Lorentz Transformations and their Inverse

[jtbell]

Where did this 8.2 come from?

A typographical error. Thanks for catching it! I've just now corrected it to 2.4.

 

[NanakiXIII]

You're welcome, jtbell, I wouldn't have guessed I'd be right about it. Anyway, if I was right in my previous post, then isn't the following statement in your explanation also false:

Apparently your position has shifted by 5.8 light-years

 

[Hootenanny]

No, I do have another question about those generalized transformations. If we can't use the Lorentz transformations between co-ordinates of S and S'', then why can we use them between intervals on S and S''? What's the difference?

Because you are looking at changes in temporal and spatial coordinates, not absolute coordinates. In the general case, we are looking at the difference between an initial position and time and a final position and time. However, in the case of the other ('non general') LT's, it is a condition that the origins of the two frames coincide at t=t''=0. This does not have to be the case in the general LT's since we are only considering the difference between two defined sets of coordinates. For example, take jtbell's general LT;

$$x_1^{\prime \prime} - x_0^{\prime \prime} = \gamma [(x_1 - x_0) - v (t_1 - t_0)]$$

Now, if we assume that the origins of the two frames (S and S'') coincide at t=t''=0 then this implies that x₀''=0 ; x₀=0 and t₀ = 0 i.e. the origin of the S'' frame lies on the origin of the S frame, both frames start from the same position at zero displacement. Does that make sense?

 

[NanakiXIII]

Yes, but to transform between S and S'', you're still just using the Lorentz transformation. You're saying:

$$\Delta x^{\prime \prime} = \gamma [\Delta x - v \Delta t]$$

And thus:

$$x''=\gamma (x-vt)$$ and $$x''_0=\gamma (x_0-vt_0)$$

Where is my error in this?

 

[Hootenanny]

Yes, but to transform between S and S'', you're still just using the Lorentz transformation. You're saying:

$$\Delta x^{\prime \prime} = \gamma [\Delta x - v \Delta t]$$

And thus:

$$x''=\gamma (x-vt)$$ and $$x''_0=\gamma (x_0-vt_0)$$

Where is my error in this?

There is no error. I never said there was.

 

[NanakiXIII]

If there is no error in what I said, then you're using the non-generalized Lorentz transformation to transform between S and S''.

$$x^{\prime \prime} = \gamma (x - vt)$$

And if that is true, the generalized transformations are redundant. So there must be an error in my previous post, mustn't there?

 

[Hootenanny]

This has already been explained, please reread the following posts

https://www.physicsforums.com/showpost.php?p=1220390&postcount=33"
https://www.physicsforums.com/showpost.php?p=1220390&postcount=34"
https://www.physicsforums.com/showpost.php?p=1220390&postcount=38"

Note what happens to the following LT when the origins coincide, that is when x₀''=0 ; x₀=0 and t₀ = 0

$$x_1^{\prime \prime} - x_0^{\prime \prime} = \gamma [(x_1 - x_0) - v (t_1 - t_0)]$$

 

[bernhard.rothenstein]

inverse LT

So, what I gather is that one way of looking at inverse Lorentz transformation is that there's just a plus sign because the velocity is negative. Am I in any way correct here? (I'm aware that the inverse transformations can be derived from the normal transformations, I'm just looking to know whether the thought described above is correct.)



Is that a Socratic question? I haven't a clue.

Socratic or not, start with
x=g(x'+Vt')
t=g(t'+Vx'/cc)[/
and with some simple algebra solve them for x' and t' in order to obtain the inverse transformaions!B]

 

[NanakiXIII]

Those links don't seem to work for me, but I found the posts anyway. I think I understand if the following is true:

The Lorentz transformation apply per definition to intervals rather than to co-ordinates.

If that is the case, however, how does that show from the derivation of the Lorentz transformations? In this case, I'll have to refer to the only derivation I know, which is Einstein's derivation as described in Relativity, a book he wrote. The derivation can also be found at http://www.bartleby.com/173/a1.html.

Or, if easier, how else could this be proven or at least be made plausible?In reply to bernhard.rothenstein: I am aware of that, I just didn't know what you meant by asking what it was called. Thanks for pointing it out in the first place, though.

 

[MeJennifer]

The Lorentz transformation apply per definition to intervals rather than to co-ordinates.

Be aware that the x and y coordinates in the presented Euclidean-Cartesian frames are not distances and time in the way the world works.

In reality, the physical distance between two objects is the amount of proper travel time taken for an object to go from one to the other.

You cannot take time and space in isolation.

"Henceforth Space by itself, and Time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality."
- Minkowski

Henceforth? Well apparently not by everyone, after more than 100 years of SR many still consider, lengths contracting and time slowing down, in isolation rather than in relation.

 

[robphy]

In reality, the physical distance between two objects is the amount of proper travel time taken for an object to go from one to the other.

Can you clarify? That statement doesn't make much sense to me.

 

[NanakiXIII]

I see your point, MeJennifer, but I'm not quite sure how to relate it to my problem.

 

[JesseM]

Yes, but to transform between S and S'', you're still just using the Lorentz transformation. You're saying:

$$\Delta x^{\prime \prime} = \gamma [\Delta x - v \Delta t]$$

And thus:

$$x''=\gamma (x-vt)$$ and $$x''_0=\gamma (x_0-vt_0)$$

Where is my error in this?

The second doesn't follow from the first--if S and S'' do not share a common origin, the first is true while the second is false. As a simple analogy, consider two cartesian coordinate systems in 2D space, with the x' axis parallel to the x-axis and the y' axis parallel to the y axis, but with the origin of S' (x' = 0, y'=0) located at coordinates x=5, y=8 in the S system. In this case, the coordinate transform is just:

x' = x - 5
y' = x - 8

However, the displacement between a given pair of points is the same in both coordinate systems:

$$\Delta x' = \Delta x$$

The Lorentz transformation apply per definition to intervals rather than to co-ordinates.

No, quite the opposite in fact.