- #1

Mentz114

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I hope to lay to rest two of the misconceptions about special relativity that are evident in the many questions asked here.

1) Why is the speed of light a constant ?

Everybody believes Pythagoras's theorem that the length of the hypotenuse of a right angle triangle is ##\sqrt{s_1^2+s_2^2}## where ##s_1,\ s_2## are the lengths of the other two sides. It a geometrical fact.

In SR the length of the hypotenuse of a right angled triangle in the ##t,x## plane is ##\sqrt{s_1^2-s_2^2}## where ##s_1## is the longer of the two sides ( unless they are equal in which case the order is irrelevant).

It is a postulate of SR that the Lorentzian length of the path of light is zero. So any two events that lie on a light path will have proper length of zero. Furthermore if these points are Lorentz transformed they will still lie on the light path.

The constancy of light speed thus follows from a postulate and a geometrical fact that is just as geometric as Pythagoras's theorem.

No physical reason for this is known. One might as well ask 'why is Pythagoras's theorem true'. Any reasons given will be purely geometric.

2) Where is 'length contraction' ?

A lot of effort has gone into 'proving the constancy of c' by using cross-frame calculations involving contracted length and dilated time. These are redundant as I hope the above demonstrates. But there is an opportunity to show what is happening in frame-independent or operational terms.

We have train and platform, with a light source in the middle of the train sending a beam to two receivers one at each end.

The diagram 'train-frame' shows this in coordinates in which the train is at rest (train coords). The blue worldline is someone on the platform receding at ##0.5c## (##\gamma=1.1547##). In the train frame the diagram is symmetric and the light beams hit the receivers at the same time on the three train clocks. The distance covered by the light is equal to the clock time that elapsed so ##c=1##.

The second diagram shows the scenario in the platform coordinates. It is no longer symmetric and the light beams do not hit the receivers at the same clock time. The distance traveled by the light has shrunk for the back receiver and grown for the front receiver. The corresponding clock times have shrunk/increased by the same factor so ##c=1## in these coordinates.

It is straightforward to show that the distances in the new coordinates ( L_1, L_2) are the Doppler shrunk/stretched lengths as measured in the train frame (these are acquired by radar distance measurement).

So - where is the contracted length or distance? All we need to balance the books is proper times and radar distances. The first is an invariant and the second is an operationally defined distance - not a fudged definition of distance.

'Contracted distance/length' exists only as a factor in a Lorentz transformation. It is not required in frame independent calculations. It is used an imaginary fudge-factor that helps get the right answer to a pointless calculation.

Persisting with this useless exercise suggests that one may be looking for a counter example which will *disprove* the postulates. There is no more hope of doing that than finding a counter example to Pythagoras. It is impossible without abandoning the geometry in which case we no longer talking about SR but some other theory.

1) Why is the speed of light a constant ?

Everybody believes Pythagoras's theorem that the length of the hypotenuse of a right angle triangle is ##\sqrt{s_1^2+s_2^2}## where ##s_1,\ s_2## are the lengths of the other two sides. It a geometrical fact.

In SR the length of the hypotenuse of a right angled triangle in the ##t,x## plane is ##\sqrt{s_1^2-s_2^2}## where ##s_1## is the longer of the two sides ( unless they are equal in which case the order is irrelevant).

It is a postulate of SR that the Lorentzian length of the path of light is zero. So any two events that lie on a light path will have proper length of zero. Furthermore if these points are Lorentz transformed they will still lie on the light path.

The constancy of light speed thus follows from a postulate and a geometrical fact that is just as geometric as Pythagoras's theorem.

No physical reason for this is known. One might as well ask 'why is Pythagoras's theorem true'. Any reasons given will be purely geometric.

2) Where is 'length contraction' ?

A lot of effort has gone into 'proving the constancy of c' by using cross-frame calculations involving contracted length and dilated time. These are redundant as I hope the above demonstrates. But there is an opportunity to show what is happening in frame-independent or operational terms.

We have train and platform, with a light source in the middle of the train sending a beam to two receivers one at each end.

The diagram 'train-frame' shows this in coordinates in which the train is at rest (train coords). The blue worldline is someone on the platform receding at ##0.5c## (##\gamma=1.1547##). In the train frame the diagram is symmetric and the light beams hit the receivers at the same time on the three train clocks. The distance covered by the light is equal to the clock time that elapsed so ##c=1##.

The second diagram shows the scenario in the platform coordinates. It is no longer symmetric and the light beams do not hit the receivers at the same clock time. The distance traveled by the light has shrunk for the back receiver and grown for the front receiver. The corresponding clock times have shrunk/increased by the same factor so ##c=1## in these coordinates.

It is straightforward to show that the distances in the new coordinates ( L_1, L_2) are the Doppler shrunk/stretched lengths as measured in the train frame (these are acquired by radar distance measurement).

So - where is the contracted length or distance? All we need to balance the books is proper times and radar distances. The first is an invariant and the second is an operationally defined distance - not a fudged definition of distance.

'Contracted distance/length' exists only as a factor in a Lorentz transformation. It is not required in frame independent calculations. It is used an imaginary fudge-factor that helps get the right answer to a pointless calculation.

Persisting with this useless exercise suggests that one may be looking for a counter example which will *disprove* the postulates. There is no more hope of doing that than finding a counter example to Pythagoras. It is impossible without abandoning the geometry in which case we no longer talking about SR but some other theory.