Linearity of Lorentz transformations

[ralqs]

I asked my prof why the Lorentz transformations had to be linear (which my textbook assumed when deriving them), and he mentioned some stuff about homogeneity and ended with "it's advanced, just believe". Can anyone offer a simple explanation?

 

[arkajad]

Well, Lorentz transformations are linear by definition. So, what is exactly your question? Probably you have something in mind that you did not tell us. Probably some special kind of a derivation of the transformation form from some particular assumptions. What are these assumptions?

 

[dextercioby]

The coordinate systems used by 2 inertial observers must be linearly related, if they are to measure the same spacetime distance between 2 points.

 

[Fredrik]

Probably some special kind of a derivation of the transformation form from some particular assumptions. What are these assumptions?

He's almost certainly talking about a "derivation" of the Lorentz transformation from Einstein's postulates.

I asked my prof why the Lorentz transformations had to be linear (which my textbook assumed when deriving them), and he mentioned some stuff about homogeneity and ended with "it's advanced, just believe". Can anyone offer a simple explanation?

Einstein's postulates are loosely stated ideas, not mathematical axioms, so they can't be used as the starting point of a proof. There are however mathematical statements that seem to capture Einstein's ideas (or aspects of them) quite well. One of them is the Lorentz transformation. If you want to, you can start with another collection of mathematical statements that can be thought of as expressing different aspects of Einstein's ideas, and derive the Lorentz transformation from them. My favorite derivation of that sort is the one I posted here. (Start reading at "The explicit..."). This is just for the 1+1 dimensional case. Note that I'm not using Einstein's postulates (which my posts refer to as "the numbered list in my previous post") as axioms, but as an inspiration for a number of mathematical assumptions that are needed along the way.

One of those assumptions is linearity. My post contains a partial motivation for it: If a function that represents a coordinate change from one inertial frame to another doesn't take straight lines to straight lines, an object that's moving with a constant velocity forever in one inertial frame would be accelerating at some point in the other. If we supplement this with the requirement that these functions are smooth (that their partial derivatives up to order n exists, for all integers n), and that they preserve the origin, we can show quite easily that the only possibility is a linear function. These assumptions are stronger than they need to be. (They're sufficient to prove linearity, but not necessary). I have never cared enough to find out what a set of necessary assumptions look like.

Another way to do these things (which I must admit I still haven't worked through myself yet, but I will, because this stuff is pretty cool) is to assume that the set of transition functions (functions that change coordinates from one inertial frame to another) form a group, which contains the rotation and translation groups as subgroups. I don't remember the exact details of the assumptions, but if someone is interested, they can be found in the paper "The rich structure of Minkowski space" by Domenico Giulini. These assumptions are supposed to make the first postulate precise. The inclusion of the translation and rotation groups as subgroups is the mathematically precise way of requiring that space is homogeneous and isotropic. These assumptions lead to the conclusion that the group of transition functions is either the Galilei group or the Poincaré group. The second postulate is then used only to select the Poincaré group out of these two options.

Edit: Here's the link to Giulini's article.

 

[bcrowell]

As the OP's prof stated, homogeneity of spacetime is one good justification. That was the justification that Einstein originally gave in his 1905 paper on SR: "In the first place it is clear that the equations must be linear on account of the properties of homogeneity which we attribute to space and time." For example, x'=(x-7)³ would give a special role to x=7.

I think Fredrik's justification also works.

IMO the right way to think about the logical basis of SR is that it's *all* about symmetry principles such as homogeneity. Here is a treatment in that style: Palash B. Pal, "Nothing but Relativity," http://arxiv.org/abs/physics/0302045v1

 

[MikeLizzi]

From:

Physics Forums > PF Library – Physics and Math Information Database > Physics > Relativity > Special Theory

Subject: Lorentz Transformation

“A combination of two Lorentz boosts in different directions is not a Lorentz boost, but is a combination of a Lorentz boost and a spatial rotation (a rotation known as "Thomas precession") in the plane of those directions.”

Question: Has anyone derived a transformation function that satisfies Einstein’s postulates but does not put in the spatial rotation?

 

[arkajad]

For example, x'=(x-7)³ would give a special role to x=7.

$x'=\gamma(x-vt)$ gives a "special role" to the line x=vt.

 

[arkajad]

Question: Has anyone derived a transformation function that satisfies Einstein’s postulates but does not put in the spatial rotation?

The "special rotation" comes from the following simple fact about the algebra of 2x2 complex matrices:

if A and B are positive definite (invertible) but do not commute, then AB is, in general, not positive definite.
It has the (unique) form CU, where C is positive definite and U is unitary.

 

[bcrowell]

$x'=\gamma(x-vt)$ gives a "special role" to the line x=vt.

No, it treats all lines with slope v the same. It also *should* give a special role to the slope v, since it is supposed to represent a boost v.

 

[arkajad]

Therefore, it is not strictly homogeneous. So, what is the exact mathematical definition of 'homogeneity" that does the job? Is there one?

 

[Fredrik]

Question: Has anyone derived a transformation function that satisfies Einstein’s postulates but does not put in the spatial rotation?

I don't think so, but I it's hard to just figure out what the above means. If you're given a group of (topological space) automorphisms of ℝ⁴, how to you determine if they "satisfy Einstein's postulates" or not? I would interpret your question as being about such a group, which has the pure boosts as a subgroups. But how do you even determine if a given transformation is a "pure boost"? I don't know how to define that term without knowing what group we're talking about. The pure boosts in the Galilei group aren't the same as the pure boosts in the Poincaré group.

 

[Fredrik]

Therefore, it is not strictly homogeneous. So, what is the exact mathematical definition of 'homogeneity" that does the job? Is there one?

As I said in my first post in this thread, I think the requirement that we're dealing with a group that has the rotation group and the translation group as subgroups does the job, and that the requirement that the translation group is a subgroup is the appropriate definition of homogeneity here. (I also mentioned that I haven't worked through the details, so you will have to check out Giulini's article if you want more information). It seems like it would be difficult to define homogeneity as a property of space in this context, since at the start of this "derivation", we haven't yet determined which slices of spacetime to call "space".

Hm, the translation group as a subgroup... Translations are precisely what differs between the Poincaré group and the Lorentz group, so if these requirements are the appropriate ones, homogeneity shouldn't play a part in the derivation that finds the Lorentz group, but isotropy should.

 

[bcrowell]

Therefore, it is not strictly homogeneous. So, what is the exact mathematical definition of 'homogeneity" that does the job? Is there one?

When you say "it is not strictly homogeneous," what does "it" refer to? If it refers to the transformation, then that's fine, but that's not the issue. If it refers to space, then your statement is incorrect.

If you want a strict mathematical definition of homogeneity of spacetime, then I would say that the correct definition is that the Lorentz transformations are linear. That is, the only mathematical structure that comes along with bare Minkowski space, and that we can inspect to see whether Minkowski space is homogeneous, is the Lorentz transformations (or actually the whole Poincare group). Physically, this is clearly the right definition to choose, because deviation from linearity would cause inertial motion in one frame to be noninertial in a second, Lorentz-boosted frame, and the discrepancy could be used to distinguish special characteristics of one portion of spacetime from another.

Do you have a different opinion on what Einstein meant when he gave homogeneity as the justification of linearity in the 1905 paper? Do you think Einstein's justification for this point was wrong or insuffucient or not rigorous?

 

[matheinste]

There is alengthy discussion on the subject of the linearity of Lorentz Transforms in Torretti, Relativity and Geometry, Section 3.6, page 71.

His general thrust of argument seems to be, although I have not plouged through it all as it it seems very detailed:--

The Lorentz Transformation is linear Mathematically by definition.

The mathematical and physical definitions are equivalent.

Therefore the physical nature nature of the Lorentz transformation is linear

I am not qualified to comment on what he has written except to say that he is usually very well informed on the subject of Relativity.

Matheinste.

 

[arkajad]

If you want a strict mathematical definition of homogeneity of spacetime, then I would say that the correct definition is that the Lorentz transformations are linear.

And that is exactly what I thought. The OP asked how homogeneity explains linearity. Now we have learned that it is just the converse: homogeneity is being explained by linearity.

 

[bcrowell]

And that is exactly what I thought. The OP asked how homogeneity explains linearity. Now we have learned that it is just the converse: homogeneity is being explained by linearity.

I don't think either one explains the other.

 

[arkajad]

I don't think either one explains the other.

At least "linearity" is mathematically precise, while I did not see a precise definition of 'homogeneity' in the framework of pure Lorentz transformations.

 

[Haelfix]

All you need is the statement that the proper time of a physical system must not change under some set of spacetime coordinate transformations (eg the speed of light is constant in all inertial reference frames).

There is only one nonsingular group of transformations that can account for this, and you can construct this group by direct computation and prove its uniqueness. That group of course is identified as the Poincare group and is linear by inspection.

Incidentally, if you partially relax the proper time transformation requirement and instead only require that it coincides for massless particles (eg those particles which move at the speed of light), you get a much larger group that is manifestly nonlinear (called the conformal group).

 

[arkajad]

Note: You can just add dilatations. They are linear and nonsingular.