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.

​

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

 

[Mike_bb]

Hello!

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.

Consider two pairs of frames and their velocities (absolute value of relative velocities are the same):

1) Frame 1: 1m/s Frame 2: 5m/s
2) Frame 3: 2m/s Frame 4: 6m/s
Absolute value of relative velocities is 4m/s. $K=4$.

Let $ds_2^2=4ds_1^2$ and $ds_4^2=4ds_3^2$. Now, I want to ask you.
Why isn't it possible that there are such intervals in frame 3 and frame 4 that $ds_4^2 = Kds_3^2$, $K \neq 4$. Why not?

Thanks.

 

[Ibix]

Hello!

Consider two pairs of frames and their velocities (absolute value of relative velocities are the same):

1) Frame 1: 1m/s Frame 2: 5m/s
2) Frame 3: 2m/s Frame 4: 6m/s
Absolute value of relative velocities is 4m/s. $K=4$.

Let $ds_2^2=4ds_1^2$ and $ds_4^2=4ds_3^2$. Now, I want to ask you.
Why isn't it possible that there are such intervals in frame 3 and frame 4 that $ds_4^2 = Kds_3^2$, $K \neq 4$. Why not?

Thanks.

You haven't defined what the $ds_i^2$ are. I presume you mean that there is some pair of events, and the spacetime interval between them is $ds^2_i$ according to frame $i$.

You can have any value of $K$ that you like; it's simply an overall scale factor. So with the factor of 4 in the $ds^2$ for frames 1 and 2 you are using (e.g.) meters and seconds in one frame and half meters and half seconds in the other. You can introduce any $K$ you like in the relationship between 3 and 4 - you simply use units that are $1/\sqrt{K}$ the size in 4 as the ones you use in 3.

$K\neq 1$ is usually taken to be precluded by the principle of relativity, so that the forward and reverse transforms are symmetric. And because if you use different units in different frames you deserve everything that happens to you. But it's perfectly allowable. The forward transforms gain a prefactor of $\sqrt{K}$ and the inverse ones a factor of $1/\sqrt K$.

 

[Mike_bb]

You haven't defined what the $ds_i^2$ are. I presume you mean that there is some pair of events, and the spacetime interval between them is $ds^2_i$ according to frame $i$.

$ds_1^2$ - interval in the frame 1
$ds_2^2$ - interval in the frame 2

$ds_3^2$ - interval in the frame 3
$ds_4^2$ - interval in the frame 4

 

[Ibix]

$ds_1^2$ - interval in the frame 1
$ds_2^2$ - interval in the frame 2

$ds_3^2$ - interval in the frame 3
$ds_4^2$ - interval in the frame 4

Yes, but interval between what? The same pair of events, I presume?

 

[Mike_bb]

Yes, but interval between what? The same pair of events, I presume?

No. Different pair of events.
For two events in the frame 1 and the frame 2:
$ds_1^2$ - interval in the frame 1
$ds_2^2$ - interval in the frame 2

For another two events in the frame 3 and the frame 4:
$ds_3^2$ - interval in the frame 3
$ds_4^2$ - interval in the frame 4

 

[Ibix]

No. Different pair of events.

As long as you mean a pair of events, the interval between which is $ds_1^2$ or equivalenly $ds_2^2$, and a different pair of events the interval between which is $ds_3^2$ or equivalently $ds_4^2$, fine, my reply stands.

 

[Mike_bb]

As long as you mean a pair of events, the interval between which is $ds_1^2$ or equivalenly $ds_2^2$, and a different pair of events the interval between which is $ds_3^2$ or equivalently $ds_4^2$, fine, my reply stands.

$ds_1^2$ - interval between two events in frame 1. $ds_2^2$ - interval between the same two events in frame 2.

$ds_3^2$ - interval between another two events in frame 3. $ds_4^2$ - interval between the same two events in frame 4.

 

[Ibix]

In that case, it's as I said. You're just picking different units in the different frames, which is a bad idea for practical reasons but does work.

 

[Mike_bb]

In that case, it's as I said. You're just picking different units in the different frames, which is a bad idea for practical reasons but does work.

martinbn wrote:

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 wrote in more details as martinbn said above:

Consider two pairs of frames and their velocities (absolute value of relative velocities are the same):

1) Frame 1: 1m/s Frame 2: 5m/s
2) Frame 3: 2m/s Frame 4: 6m/s
Absolute value of relative velocities is 4m/s. $K=4$.

Let $ds_2^2=4ds_1^2$ and $ds_4^2=4ds_3^2$. Now, I want to ask you.
Why isn't it possible that there are such intervals in frame 3 and frame 4 that $ds_4^2 = Kds_3^2$, $K \neq 4$. Why not?

 

[Ibix]

Normally, you pick $K=1$ for all pairs of frames so $ds^2$ is the same gor a pair of events described in any frame. You have chosen otherwise, and you are free to choose any value of $K$ that you like. There is no restriction except that $K>0$.

As I say, all picking $K\neq 1$ does is introduce a unit change at the same time as the frame change. That's perfectly fine mathematically. It's just a bad idea because of the extra book-keeping you need.

 

[Sagittarius A-Star]

martinbn wrote:

I wrote in more details as martinbn said above:

1. The absolute value of relative velocities is in both cases not $4m/s$. If the individual velocities of your 4 frames refer to a certain unnamed inertial reference frame, then you must use the relativistic velocity addition formula.
2. From equation $(2.6)$ in the scan in the OP (posting #1) follows $ds_2^2=ds_1^2$ and $ds_3^2=ds_4^2$.