the direction of propagation must not be affected by the motion of the emitting source. therefore according to postulate 2 the motion of the emitter will have no effect on the direction of the light it emits and therefore the light will have no vector component induced by the motion of the emitter and will not travel parallel to the emitting source. the light will travel in a straight line from the point in space where it is formed, and propagate in the direction the emitter is pointed at that instant. only a particle with mass, can and will, have momentum transferred to it, thereby causing a vector angle path whereby the particle will travel parallel to the emitter. light must travel in a straight line from the point in space where it is created. so, any frame of reference that is moving with respect to that point in space where the photon is created will see an angled path for perpendicular movement and Doppler effect for parallel movement(same or opposite direction). creating a photon in free space with a moving emitter is like slapping the surface of the water on a lake while traveling along in a canoe, where the created waves propagation is independent of the emitters/slappers motion. creating an EM wave with an omnidirectional antenna while traveling in a spaceship at high velocity would give a similar visualization as the canoe experiment, but since light is a unidirectional EM wave it would travel from the point of creation in space and propagate in the direction the emitter was pointing(perpendicular to ship travel) and not in a vector angle direction from the emitter. so, the light clock could not work as described by Einstein et al if postulate 2 is true..... Either the postulate or its application in the light clock thought experiment is incorrect...

Einstein was wrong, one way or the other...

however, to answer your question, time dilation is the Pythagorean ratio of velocity of an object to c within a given medium. you cannot compare the velocity of light between mediums and infer a dilation.

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