Derivation of Gamma: Is it Correct?

[cryptic]

Einstein's differential equation above is a general relation between arbitrary sets of tau, x', and t' coordinates; if you find the correct expression for tau as a function of x' and t' (it ends up being given by the function \tau = \frac{1}{\sqrt{1 - v^2/c^2}} [t' - \frac{v*(x' + vt')}{c^2}]), so you can find dtau/dx' and dtau/dt' at any event (x',t'), then the differential equation will always be satisfied. In contraxt, x'=(c-v)*t' is not a general relation between arbitrary x' and t' coordinates, it's an equation of motion for a single light pulse: only combinations of x' and t' coordinates that lie along this light pulse's worldline will satisfy the equation x'=(c-v)t'. There are plenty of other perfectly valid combinations of x' and t' where x is not equal to (c-v)t'.

No, it is not true, \frac{1}{\sqrt{1 - v^2/c^2}} is an arbitrary "constant(v)" which can not be derived from Einstein's equation.

 

[JesseM]

No, it is not true, \frac{1}{\sqrt{1 - v^2/c^2}} is an arbitrary "constant(v)" which can not be derived from Einstein's equation.

I didn't say that the final answer <br /> \tau = \frac{1}{\sqrt{1 - v^2/c^2}} [t&#039; - \frac{v*(x&#039; + vt&#039;)}{c^2}] could be derived from the equation <br /> \frac{1}{2}\left(\frac{1}{c-v}+\frac{1}{c+v}\right) \frac{\partial\tau}{\partial t&#039;}= \frac{\partial \tau}{\partial x&#039;}+\frac{1}{c-v}\frac{\partial\tau}{\partial t&#039;} alone; you could probably find other functions tau(x',t') which satisfy Einstein's equation there. However, Einstein placed additional constraints on the function tau(x',y',z',t') besides just that it satisfied that equation; for example, he required that tau be a linear function which could be expressed in the form tau = Ax' + By' + Cz' + Dt'. From this he is able to show that tau(x',y',z',t') must have the form (unknown function of v)*(t' - vx'/(c^2 - v^2)). Then later he places some additional constraints on the unknown function of v, like the fact that phi(v) = phi(-v). With the combination of constraints he assumes, he is able to prove that there is only one possible function tau that satisfies all the constraints, and it's the one I mention above.

 

[cryptic]

I didn't say that the final answer <br /> \tau = \frac{1}{\sqrt{1 - v^2/c^2}} [t&#039; - \frac{v*(x&#039; + vt&#039;)}{c^2}] could be derived from the equation <br /> \frac{1}{2}\left(\frac{1}{c-v}+\frac{1}{c+v}\right) \frac{\partial\tau}{\partial t&#039;}= \frac{\partial \tau}{\partial x&#039;}+\frac{1}{c-v}\frac{\partial\tau}{\partial t&#039;} alone; you could probably find other functions tau(x',t') which satisfy Einstein's equation there. However, Einstein placed additional constraints on the function tau(x',y',z',t') besides just that it satisfied that equation; for example, he required that tau be a linear function which could be expressed in the form tau = Ax' + By' + Cz' + Dt'. From this he is able to show that tau(x',y',z',t') must have the form (unknown function of v)*(t' - vx'/(c^2 - v^2)). Then later he places some additional constraints on the unknown function of v, like the fact that phi(v) = phi(-v). With the combination of constraints he assumes, he is able to prove that there is only one possible function tau that satisfies all the constraints, and it's the one I mention above.

Einstein's derivation has no mathematical justification and he used an arbitrary function phi(v)=1 or an arbitrary constant a. His tau(x',y',z',t')=(t' - vx'/(c^2 - v^2)) is nothing else but Voigt's, Lorentz's and Poincare's derivation of "local time", which is only Galilean transformation divided by c'=sqrt(c²-v²). That is:

x'/c'=x/c'-vt/c' and with t=x/c'

t'=t-vx/c'²=t-vx/(c²-v²).

Physical meaning of this transformation is only this: The light needs less time to traverse a shorter distance.

 

[JesseM]

Einstein's derivation has no mathematical justification

Please point out the specific step that is not mathematically justified.

and he used an arbitrary function phi(v)=1 or an arbitrary constant a.

Yes, he introduces an unknown function and then figures out what that function must be if various constraints apply. Do you object to the practice of algebra of introducing unknown variables like "x", and then figuring out what numerical value they must have using various combinations of constraints (like y=3x + 2 and z=(y-1)/2 and z=2x - 2, which together imply that x=5)? Introducing unknown functions and then solving for them is fundamentally no different.

His tau(x',y',z',t')=(t' - vx'/(c^2 - v^2))

That is not the equation he derived. He found that tau as a function of the x, t coordinates of inertial frame K was:

tau(x,y,z,t) = (1/sqrt(1 - v^2/c^2))*(t - vx/c^2), which can also be written as tau(x,y,z,t) = (c/sqrt(c^2 - v^2))*(t - vx/c^2)

If you want to find tau as a function of the primed coordinate x',t' which I used for the coordinate system Kg, then you have to remember that Kg was just constructed by doing a Galilei transform on K, so x' = x - vt and t'=t (which means x = x' + vt'). Plug that into the above equation and you get:

tau(x',y',z',t') = (1/sqrt(1 - v^2/c^2))*(t' - v(x' + vt')/c^2)

is nothing else but Voigt's, Lorentz's and Poincare's derivation of "local time", which is only Galilean transformation divided by c'=sqrt(c²-v²).

It's true that Lorentz had already derived the coordinate transformation for tau(x,y,z,t) which I wrote above (along with xi(x,y,z,t) and so forth), which is why it's called the "Lorentz transformation" and not the "Einstein transformation". But neither Lorentz nor Poincare had shown that it could be derived in Einstein's particular way: assuming that every inertial observer defined his own coordinate system using a system of rulers and clocks at rest in his own frame, with the clocks set using the Einstein synchronization convention (so if a light signal is sent from clock A when it reads t0, bounces back from clock B when it reads t1, and returns to clock A when it reads t2, then A and B are defined as synchronized if 1/2*(t0 + t2) = t1), and with the two basic postulates of relativity holding: that all laws of physics will obey the same equations in both inertial coordinate systems, and that light will move at c in both inertial coordinate systems. All the constraints on the coordinate transformation that he introduces in the paper follow from these basic assumptions.

x'/c'=x/c'-vt/c' and with t=x/c'

But here you are only dividing the equation x' = x - vt by c', which does *not* give the transformation Einstein derived (the Lorentz transformation which I wrote above)...were you saying this equation was supposed to be equivalent to the Lorentz transformation when you said "tau(x',y',z',t')=(t' - vx'/(c^2 - v^2)) is nothing else but ... Lorentz's ... derivation of "local time", which is only Galilean transformation divided by c'=sqrt(c²-v²)" ? If the x',t' coordinate system was an inertial frame constructed according to Einstein's assumptions (and remember that in my version of Einstein's proof, the coordinate system Kg which used coordinates x',y',z',t' was not an SR inertial frame, it was just a coordinate system created by doing a Galilei transform on the inertial frame K), then the relation would be x' = (1/sqrt(1 - v^2/c^2))*(x - vt), or x' = (c/c')*(x - vt). This is obviously not equivalent to x'/c'=x/c'-vt/c'.

Also, what do you mean when you say t=x/c'? That can't be a coordinate transformation since both t and x are unprimed and thus presumably part of the same coordinate system, so is it supposed to be the equation of motion for some object?

t'=t-vx/c'²=t-vx/(c²-v²).

Physical meaning of this transformation is only this: The light needs less time to traverse a shorter distance.

What is the physical meaning of the x', t' coordinates supposed to be? Your equations are not equivalent to the Lorentz transformation so they cannot be the coordinates of an inertial frame constructed according to Einstein's assumptions, with the assumption that the two postulates of SR hold; but they also don't correspond to a Galilei transform on the x,t coordinates, as with the coordinate system I labeled Kg in my version of Einstein's proof.

 

[tiny-tim]

\gamma = csc(cos^{-1}(v/c))

\gamma^{-1} = sin(cos^{-1}(v/c))

Is there something wrong with this derivation? If not, why isn't this the more formal way of writing it, it seems a lot more simple?

Hi epkid08!

(I'm going to use c = 1 )

The substitution usually made is the hyperbolic trig substitution:

v = tanh, 1/γ = sech​

(wikipedia calls tanh-¹ the rapidity)

Yes, you can define v = cos, 1/γ = sin instead (or vice versa).

This is because sin² + cos² = 1, just as sech² + tanh² = 1.

Mathematicians can define anything they like, and your definition is certaily valid.

But some definitions are more useful than others, and rapidity has the advantae that the rule for addition of velocities in one dimension reads:

tanhw = (tanhu + tanhv)/(1 + tanhutanhv) = tanh(u+v)​

in other words, you just add the rapidities.

Your definition would give:

cosw = (cosu + cosv)/(1 + cosucosv)​

which isn't any help at all.

 

[cryptic]

@ JesseM
It is clearly evident that Einstein's derivation of SR is mathematically wrong, as shown in my recent posts. SR is based on some ad hoc assumptions, introduced by Lorentz and Einstein without any physical reason.

Instead of trying to confuse with "so-called" partial differential equation, Einstein could simply write L'/(c-v)+L'/(c+v)=2L/c, with L, L'=length of rigid rod in both systems of reference, to result in:

L'*2c/(c²-v²)=2L/c and

L'/(c²-v²)=L/c², or

L'=L* (1-v²/c²).

If you apply Lorentz ad hoc assumption L'=L*sqrt(1-v²/c²), you can see, that this is not consistent with the "constance of the speed of light" in all inertial frames ("missing" Einstein's Beta).

 

[JesseM]

@ JesseM
It is clearly evident that Einstein's derivation of SR is mathematically wrong, as shown in my recent posts.

And yet each time you have introduced a specific objection, I have countered it, and you haven't responded to show what it is wrong with my counter. For example, I pointed out you couldn't use the equation x'=(c-v)t to substitute ∂x'=(c-v)∂t into the differential equation Einstein derived relating dtau/dx' to dtau/dt', because x'=(c-v)t was just the equation of motion for a single light pulse, not a general relation between arbitrary x' and t coordinates. I don't know if you accepted that your line of argument here was incorrect since you never responded; if you didn't, I'd be happy to show in more explicit detail how it's possible to find a coordinate tranformation that satisfies Einstein's differential equation but which doesn't satisfy the new equation you obtained with the substitution ∂x'=(c-v)∂t. But it's very hard to have a reasoned discussion with you about Einstein's proof when you mostly just make sweeping nonspecific statements like "Einstein's derivation of SR is mathematically wrong" and refuse to say specifically what you're objecting to most of the time, and when you do occasionally provide specific objections and I rebut them, you don't tell me whether you accept my rebuttal or not (and if not, you need to tell me what specifically is wrong with my argument in these cases).

SR is based on some ad hoc assumptions, introduced by Lorentz and Einstein without any physical reason.

Einstein's 1905 paper is not intended to "prove" that the two postulates of SR are correct (only experiment can support physical postulates, and plenty of subsequent experiments have agreed with SR's predictions), it's just showing mathematically that if the two postulates are true, then mathematically that means the coordinate systems of different inertial observers must be related by the Lorentz transformation. One can accept Einstein's reasoning about the logical consequences of the postulates without necessarily believing that the postulates are, in fact, correct. Of course the postulates were not totally ad hoc either--the Michelson-Morley experiment had failed to find evidence that light moves at different speeds in different directions when we're moving relative to the ether, and the principle that the laws of physics should look the same in every inertial frame had long been a principle of Newtonian mechanics.

Instead of trying to confuse with "so-called" partial differential equation, Einstein could simply write L'/(c-v)+L'/(c+v)=2L/c, with L, L'=length of rigid rod in both systems of reference, to result in:

L'*2c/(c²-v²)=2L/c and

L'/(c²-v²)=L/c², or

L'=L* (1-v²/c²).

The equation L'/(c-v)+L'/(c+v)=2L/c doesn't make much sense because it ignores time dilation. If the rod has length L' in the frame where the rod is moving at speed v, and we assume the light moves at c in this frame, then when the rod is moving from back to front the front will be retreating from the light pulse at speed v, so the distance between the light pulse and the front will be decreasing at a "closing velocity" of (c-v), so the light will take a time of L'/(c-v) to reach the front in this frame. Then when the light is going from front to back, the back is moving towards the pulse at speed v so the distance between the light and the back is decreasing with a closing velocity of (c+v), so the light will take a time of L'/(c+v) to reach the back in this frame. So, in this frame the total time for the light to go from back to front to back again will be L'/(c-v) + L'/(c+v), just like the left side of the equation. And of course, if the rod has length L in its own rest frame and the light moves at c in both directions in this frame, the time in this frame for the light to go from back to front to back again will be 2L/c, like the right side of your equation. But you cannot assume the time as measured in the first frame will be equal to the time as measured in the second frame unless you assume that clocks in both frames tick at the same rate, which of course is not true in relativity.

Also, it's fairly easy to show that if length contraction existed but there was no time dilation, then the two postulates that Einstein started from in his 1905 paper could not both be satisfied. For instance, consider two methods of synchronizing clocks: one method is just to set the clocks to the same time when they are at the same location, then move them apart, while the other method would be to set off a light pulse at the midpoint between the clocks and set them both to read the same time when the light hit them. In a universe with length contraction but no time dilation, there would be a unique frame where both methods would lead to clocks ending up synchronized in the same way, while in other frames a pair of clocks synchronized by the first method and at rest in the frame would disagree with a pair of clocks synchronized by the second method and sitting next to them. If you don't see why this would be true, just ask and I can elaborate; if you agree this would be true in a universe with length contraction but no time dilation, then you can see that this would violate the first postulate of SR, which says that any physical experiment must yield the same result when performed in different inertial frames.

If you apply Lorentz ad hoc assumption L'=L*sqrt(1-v²/c²), you can see, that this is not consistent with the "constance of the speed of light" in all inertial frames ("missing" Einstein's Beta).

If you take into account length contraction, time dilation, and the relativity of simultaneity (all of which can be derived from the Lorentz transformation, which itself can be derived from Einstein's two postulates), you do find that different inertial frames all agree that light moves at c. If you don't believe me, here's a numerical example I wrote up here:

To measure the speed of anything, you need to measure its position at one time and its position at another time, with the time of each measurement defined in terms of a local reading on a synchronized clock in the same local region as the measurement; then "speed" is just (change in position)/(change in time). So, time dilation, length contraction, and the relativity of simultaneity all come into play. From the stationary observer's perspective, the ruler which the moving observer used to measure the distance was shrunk by a factor of \sqrt{1 - v^2/c^2}, the time between ticks on the moving clocks is expanded by 1/\sqrt{1 - v^2/c^2}, and the two clocks are out-of-sync by vx/c^2 (where x is the distance between the clocks in their own rest frame, where they are synchronized). On another thread I posted an example of how these factors come together to ensure both observers measure the same light beam to move at c:

Say there's a ruler that's 50 light-seconds long in its own rest frame, moving at 0.6c in my frame. In this case the relativistic gamma-factor (which determines the amount of length contraction and time dilation) is 1.25, so in my frame its length is 50/1.25 = 40 light seconds long. At the front and back of the ruler are clocks which are synchronized in the ruler's rest frame; because of the relativity of simultaneity, this means that in my frame they are out-of-sync, with the front clock's time being behind the back clock's time by vx/c^2 = (0.6c)(50 light-seconds)/c^2 = 30 seconds.

Now, when the back end of the moving ruler is lined up with the 0-light-seconds mark of my own ruler (with my own ruler at rest relative to me), I set up a light flash at that position. Let's say at this moment the clock at the back of the moving ruler reads a time of 0 seconds, and since the clock at the front is always behind it by 30 seconds in my frame, then in my frame the clock at the front must read -30 seconds at that moment. 100 seconds later in my frame, the back end will have moved (100 seconds)*(0.6c) = 60 light-seconds along my ruler, and since the ruler is 40 light-seconds long in my frame, this means the front end will be lined up with the 100-light-seconds mark on my ruler. Since 100 seconds have passed, if the light beam is moving at c in my frame it must have moved 100 light-seconds in that time, so it will also be at the 100-light-seconds mark on my ruler, just having caught up with the front end of the moving ruler.

Since 100 seconds passed in my frame, this means 100/1.25 = 80 seconds have passed on the clocks at the front and back of the moving ruler. Since the clock at the back read 0 seconds when the flash was set off, it now reads 80 seconds; and since the clock at the front read -30 seconds, it now reads 50 seconds. And remember, the ruler was 50 light-seconds long in its own rest frame! So in its frame, where the clock at the front is synchronized with the clock at the back, the light flash was set off at the back when the clock there read 0 seconds, and the light beam passed the clock at the front when its time read 50 seconds, so since the ruler is 50-light-seconds long, the beam must have been moving at 50 light-seconds/50 seconds = c as well! So you can see that everything works out--if I measure distances and times with rulers and clocks at rest in my frame, I conclude the light beam moved at 1 c, and if a moving observer measures distance and times with rulers and clocks at rest in his frame, he also concludes the same light beam moved at 1 c.

If you want to also consider what happens if, after reaching the front end of the moving ruler at 100 seconds in my frame, the light then bounces back towards the back in the opposite direction towards the back end, then at 125 seconds in my frame the light will be at a position of 75 light-seconds on my ruler, and the back end of the moving ruler will be at that position as well. Since 125 seconds have passed in my frame, 125/1.25 = 100 seconds will have passed on the clock at the back of the moving ruler. Now remember that on the clock at the front read 50 seconds when the light reached it, and the ruler is 50 light-seconds long in its own rest frame, so an observer on the moving ruler will have measured the light to take an additional 50 seconds to travel the 50 light-seconds from front end to back end.