# Is SR really internally consistent?

Nugatory
Mentor
I have found an article where the author says the following about einstein's SR: If we have two reference frames, S and S', with S stationary and the S' moving along the 'x' axis with speed 'v' of S, then as per SR there is length contraction in the 'x' direction but no change in the 'y' or the 'z' directions of objects in S'. That is y=y' and z=z'. If we take just the y=y', then the length y2-y1 = y'2-y'1. However, SR says that the time is slower in S' than in S. This means t2-t1 is < t'2-t'1. So, if we have a beam of light traveling along the y axis in S then the speed of the light is C= y2-y1/t2-t1. For the observer in S' who is also looking at the same beam of light it will be C'= y'2-y'1/t'2-t'1. Now, the numerators are equal but the denominator in S' is larger than in S due to time dilation. This means C'<C, which contradicts the constancy of the speed of light postulate of SR!!!! Also, for light traveling in x-axis direction we have from length contraction x2-x1 > x'2-x'1 and from time dilation we have t2-t1 < t'2-t'1, then the speed of light in S is C=x2-x1/t2-t1 and in S' it is C'= x'2-x'1/t'2-t'1. So, for C' we have the numerator that is smaller than in S and the denominator that is greater than that in S. This again leads to C' < C !!!!. So, how do we resolve this? thanks.
Either you've misunderstood the article or it's wrong.

A beam of light traveling along the y-axis in one frame is not travelling along the y-axis in the other frame - it has an x-axis component as well, and therefore travels a longer distance. The time dilation and length contraction is exactly enough to balance that effect, keep the speed of light the same.

Although a beam of light traveling along the x-axis in one frame is also traveling along the x-axis in the other frame, the distance covered from the origin of one frame as viewed from the other is different in the two frames. Again, length contraction and time dilation together exactly balance that effect.

You can see this for yourself if you choose either reference frame, imagine two pulses of light leaving the origin of that frame at time zero. After one second, the (x,y,t) coordinates of the pulse directed along the y-axis will be (0,1,1) and the coordinates of the pulse directed along the x-axis will be (1,0,1). Use the Lorentz transforms (not the contraction and dilation formulas!) to convert these into coordinates in the other frame, then calculate the distance covered by the light in one second in that frame. It'll come out to be c.

I think you are incorrectly assuming that the light will be travelling along the y axis in both both S and S'. If it is travelling along the y axis in S, then it will be travelling along a diagonal path in S'. You need use the resultant path length in S' taking the both the horizontal and vertical motion. Do a search for light clock to see what I mean.
The light clock example is just to show the phenomenon of time dilation in S' compared to S. The person in S' who is moving with speed 'v' will not see the light beam traveling diagonally as the horizontal component equal to 'vt' will be cancelled out by the actual movement of S' along the x'-axis. Hence, the S' person will also see the light beam traveling along y'. But the person in S will see the light beam travel diagonally in S'. In this way the light travels along y in S and y' in S'. This is the relativistic principle i.e that any phenomenon in S has to be exactly the same in S'. thanks.

in the above post i should have said:...the horizontal component equal to '-vt' will be cancelled.....movement of S' along the x'-axis equal to '+vt'.

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Either you've misunderstood the article or it's wrong.

A beam of light traveling along the y-axis in one frame is not travelling along the y-axis in the other frame - it has an x-axis component as well, and therefore travels a longer distance. The time dilation and length contraction is exactly enough to balance that effect, keep the speed of light the same.

Although a beam of light traveling along the x-axis in one frame is also traveling along the x-axis in the other frame, the distance covered from the origin of one frame as viewed from the other is different in the two frames. Again, length contraction and time dilation together exactly balance that effect.

You can see this for yourself if you choose either reference frame, imagine two pulses of light leaving the origin of that frame at time zero. After one second, the (x,y,t) coordinates of the pulse directed along the y-axis will be (0,1,1) and the coordinates of the pulse directed along the x-axis will be (1,0,1). Use the Lorentz transforms (not the contraction and dilation formulas!) to convert these into coordinates in the other frame, then calculate the distance covered by the light in one second in that frame. It'll come out to be c.
The x which is equal to +vt, where t is the time in S, will be cancelled out by the movement of S' with respect to S by -vt where again t is the time in S, thereby resulting in the light traveling straight along y' with respect to the person in S', even though it is diagonal for a person in S looking at the beam in S'. This has to be the case according to the principle of relativity.

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Referring to post#51. The article is not published in a peer-reviewed scientific journal. A quick reading leads me to conclude that it would not be accepted.

The author has some misunderstanding of the difference between coordinate effects and invariants.
I like you to read the article by Weinberg regarding the article by de Broglie. He says that, and i am paraphrasing, given the political environment today regarding publishing it would not have been possible that de Broglie's article would have been accepted for publication by any journal. thanks.

Ibix
The x which is equal to +vt, where t is the time in S, will be cancelled out by the movement of S' with respect to S by -vt where again t is the time in S, thereby resulting in the light traveling straight along y' with respect to the person in S', even though it is diagonal for a person in S looking at the beam in S'. This has to be the case according to the principle of relativity.
I think you are confusing your frames. If the light ray is parallel to the y axis it cannot be parallel to the y' axis because the two axes are moving apart. If the ray remains on one axis it must be getting further away from the other.

An analogy might help. A train crosses a bridge over a road just as a car goes under. In the frame of a person standing by the roadside, the train is moving in the +x direction and the car in the +y direction. In the frame of an observer on the train, however, the car is moving diagonally: it is moving along the road in the +y' direction, and also both the car and the road are moving in the -x' direction.

The extra effects in relativity conspire so that if the car is replaced by a photon, its speed is equal in any frame. The direction is not.

Dale
Mentor
I have found an article where the author says the following about einstein's SR: If we have two reference frames, S and S', with S stationary and the S' moving along the 'x' axis with speed 'v' of S, then as per SR there is length contraction in the 'x' direction but no change in the 'y' or the 'z' directions of objects in S'. That is y=y' and z=z'. If we take just the y=y', then the length y2-y1 = y'2-y'1. However, SR says that the time is slower in S' than in S. This means t2-t1 is < t'2-t'1. So, if we have a beam of light traveling along the y axis in S then the speed of the light is C= y2-y1/t2-t1. For the observer in S' who is also looking at the same beam of light it will be C'= y'2-y'1/t'2-t'1. Now, the numerators are equal but the denominator in S' is larger than in S due to time dilation. This means C'<C, which contradicts the constancy of the speed of light postulate of SR!!!!
The article referenced is not from an acceptable source, which is not surprising given how eggregiously flawed the reasoning is. You cannot use the time dilation formula for light since there is no frame in which it is at rest, nor can you use the length contraction formula for the same reason. You must use the full Lorentz transform which includes all effects including length contraction, time dilation, relativity of simultaneity, abberation, Doppler, etc.

The equation:
##c^2 t^2 = x^2 + y^2 + z^2##
is the equation for a sphere of radius ct about the origin. I.e. it is a sphere expanding at c in all directions. If you use the Lorentz transform on that it takes only a couple of minutes worth of algebra to show that in any other frame you get
##c^2 t'^2 = x'^2 + y'^2 + z'^2##
which is also a sphere expanding at c in all directions.

Any insinuation that SR preserves c in one direction but not in another direction is nonsense.

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