Why can coefficient "a" between spacetime intervals depend on velocity between systems?

[Mike_bb]

Hello!

I read Landau & Lifshitz' Classical Theory of Fields [Link to copyrighted textbook redacted by the Mentors] (see pic below) and I was confused when I saw in proof that coefficient "a" between spacetime interval (ds)² and (ds')² can only depend on the absolute relative velocity between the systems. I.e. (ds)²=a(ds')²

Why can coefficient "a" depend on absolute relative velocity between the systems? i.e. a=a(V)?

I have the similar problem as in this topic.

Thanks!

 

[Ibix]

The link doesn't work. I assume these are inertial coordinate systems? The point about inertial coordinates $(x,t)$ is that any inertially moving object has equation of motion $x=x_0+vt$. The same is true in a primed coordinate system.

What things can possibly be in the transformation? The only quantities available are the frame velocity and the coordinates themselves. What would that linear equation of motion look like if transformed with a transform depending on the coordinates?

 

[Mike_bb]

I have the similar problem as in this topic.

 

[martinbn]

The point they make is that it can depend on the absolute value i.e. the magnitude of the velocity but it cannot depend on the direction of the velocity. Otherwise space wouldn't be isotropic.

 

[Mike_bb]

The point they make is that it can depend on the absolute value i.e. the magnitude of the velocity but it cannot depend on the direction of the velocity. Otherwise space wouldn't be isotropic.

I mean another. I can't understand why coefficient "a" must depend on V. Why is it so?

 

[martinbn]

I mean another. I can't understand why coefficient "a" must depend on V. Why is it so?

What else for example?

 

[Mike_bb]

What else for example?

I proved that coefficient "a" should be. But I don't understand what should it depend on and why?

 

[martinbn]

I proved that coefficient "a" should be. But I don't understand what should it depend on and why?

I don't understand your question. The relative velocity is the only thing given. What else could $a$ depend on? Can you give one example?

 

[Mike_bb]

What else could $a$ depend on? Can you give one example?

I only know that there is coefficient "a" but I don't know what it depends on. How do I know that coefficient "a" should depend on relative velocity?

 

[martinbn]

I only know that there is coefficient "a" but I don't know what it depends on. How do I know that coefficient "a" should depend on relative velocity?

There is a relative velocity between the two frames. So ti could depend on that. If it doesn't you need to explain why not.

 

[Mike_bb]

There is a relative velocity between the two frames.

I know it. But what is relation between relative velocity and spacetime interval?

 

[martinbn]

I know it. But what is relation between relative velocity and spacetime interval?

What?!

 

[berkeman]

Thread closed temporarily for Moderation.

 

[berkeman]

A link to a copyrighted textbook in the OP has been redacted, and the thread is back open. Thanks for your patience.

 

[PeterDonis]

Why can coefficient "a" depend on absolute relative velocity between the systems? i.e. a=a(V)?

If you read the rest of the proof, you will see that it actually doesn't.

Their reasoning is as follows: first narrow things down to what "a" could possibly depend on: the only such thing is the magnitude of the relative velocity, $V$. It can't depend on the direction of the relative velocity because, as @martinbn pointed out, that would mean space would not be isotropic. But a priori there is no reason why it can't depend on the magnitude of the relative velocity, so we have to start with the premise that it might depend on the magnitude $V$ and see where that leads. And where it leads is that, when you actually work through the details, it can't depend on the magnitude $V$ of the relative velocity either. That is what Landau and Lifshitz are doing.

 

[Sagittarius A-Star]

Why can coefficient "a" depend on absolute relative velocity between the systems? i.e. a=a(V)?

On page 5 according to your screenshot you can conclude from the first sentence, that $a$ must be constant. L&L doesn't explain this, but in the 1960 book "Special Relativity" of Wolfgang Rindler, he did it in §8, page 16:

Source:
https://www.amazon.com/-/de/dp/101342879X/?tag=pfamazon01-20

A more detailed explanation by Rinder is cited in:
https://www.physicsforums.com/threa...simal-spacetime-interval.1064642/post-7109840

The factor $a$ would be still constant if it were a function only of the magnitude of the relative velocity ^**, because the velocity between inertial frames itself is always constant.

^** what it isn't, as later shown

 

[Mike_bb]

On page 5 according to your screenshot you can conclude from the first sentence, that $a$ must be constant. L&L doesn't explain this, but in the 1960 book "Special Relativity" of Wolfgang Rindler, he did it in §8, page 16:

View attachment 353291​

Thanks! Could you explain how does theorem of algebra work in this case?

 

[Sagittarius A-Star]

Thanks! Could you explain how does theorem of algebra work in this case?

In his sold-out 1st edition (1982) of the book "Introduction to Special Relativity", Rindler described this. Unfortunately, this was removed from the 2nd edition of the book, in which the LT is derived differently.

https://www.amazon.com/Introduction-to-Special-Relativity/dp/0198531818?tag=pfamazon01-20

​

 

[Mike_bb]

In his sold-out 1st edition (1982) of the book "Introduction to Special Relativity", Rindler described this. Unfortunately, this was removed from the 2nd edition of the book, in which the LT is derived differently.

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Big thanks!!! Your answers are very interesting and very useful! I fully understand how it works!