Is the Constancy of Light in Vacuum a Proof or a Postulate?

[Boorglar]

Einstein postulated that the speed of light in vacuum is constant and is the same for all observers. It this related to the fact that in Maxwell's equations for electromagnetic waves in a vacuum,

c = \frac {1} {\sqrt{\mu_0 \epsilon_0}}?

The electric and magnetic constants, which are properties of free space, should indeed remain constant no matter how you are observing.

In this case, why do we say that he "postulated", when in fact it must be true by Maxwell's equations? Shouldn't we say that he actually prove it?EDIT:

Actually now that I think about it, its the other way around. I'm confused, now, because since no matter at what speed you move, you obviously will measure the same electric and magnetic constants, and so light ought to have the same speed. But as you move faster, your speed is added to that of the light and so you should see light moving faster. Since we know this is false, it would follow that the e/m constants are not, in fact, constants?
Can someone clarify that to me?

 

[Integral]

Yes, that is precisly where the postulate originates, and why AE was able to make that postulate and no one raised an eyebrow.

That result of Maxwells was the initiator of the Great Schism of physics which lasted from the publishing of Maxwells work till AE's 1905 paper on relativity. This used to be a topic taught in elementary school, guess not anymore.

 

[Bill_K]

Einstein postulated that Maxwell's Equations remain unmodified regardless of the velocity of the rest frame. Also that the group of linear transformations that leave Maxwell's Equations unchanged also leave the rest of physics unchanged, including mechanics.

Yes, that is precisly where the postulate originates, and why AE was able to make that postulate and no one raised an eyebrow.

Many eyebrows were raised.

 

[TSny]

It could be that the motivation for the light speed postulate was that Einstein felt that Maxwell's equations should be valid in all inertial frames. However, in the first 1905 paper, he took the light speed postulate as an independent postulate. The other postulate was of course the relativity postulate which implied that the true laws of electromagnetism would have to be the same in all inertial frames.

In the paper, he did not postulate that Maxwell's equations satisfy the relativity postulate. Rather, he first derived the Lorentz transformations from the light postulate. Then he spent a good deal of the paper proving (not postulating) that Maxwell's equations do indeed take on the same form in all inertial frames if the electric and magnetic fields transform in certain ways.

 

[TSny]

My previous remark was based on memory and is not correct. After digging out the paper I see that after deriving the Lorentz transformations from the light postulate, Einstein goes on to show that if you assume (postulate) that the Maxwell equations have the same form in all inertial frames, then the electric and magnetic fields must transform in certain ways. Overall, he's proving the consistency of Maxwell's equations with the postulates of relativity. And this only takes up 2 or 3 pages of the paper.

Nevertheless, I believe that in the paper Einstein was taking the light postulate as an independent postulate on which to base the theory of relativity.

 

[cosmik debris]

To be picky Einstein's postulate was that, the speed of light was independent of the speed of the source. They amount to the same thing in the end I guess.

 

[Histspec]

In this case, why do we say that he "postulated", when in fact it must be true by Maxwell's equations? Shouldn't we say that he actually prove it?

Note that when you talk about "Maxwell's equations" of that time, then you are actually talking about Maxwell's theory as formulated by Lorentz in 1895. Here, only in one "aether" frame is the speed of light constant in all directions and independent of the speed of the source. Einstein took this as a postulate, generalized it to all inertial frames, and abandoned the aether. Einstein wrote in 1912 ("Relativity and Gravitation. Reply to a remark of M. Abraham", in: The Collected Papers of Albert Einstein, Volume 4: The Swiss Years: Writings, 1912–1914:)

It is impossible to base a theory of the transformation laws of space and time on the principle of relativity alone. As we know, this is connected with the relativity of the concepts of "simultaneity" and "shape of moving bodies." To fill this gap, I introduced the principle of the constancy of the velocity of light, which I borrowed from H. A. Lorentz’s theory of the stationary luminiferous ether, and which, like the principle of relativity, contains a physical assumption that seemed to be justified only by the relevant experiments (experiments by Fizeau, Rowland, etc.)

and in 1928 Einstein said about Lorentz: (in: A. Pais, Subtle is the Lord, p. 169):

The enormous significance of his work consisted therein, that it forms the basis for the theory of atoms and for the general and special theories of relativity. The special theory was a more detailed exposé of those concepts which are found in Lorentz's research of 1895.

Here is Lorentz's 1895 paper that Einstein referred to:
http://en.wikisource.org/wiki/Attempt_of_a_Theory_of_Electrical_and_Optical_Phenomena_in_Moving_Bodies

 

[lugita15]

There's one thing that should be noted. It is indeed true that special relativity, in particular the second postulate, follows from the assumption that Maxwell's equations are true in all inertial reference frame. However, this assumption would have seemed absurd to the physicists of the nineteenth century, including Maxwell himself. When Maxwell completed his four equations, he realized that they lead to the implication that electromagnetic waves (light) propagate at a speed c. Now Maxwell was a firm believer in Newtonian mechanics, so he reasoned that since his equations were clearly not invariant under Galilean transformations, by the Principle of Relativity they could not be "real" laws of physics, i.e. they could only be true in one reference frame. And he assumed that this was the rest frame of the aether. In any other frame, he assumed that the correct equations describing the electromagnetic field would not be his original equations, but rather equations that were slightly different because they would include terms dependent on the velocity of the aether with respect to that frame. These are the so-called modified Maxwell equations (formulated by Hertz), and they include aether velocity terms added to the Ampere-Maxwell Law and Faraday's Law. According to these modified equations, the speed of light in a frame moving with respect to the aether is either c+v or c-v depending in which way the light is headed, where v is the speed of the aether with respect to your frame. This is exactly analogus to sound waves: the sound wave equation is only exactly true in the rest frame of the air, and in moving frames it must be replaced by another equation that includes a term dependent on the speed of the air.

This was the context in which the Michelson-Morley experiment was done: they believed that the speed of light would not be c as predicted by the original Maxwell equations, but rather would be c+v or c-v as predicted by the modified equations, and they hoped to find that value v. But instead, of course, they found that the speed of light seemed to be the same in all inertial reference frames, and thus Maxwell's equations true in all frames. (In stark contrast to the wave equation for sound.) But as mentioned above, Maxwell's equations are not Galilean-invariant, so this posed a perceived threat to the Principle of Relativity. Lorentz's solution to this problem was to say that objects moving with respect to the aether experienced length contraction and time dilation (due to electromagnetic effects), which made the speed of light appear to be constant even though it "really" wasn't. He assumed that if we had "accurate" rulers and clocks, which did not experience these effects, then they would show that the speed of light IS c+v or c-v in frames moving with respect to the aether, and thus the modified Maxwell equations are what are "really" true in all frames, even Maxwell's original equations appear to be true.

Now although Lorentz believed that the "errors" caused by length contraction and time dilation prevented accurate measurement of the speed of light and thus the speed of the aether, he still held out hope that the speed of the aether could in principle be determined. Perhaps if A and B were moving at different speeds with respect to the aether, A could measure how much B's ruler contracted, and B could measure how much A's ruler contracted, and by comparing results they could somehow deduce their speeds relative to the aether. But then Poincare demonstrated that the Lorentz transformations relate not only the aether frame to a moving frame, but also related moving frames to each other. (Specifically, he showed that the composition of LT's is an LT, and the inverse of an LT is an LT.) However, Poincare just viewed this as a curious phenomenon. It took Einstein to realize the real significance of all this: that the invariance of Maxwell's equations was fully compatible with the Principle of Relativity, and that it was our traditional notions of space and time that needed rethinking.

 

[Nugatory]

Lugita's comments about the historical perspective are spot-on, and it's easy to lose sight of just how strange and perplexing the electrodynamics of moving bodies once seemed.

To the modern ear, the second postulate sounds redundant ("Postulate 2: We really mean what we're saying in postulate 1" would be a fair paraphrase). It didn't sound that way at the turn of the last century.

 

[PhilDSP]

If the answer is "yes" then it can only apply in a vacuum. I believe Maxwell was fully cognizant that the speed of light is highly variable when charges are interspersed in the space that light travels but it will take a bit of sorting through his material to find an appropriate quote.

His contemporary, Helmholtz, had already worked out the specifics of EM dispersion and how the speed of EM propagation is effected in a most sophisticated manner. All of that really is quite apart from whether there is or isn't an aether and whether a theoretical aether does or doesn't move with respect to a particle. Both Hertz and Lorentz benefited from Helmholtz' work.

The presence of charges and current density causes the constants \epsilon_0 and \mu_0 to become the variables \epsilon and \mu.

 

[harrylin]

Einstein postulated that the speed of light in vacuum is constant and is the same for all observers.

Not exactly: you refer here to two different postulates. However, getting back to your title: Yes, Einstein's postulate for the speed of light (see posts #6 and #7) was a consequence of Maxwell's theory, which had been firmly established by then. The issue was how to combine it with the relativity postulate, which seemed to be incompatible with it. Einstein explained that succinctly as follows:

"We will raise this conjecture (the purport of which will hereafter be called the “Principle of Relativity”) to the status of a postulate, and also introduce another postulate, which is only apparently irreconcilable with the former, namely, that light is always propagated in empty space with a definite velocity c which is independent of the state of motion of the emitting body. These two postulates suffice for the attainment of a simple and consistent theory of the electrodynamics of moving bodies based on Maxwell's theory for stationary bodies." (emphasis mine).
- http://www.fourmilab.ch/etexts/einstein/specrel/

[..] I'm confused, now, because since no matter at what speed you move, you obviously will measure the same electric and magnetic constants, and so light ought to have the same speed. But as you move faster, your speed is added to that of the light and so you should see light moving faster. Since we know this is false, it would follow that the e/m constants are not, in fact, constants?
Can someone clarify that to me?

That's exactly why Einstein wrote "apparently irreconcilable" - that was the main riddle that the new theory was meant to solve (and it did)! Lorentz in 1904 (with one or two glitches) and next Einstein in 1905 (very smoothly) showed how it is possible that one can observe the same natural phenomena even if the speed of light in vacuum is, as Maxwell's theory has it, independent of the motion of the source. The essential change was that Newton's theory had to be modified, so that the measurement of more physical quantities than before had to be made "relative" - in particular measures of distance and time.

 

[harrylin]

[..] Now although Lorentz believed that the "errors" caused by length contraction and time dilation prevented accurate measurement of the speed of light and thus the speed of the aether, he still held out hope that the speed of the aether could in principle be determined. Perhaps if A and B were moving at different speeds with respect to the aether, A could measure how much B's ruler contracted, and B could measure how much A's ruler contracted, and by comparing results they could somehow deduce their speeds relative to the aether. [..]

Actually, such effects were his explanation in 1904 of why experiments like MMX could bring nothing at all - up to any precision. Perhaps we should explain how light from a spherical light source is measured as propagating at c in all directions according to relatively moving inertial reference systems. Boorglar, is that the issue that you have?

 

[grav-universe]

In addition to the excellent post by lugita15, the aberration of stars was the first important observation. It showed that the speed of light was the same regardless of the motion of the source, the stars in this case.

 

[grav-universe]

Quote of nonsense post that refutes relativity deleted.[/color]

Yes, but the M-M experiment showed that light pulses will travel identical perpendicular distances and back in the same time, even when the apparatus is turned in any direction, so the two way time of propagation of the light in any direction away and back to a single clock is always the same, and from this it is then possible to synchronize clocks within the observing frame such that they measure the same one way speed of light in any direction using the Einstein simultaneity convention. But if that is the result of an experiment performed within our own frame, then since there is nothing special about our own (inertial) frame, for instance we cannot consider that only our frame just happens to be always at rest with the field, then it must be true for all inertial frames, so all inertial frames must measure a constant speed of light in any direction when stationary clocks within those frames are synchronized according to the Einstein simultaneity convention.

 

[lugita15]

However, getting back to your title: Yes, Einstein's postulate for the speed of light (see posts #6 and #7) was a consequence of Maxwell's theory, which had been firmly established by then.

I would word things slightly differently. The invariance of Maxwell's equations implies the invariance of the speed of light, but in the nineteenth century no one believed in the invariance of Maxwell's equations, precisely because it DID imply the invariance of the speed of light, and they believed that contradicted the principle of Relativity, and so they concluded that Maxwell's equations only held in one reference frame. That is why the null result of the Michelson-Morley experiment was so surpising (except for the believers in the aether drag hypothesis and the contraction hypothesis, who of course would have their own problems later on).

 

[lugita15]

Actually, such effects were his explanation in 1904 of why experiments like MMX could bring nothing at all - up to any precision.

My understanding is that Lorentz didn't realize the symmetry of the Lorentz transformations until Poincare pointed it out, and that Poincare didn't realize the significance of that symmetry until Einstein explained it. So I'm guessing that by 1904 Poincare had already done his work, and Lorentz was building off of it. But I think in Lorentz's original aether theory, he still believed there was possibility that some combination of experiments could somehow find the aether frame. Do I have that right?

 

[DrGreg]

I can't claim to be an expert in the history of relativity, but everything lugita15 has said so far agrees with my understanding.

I would add (I don't think anyone's explicitly said this yet) that Lorentz came up with the Lorentz transform (between the aether frame and any other inertial frame) as exactly what was needed to make Maxwell's equations valid in all inertial frames.

 

[ardenmann0]

I can't claim to be an expert in the history of relativity, but everything lugita15 has said so far agrees with my understanding.

I would add (I don't think anyone's explicitly said this yet) that Lorentz came up with the Lorentz transform (between the aether frame and any other inertial frame) as exactly what was needed to make Maxwell's equations valid in all inertial frames.

I would add that Lorentz came up with the right equations, Lorentz transform. But his interpretation of the contraction is very different from that of Einstein's.

 

[pervect]

My impression of the times is that Michelson was motivated by wanting to replace the physical meter standard with an interferometric one, based on the wavelength of some particular light source.

(my source for this impresion isn't based on much reading, just the wiki article on the history of the meter: http://en.wikipedia.org/w/index.php?title=Metre&oldid=502620767)

In 1893, the standard metre was first measured with an interferometer by Albert A. Michelson, the inventor of the device and an advocate of using some particular wavelength of light as a standard of length

This goal - replacing the physical meter with one based on the wavelength of light - wasn't actually accomplished until the 1960's, however.

To make progress towards this goal, Michelson needed to pin down experimental sources of error. One source of expected error was "ether wind". He , and everyone else, was rather surprised to find out that the expected effect didn't exist.

 

[harrylin]

I would word things slightly differently. The invariance of Maxwell's equations implies the invariance of the speed of light, but in the nineteenth century no one believed in the invariance of Maxwell's equations, precisely because it DID imply the invariance of the speed of light, and they believed that contradicted the principle of Relativity, and so they concluded that Maxwell's equations only held in one reference frame.[..].

Maxwell's equations did not imply the invariance of the speed of light - but yes, I should have written that Maxwell's theory for stationary systems was firmly established (which is of course what Einstein meant with "stationary bodies"; at the time he evidently did not distinguish bodies from systems).

My understanding is that Lorentz didn't realize the symmetry of the Lorentz transformations until Poincare pointed it out [..] I think in Lorentz's original aether theory, he still believed there was possibility that some combination of experiments could somehow find the aether frame. Do I have that right?

I think that the part that I retained here is correct. However, such a discussion is a bit besides the topic.

PS. Strange: the OP did not react on our clarifications... I won't comment anymore if he/she isn't really interested in his/her topic!

 

[PhilDSP]

... but in the nineteenth century no one believed in the invariance of Maxwell's equations, precisely because it DID imply the invariance of the speed of light ...

Or rather in the nineteenth century anyone who properly understood the Maxwell equations realized they applied to stationary charges only and therefore could not be invariant when applied to a situation with moving charges. Maxwell himself developed a preliminary augmentation of the equations to accommodate moving charges (with no reference to an "aether") but apparently didn't concentrate on further developments. Helmholtz, Heaviside, Hertz, Ritz and Cohn all developed their own variants of the Maxwell equations for moving charges. Each of them are form invariant.

 

[bcrowell]

There are really two questions here. (1) How did Einstein originally present his theory in relation to the experimental and theoretical work that came before? (2) In Einstein's young brain, what thought process led him to come up with his theory?

If the OP has in mind #1, then the (oversimplified) answer is yes, and the complete answer can be determined with certainty by reading Einstein's 1905 paper.

If #2, then it's more complicated, and nobody will ever really know the answer for sure: http://arxiv.org/abs/0908.1545

 

[Histspec]

Now Maxwell was a firm believer in Newtonian mechanics, so he reasoned that since his equations were clearly not invariant under Galilean transformations, by the Principle of Relativity they could not be "real" laws of physics, i.e. they could only be true in one reference frame. And he assumed that this was the rest frame of the aether.

Maybe it should be stated that the term "aether frame" wasn't well defined at this time period:

1) Fresnel in 1818 proposed an aether almost completely at rest, as necessary to explain the aberration of light. Here, light is a transverse wave in the aether. However, he had do include "partial aether dragging" to explain the negative outcome of the Arago experiment. So the speed of light is constant in vacuum, but variable within matter in accordance with Fresnel's dragging coefficient (later confirmed in the Fizeau experiment in 1851).

2) Stokes (1844) proposed that the aether is completely dragged by Earth. He had to invent certain auxiliary hypothesis to derive Fresnel's dragging coefficient and the aberration of light (which wasn't very convincing).

3) Maxwell (1865) derived his equations (actually twenty of them). But he made no comment on the state of motion of the luminiferous aether in matter or its vicinity. Later (shortly before his death) he argued that if there is an aether wind, then it is of second order in v/c when two-way measurements were made. (See Maxwell's own Ether entry in the encyclopedia britannica, section "Relative motion of the aether", 1878).
Encyclopædia Britannica, Ninth Edition/Ether.

4) Michelson (1881) and Michelson&Morly (1887) followed Maxwell's suggestion and executed their famous second order experiment - no aether wind was found. A contradicting situation arose: The Fizeau experiment and the aberration of light "proved" an almost resting aether, while the Michelson-Morley experiment seemed to imply complete aether dragging.

5) Heaviside and Hertz (1890) brought Maxwell's (now four) equations into their modern form - but Hertz believed in complete aether dragging (even though Hertz knew that this was at variance with the Fizeau experiment).

6) Lorentz was the first (1892 and 1895) to argue that the aether is completely motionless and thus totally unaffected by the motion of matter. Now we can talk about one (and only one) "aether frame", in which the speed of light is the same in all directions. To avoid "aether wind" effects, he (together with Larmor and Poincaré) had to include "local time" and "length contraction", and later the complete Lorentz transformation (1904).

7) Einstein used the Maxwell-Lorentz theory of 1895, in which the speed of light was constant in only one frame. Per the relativity principle, he argued that this constancy and Maxwell's equations must apply to all reference frames, none of them should be called "aether" any more. Special relativity was born.

 

[Boorglar]

Wow I did not expect to get so many answers! I was away and didn't read this thread for a while. Unfortunately I am not really knowledgeable on the history and the mathematics behind Einstein's idea and the Lorentz transformations.

Thanks to lugita15 for the first post with the historical facts. You mentioned that in order to reconcile electromagnetism with the relativity principle, Lorentz had to conclude that length and time could contract.

But how exactly were the time/space dilation effects proved so as to keep electromagnetism consistent with relativity? I have seen the weird-looking formulas involving square roots of 1 - (v/c)^2 but I've never understood how they came up with precisely this expression...

Also, how do you make something "invariant" with respect to something else?
I mean, answers mention a lot the "invariance" of the equations, but I don't really get what it means. How did they translate the fact that the equations should work in all reference frames mathematically?

 

[PhilDSP]

In not so technical terms the invariance of the Maxwell equations effectively means that they provide proper solutions for any conceivable situation. Since they don't provide valid solutions for moving charges in Euclidean space (outside of the original inertial frame using a Galilean transformation), they are broken (i.e. not invariant).

Crudely speaking Fitzgerald, Larmor and Lorentz conceived of a game-changing means of allowing the equations to work for moving charges by manipulating time and space variables (which don't directly appear in the equations but are required for interpreting solutions). Normally, when solving differential equations, the relationships between time and space are frozen during the iteration of determining a solution. Poincare investigated the meaning of manipulating the relationships and worked out a lot of details and principles on which SR is based but may have fell short of being able to predict an actual experimental observation. Einstein very definitely declared that there is legitimacy in basing the interpretation of solutions on the translation of time and space variables and further tidied up some loose ends to bring forth the core of SR.

 

[harrylin]

Maybe it should be stated that the term "aether frame" wasn't well defined at this time period:

1) Fresnel in 1818 proposed an aether almost completely at rest, as necessary to explain the aberration of light. Here, light is a transverse wave in the aether. However, he had do include "partial aether dragging" to explain the negative outcome of the Arago experiment. So the speed of light is constant in vacuum, but variable within matter in accordance with Fresnel's dragging coefficient (later confirmed in the Fizeau experiment in 1851).

2) Stokes (1844) proposed that the aether is completely dragged by Earth. He had to invent certain auxiliary hypothesis to derive Fresnel's dragging coefficient and the aberration of light (which wasn't very convincing).

3) Maxwell (1865) derived his equations (actually twenty of them). But he made no comment on the state of motion of the luminiferous aether in matter or its vicinity. Later (shortly before his death) he argued that if there is an aether wind, then it is of second order in v/c when two-way measurements were made. (See Maxwell's own Ether entry in the encyclopedia britannica, section "Relative motion of the aether", 1878).
Encyclopædia Britannica, Ninth Edition/Ether.

4) Michelson (1881) and Michelson&Morly (1887) followed Maxwell's suggestion and executed their famous second order experiment - no aether wind was found. A contradicting situation arose: The Fizeau experiment and the aberration of light "proved" an almost resting aether, while the Michelson-Morley experiment seemed to imply complete aether dragging.

5) Heaviside and Hertz (1890) brought Maxwell's (now four) equations into their modern form - but Hertz believed in complete aether dragging (even though Hertz knew that this was at variance with the Fizeau experiment).

6) Lorentz was the first (1892 and 1895) to argue that the aether is completely motionless and thus totally unaffected by the motion of matter. Now we can talk about one (and only one) "aether frame", in which the speed of light is the same in all directions. To avoid "aether wind" effects, he (together with Larmor and Poincaré) had to include "local time" and "length contraction", and later the complete Lorentz transformation (1904).

7) Einstein used the Maxwell-Lorentz theory of 1895, in which the speed of light was constant in only one frame. Per the relativity principle, he argued that this constancy and Maxwell's equations must apply to all reference frames, none of them should be called "aether" any more. Special relativity was born.

Very nice summary!

 

[harrylin]

[..] But how exactly were the time/space dilation effects proved so as to keep electromagnetism consistent with relativity? I have seen the weird-looking formulas involving square roots of 1 - (v/c)^2 but I've never understood how they came up with precisely this expression...

It cannot be proved in a strong sense; instead a theory is a creative solution or invention that is meant to explain observations. However, in this case there is not much room for alternative solutions, so to that extent it can be considered "proved".
Do you understand why according to Lorentz and Einstein a "moving" interferometer such as that of Michelson and Morley must "be" contracted in length by that factor? If not, that's a good point to start with. Next there is a similar example with a "light clock". You can "google" for those - and you may find hits of this forum.

Also, how do you make something "invariant" with respect to something else?
I mean, answers mention a lot the "invariance" of the equations, but I don't really get what it means. How did they translate the fact that the equations should work in all reference frames mathematically?

That's exactly what is meant with "invariant"! The speed of a light ray or wave is invariant in the sense that it is "observed" as going at c by means of any inertial reference frame that has been set up in a certain way. And to understand how it works, it may be good to start with the things that I mentioned just here above.

 

[Ibix]

Also, how do you make something "invariant" with respect to something else?
I mean, answers mention a lot the "invariance" of the equations, but I don't really get what it means. How did they translate the fact that the equations should work in all reference frames mathematically?

Following Newton:

You are standing in a stationary train which is on a track parallel to the x-axis and I am on the platform outside. We are each holding a ball, and we drop them simultaneously at time t=0. The initial positions of the balls are (x_0,y_0) (there is a z-coordinate perpendicular to the track which would have different values, but I am ignoring here because it doesn't matter). After time t, the positions are both (x_0,y_0-gt^2/2).

Now, we repeat the experiment but with your train passing through the station at speed v in the x direction. We drop the balls at time t=0, as we pass one another. My ball still follows the same path: (x_0,y_0-gt^2/2). But your ball obviously won't. The principle of relativity states, though, that you are entitled to consider yourself at rest, so you must see the same path you did when you were at rest. If you are to see (x_0,y_0-gt^2/2), but from my perspective you are moving at speed v in the x direction then the path I see must be (x_0+vt,y_0-gt^2/2). You can make the exact same argument (except that to you, I am moving at -v): your ball follows (x_0,y_0-gt^2/2); I am entitled to believe I am stationary so my ball must (to you) follow (x_0-vt,y_0-gt^2/2) so that I can continue to believe that.

That is invariance; the equations are the same for you and for me. Nothing different happens just because we are moving with respect to one another. The maths I use to describe me are the same you use to describe you; the maths I use to describe you are the same as the maths you use to describe me. I leveraged that observation to derive something useful: if I want to know the position of a ball according to somebody else, all I need to do is take my observations and add the current position ((vt,0) in the example above) of the other person.

What we are discussing on this thread is a fifty-year investigation that lead to the conclusion that our watches won't agree (although, again, my complaint about your watch will be the same as your complaint about mine). Since I have assumed above that we will always agree on the t coordinate, what I have written is subtly wrong. This would be obvious to everybody if we typically moved at 10% of lightspeed, or thereabouts, but we move a lot slower than that so we never notice. Maxwell's equations, describing things that do move at light speed, noticed.

Does that kind of make sense?

 

[Boorglar]

Yes this makes perfect sense. But then what bothers me is that when the speed of the train approaches c, then you can no longer add v*t in the equation, and I don't know how to reason it out.

Say you are in a spaceship moving with speed v. Next to you comes a ray of light in vacuum with speed c. Now normal intuition would suggest that you will see the ray racing against you at speed c - v. Relativity theory says this is impossible.

So somehow, we need to change another parameter so that you end up seeing light moving with speed c no matter how fast you move. Suppose you and the light ray started at x = 0 in Jupiter. On the clock in your spaceship, you measure t seconds, and find that you are at position v*t next to planet Mars, and light at position x, next to Earth. Then the speed of light as measured by you is c = (x - vt)/t.

From an observer at rest in Jupiter, when you reach Mars, light is at position c*t'. (t' is another time parameter). Now I think that when you reach Mars. both you and the observer see the light ray next to Earth, at position x, am I right? So I can say that x = c*t'.
So c = (c*t' - vt)/t = c*(t'/t) - v so t' = t * (v+c)/c = t*(1+v/c).

This would mean that the observer at rest has lived 1 + v/c times longer than you (from your perspective) when you reached Mars (time dilation). But this is not the true factor which is used in relativity theory, so I get confused even more...

 

[lugita15]

But how exactly were the time/space dilation effects proved so as to keep electromagnetism consistent with relativity? I have seen the weird-looking formulas involving square roots of 1 - (v/c)^2 but I've never understood how they came up with precisely this expression...

See the attached excerpt from the Feynman Lectures on Physics, specifically the last two pages.