Proving that interval preserving transformations are linear

In almost all proofs I've seen of the Lorentz transformations one starts on the assumption that the required transformations are linear. I'm wondering if there is a way to prove the linearity:

Prove that any spacetime transformation $\left(y^0,y^1,y^2,y^3\right)\leftrightarrow \left(x^0,x^1,x^2,x^3\right)$ that preserves intervals, that is, such that

$$\left(dy^0\right)^2-\left(dy^1\right)^2-\left(dy^2\right)^2-\left(dy^3\right)^2=\left(dx^0\right)^2-\left(dx^1\right)^2-\left(dx^2\right)^2-\left(dx^3\right)^2$$

is linear (assuming that the origins of both coordinates coincide). That is, show that $\frac{\partial y^i}{\partial x^j}=L_j^i$ is constant throughout spacetime (that is, show that $\frac{\partial L_j^i}{\partial x^k}=0$).

Thus far all I've been able to prove is that $g_{ij}L_p^iL_q^j=g_{pq}$ (where $g_{ij}$ is the metric tensor of special relativity) and that $\frac{\partial L_j^i}{\partial x^k}=\frac{\partial L_k^i}{\partial x^j}$. Any further ideas?

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For the group of length-preserving transformations of 3D space, the usual argument for linearity is that they preserve triangles. Since the vector sum (difference) is given by triangles, the uniqueness theorems for triangles prove linearity. A similar argument might work for the Poincaré group. –  Greg Graviton Jul 23 '11 at 20:41
I don't think becko's conjecture is rigorously stated, and if it were, I think it would be false. We're omitting the assumption of homogeneity and isotropy, but it's not clear what other assumptions we are making. If the claim is going to hold in any meaningful sense, then it has to hold for spacetimes that are inhomoheneous and/or anisotropic. Then what theory are we operating in? Is it a manifold with a certain kind of connection? Do we allow torsion? In that kind of generalized context, we don't even expect to be able to cover the whole manifold with a single coordinate chart. –  Ben Crowell Jul 24 '11 at 16:04
Becko, it sounds like you're asking for a proof that the Lorentz transformation is linear in curved spacetime, where homogeneity and isotropy fail. But in a curved spacetime, there is not even a notion of a global Lorentz transformation; it only exists within an infinitesimal neighborhood in spacetime. If you restrict yourself to an infinitesimal neighborhood, then you essentially are assuming homogeneity and isotropy, since curvature becomes undetectable on small scales. This is essentially what the equivalence principle says. –  Ben Crowell Jul 24 '11 at 16:14
@Ben Crowell: As I read becko's question, becko is asking for a proof that a local coordinate transformation between two local coordinate systems (on a $3+1$ dimensional Lorentzian manifold) must be affine, if the metric $g_{\mu\nu}$ in both local coordinate systems happen to be on Minkowski form $\eta_{\mu\nu}={\rm diag}(-1,+1,+1,+1)$. –  Qmechanic Jul 24 '11 at 16:48
@Qmechanic: OK, maybe I'm just missing something, but this still doesn't make much sense to me. If the spacetime is assumed to be flat, then isn't that a stronger assumption than an assumption of homogeneity and isotropy? –  Ben Crowell Jul 24 '11 at 18:02

In hindsight, here is a short proof.

The metric $g_{\mu\nu}$ is the flat constant metric $\eta_{\mu\nu}$ in both coordinate systems. Therefore, the corresponding (uniquely defined) Levi-Civita Christoffel symbols

$$\Gamma^{\lambda}_{\mu\nu}~=~0$$

are zero in both coordinate systems. It is well-known that the Christoffel symbol does not transform as a tensor under a local coordinate transformation $x^{\mu} \to y^{\rho}=y^{\rho}(x)$, but rather with an inhomogeneous term, which is built from the second derivative of the coordinate transformation,

$$\frac{\partial y^{\tau}}{\partial x^{\lambda}} \Gamma^{(x)\lambda}_{\mu\nu} ~=~\frac{\partial y^{\rho}}{\partial x^{\mu}}\, \frac{\partial y^{\sigma}}{\partial x^{\nu}}\, \Gamma^{(y)\tau}_{\rho\sigma}+ \frac{\partial^2 y^{\tau}}{\partial x^{\mu} \partial x^{\nu}}.$$

Hence all the second derivatives are zero,

$$\frac{\partial^2 y^{\tau}}{\partial x^{\mu} \partial x^{\nu}}~=~0,$$

i.e. the transformation $x^{\mu} \to y^{\rho}=y^{\rho}(x)$ is affine.

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+1 Nice proof.. –  becko Jul 28 '11 at 1:04

Here I just want to mention that there exists a direct proof in $1+1$ dimensions using elementary arguments. Let the two coordinate patches $U_x$ and $U_y$ (which are, say, both convex sets in $\mathbb{R}^2$, containing the origin) have light-cone coordinates $x^{\pm}$ and $y^{\pm}$, respectively. The metric reads

$$dy^{+}dy^{-} ~=~ dx^{+}dx^{-}.$$

$$\frac{\partial y^{+}}{\partial x^{+}} \frac{\partial y^{-}}{\partial x^{+}} ~=~0 \qquad \qquad\Leftrightarrow\qquad \qquad\frac{\partial y^{+}}{\partial x^{+}}~=~0 \qquad\mathrm{or}\qquad \frac{\partial y^{-}}{\partial x^{+}} ~=~0 ,$$ $$\frac{\partial y^{+}}{\partial x^{-}} \frac{\partial y^{-}}{\partial x^{-}} ~=~0\qquad \qquad\Leftrightarrow \qquad\qquad\frac{\partial y^{+}}{\partial x^{-}}~=~0 \qquad\mathrm{or}\qquad \frac{\partial y^{-}}{\partial x^{-}} ~=~0,$$ $$\frac{\partial y^{+}}{\partial x^{+}} \frac{\partial y^{-}}{\partial x^{-}} +\frac{\partial y^{+}}{\partial x^{-}} \frac{\partial y^{-}}{\partial x^{+}} ~=~1.$$

Since $\det \frac{\partial y}{\partial x}\neq 0$, there are really only two possibilities. Either

$$\frac{\partial y^{-}}{\partial x^{+}}~=~0 ~=~\frac{\partial y^{+}}{\partial x^{-}},$$

or

$$\frac{\partial y^{+}}{\partial x^{+}}~=~0 ~=~\frac{\partial y^{-}}{\partial x^{-}}.$$

By possibly relabeling $x^{+} \leftrightarrow x^{-}$, we may assume the former. So

$$y^{+}~=~f^{+}(x^{+})\qquad \mathrm{and} \qquad y^{-}~=~f^{-}(x^{-}).$$

From the third PDE, we conclude that

$$\frac{\partial f^{+}}{\partial x^{+}}\frac{\partial f^{-}}{\partial x^{-}} ~=~1.$$

By separation of variables, this is only possible if $\frac{\partial f^{\pm}}{\partial x^{\pm}}$ is independent of $x^{\pm}$. It follows that $y^{\pm}~=~f^{\pm}(x^{\pm})$ are affine functions. Q.E.D.

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+1 for a general approach that can be e.g. used to derive the class of (infinitesimal) conformal transformations. –  Marek Jul 24 '11 at 5:59

Let's first assume that the scalar product that is preserved has positive signature to show the main idea. Also, you say you don't want to assume homogeneity but this is already implicit in your equation since to form intervals differences of space-time points are used so we might as well take one of those points to be zero of a vector space (equivalently, you might be talking about the preservation of a scalar product on a tangent space to a point but this is also linear, not affine).

Let $$f : \mathbb R^2 \to \mathbb R^2, \quad (x,y) \mapsto (A(x,y), B(x,y))$$ be length-preserving and suppose $f$ is analytic with $$A(x,y) = \sum_{n,m=0}^{\infty} {a_{n,m} \over n! m!} x^n y^m, \quad B(x,y) = \sum_{n,m=0}^{\infty} {b_{n,m} \over n! m!} x^n y^m.$$

Then we have $x^2 + y^2 = A(x,y)^2 + B(x,y)^2$ for all $x,y \in \mathbb R^2$ or explicitly first $$x^2 + y^2 = \left(\sum_{n,m=0}^{\infty} {a_{n,m} \over n! m!} x^n y^m \right)^2 + \left( \sum_{n,m=0}^{\infty} {b_{n,m} \over n! m!} x^n y^m \right)^2.$$ This immediately shows that the only non-vanishing coefficients occur when $n+m \leq 1$. We just need to investigate $n=m=0$ case but this is trivial since $f(0,0) = (a_{0,0}, b_{0,0})$.

For the $n$D case the discussion is completely analogous. For arbitrary signature some care needs to be taken since we can't use $x^2 + y^2 = 0 \rightarrow x = y =0$ anymore (perhaps one can work in $\mathbb C$ instead of $\mathbb R$ and use analytic continuation).

The last remaining ingredient of this argument is the analyticity of $f$. But this is trivial since $||f(x,y)||^2 = x^2 + y^2$ and $||\cdot||^2$ are analytic around any $(x,y) \in \mathbb R^2$.

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The OP says s/he doesn't want to invoke the homogeneity or isotropy of space. By assuming that the space is Euclidean, you're assuming homogeneity and isotropy. –  Ben Crowell Jul 24 '11 at 15:55
@Ben: I should clarify that I wasn't assuming that the space is Euclidean. All I assumed was that the scalar product with positive signature is preserved, or more generally that any scalar product is preserved (which is all OP asked for). But I'll edit the answer to make this explicit. –  Marek Jul 24 '11 at 16:01
And why a downvote? This answer is certainly correct perhaps except for some technical details but if their are some minor bugs please point them out. –  Marek Jul 24 '11 at 16:14
OK, I've undone the downvote. As discussed in the comments on the question, it seems to me that the OP was asking an ill-defined or self-contradictory question, but I could be wrong. –  Ben Crowell Jul 24 '11 at 18:00
@Ben: after rereading the question, I agree with you. I just proved that invariance of scalar product implies linearity but I wonder whether OP wanted this or something little different. –  Marek Jul 24 '11 at 18:13

The first condition implies that the Jacobian matrix $L^i_j=\frac{\partial y^i}{\partial x^j}$ is a Lorentz transformation. By substitution of the definition of the Jacobian in this condition, we obtain:

$g^{ij}\frac{\partial y^k}{\partial x^i}\frac{\partial y^l}{\partial x^j} = g^{kl}$

In particular, taking the diagonal equations equating $l=k$, we have

$g^{ij}\frac{\partial y^k}{\partial x^i}\frac{\partial y^k}{\partial x^j} = g^{kk}= \pm 1$

(The plus sign for the time coordinate and the minus sign for the space coordinates).

But this is just the Hamilton-Jacobi equation for a free relativistic particle, whose unique solution can be obtained by separation of variables:

$y^k = \sum_i f^{(k)}_i(x^i)$

By Substitution, we obtain:

$\frac {df^{(k)}_i(x^i)}{dx^i} = const$

Thus, the new coordinates are linear functions of the old coordinates. The constant coefficients are not independent, since the Jacobian matrix must be a Lorentz transformation.

Update:

Upon lurscher's suggestion, here are two references containing the Hamilton-jacobi equation of a relativistic particle. (Both references refer to a particle in an external electromagnetic field. In order to obtain the Hamilton-Jacobi equation for the free particle one needs the particualr case with a vanishing vector potential): reference-1 (by A. granik), reference-2

(The needed version appears in equation (33) of the first reference, the second reference contains the (proper) time dependent version).

In addition, I'll give here an other derivation based on the WKB approximation of the Klein- Gordon equation:

$\frac {1}{c^2}\frac {\partial^2\psi}{\partial t^2}-\nabla^2 \psi + \frac{m^2 C^2}{\hbar^2}\psi = 0$

The plane wave solutions are given by:

$\psi = C \exp(i\frac{\mathbf{p}.\mathbf{x}-\sqrt{m^2c^4+p^2c^2}t}{\hbar})$

To perform a WKB approximation, we seek a solution of the form:

$\psi = A(x,t)\exp(\frac{iS(x,t)}{\hbar})$

and take the leading terms in the limit $\hbar \rightarrow 0$. ($S$ is sometimes called the Hamilton-Jacobi phase function)

By subtitution, we obtain:

$((\frac {1}{c^2}\frac {\partial^2A}{\partial t^2}-\nabla^2A)+\frac{2i}{\hbar}(\frac {1}{c^2}\frac {\partial A}{\partial t} \frac {\partial S}{\partial t} -\mathbf{\nabla}A.\mathbf{\nabla}S) -\frac{A}{\hbar^2}(\frac {1}{c^2}\frac {\partial^2S}{\partial t^2}-\nabla^2S - m^2 c^2 )) = 0$

The leading term is the hamilton-Jacobi equation:

$\frac {1}{c^2}\frac {\partial^2S}{\partial t^2}-\nabla^2S - m^2 c^2 = 0$

Which can be seen to be equivalent to each equation on the main diagonal of the matrix equation written in the original answer.

Now, it is also easy to see the uniqueness of the solution. For the free particle, one can see that the non-leading terms actually vanish. i.e., the WKB approximation is exact.

The Hamilton-Jacobi phase function $S$ is just the phase of the plane wave solutions of the Klein-Gordon equation:

$S = \mathbf{p}.\mathbf{x}-\sqrt{m^2c^4+p^2c^2}t$

On $\mathbb{R}^4$, all solutions of the free Klein-Gordon equation in Cartesian coordinates are of the form of the plane waves, which implies that the Hamilton-Jacobi phase function is linear in the Cartesian coordinates.

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can you elaborate on how you go from the equation you obtain after doing a trace, to saying that it is equivalent to a Hamilton-Jacobi equation? –  lurscher Jul 24 '11 at 15:51
I have added an update containing the required elaboration and references. I also corrected an error: Each diagonal term of the matrix equation is equivalent to a Hamilton-Jacobi equation (no need to take the trace). –  David Bar Moshe Jul 25 '11 at 9:22
Dear David Bar Moshe, as I'm sure you are aware, the solution isn't unique, as you seem to write (v2). We are trying to prove that it must be affine. You also seem to claim that solution necessarily must obey (additive) separation of variables. Could you elaborate why, preferably backed by a reference? –  Qmechanic Jul 25 '11 at 20:01
@Qmechanic Of course the solution is not unique because you need boundary conditions to make it unique, but the solution form (Linear in the Cartesian coordinates) is unique. The solution of the four Hamilton-Jacobi equations on the diagonal will be with different constant coefficients which have to further satisfy the of-diagonal relations to be matrix coefficients of a Lorentz transformation. I gave a reasoning that the Hamilton-Jacobi phase function is the phase of a plane wave solution of the Klein-Gordon equation. –  David Bar Moshe Jul 26 '11 at 3:35
The form uniqueness of the solution is dictated by the linearity of the solutions of the equation of motion of a free particle. Nevertheless, I'll try to find a more rigorous proof of the uniqueness of the separation of variables solution. –  David Bar Moshe Jul 26 '11 at 3:37

Let us reformulate OP's question as follows:

Give a proof that a local coordinate transformation $x^{\mu} \to y^{\rho}=y^{\rho}(x)$ between two local coordinate systems (on a 3+1 dimensional Lorentzian manifold) must be affine if the metric $g_{\mu\nu}$ in both coordinate systems happen to be on constant flat Minkowski form $\eta_{\mu\nu}$.

Here we will present a proof that works both with Minkowski and Euclidean signature; in fact for any signature and for any finite non-zero number of dimensions, as long as the metric $g_{\mu\nu}$ is invertible.

1) Let us first recall the transformation property of the inverse metric $g^{\mu\nu}$, which is a contravariant $(2,0)$ symmetric tensor,

$$\frac{\partial y^{\rho}}{\partial x^{\mu}} g^{\mu\nu}_{(x)}\frac{\partial y^{\sigma}}{\partial x^{\nu}}~=~g^{\rho\sigma}_{(y)},$$

where $x^{\mu} \to y^{\rho}=y^{\rho}(x)$ is a local coordinate transformation. Recall that the metric $g_{\mu\nu}=\eta_{\mu\nu}$ is the flat constant metric in both coordinate systems. So we can write

$$\frac{\partial y^{\rho}}{\partial x^{\mu}} \eta^{\mu\nu}\frac{\partial y^{\sigma}}{\partial x^{\nu}}~=~\eta^{\rho\sigma}. \qquad (1)$$

2) Let us assume that the local coordinate transformation is real analytic

$$y^{\rho} ~=~ a^{(0)\rho} + a^{(1)\rho}_{\mu} x^{\mu} + \frac{1}{2} a^{(2)\rho}_{\mu\nu}x^{\mu}x^{\nu} + \frac{1}{3!} a^{(3)\rho}_{\mu\nu\lambda}x^{\mu} x^{\nu} x^{\lambda} + \ldots.$$

By possibly performing an appropriate translation we will from now on assume without loss of generality that the constant shift $a^{(0)\rho} =0$ is zero.

3) To the zeroth order in $x$, the equation $(1)$ reads

$$a^{(1)\rho}_{\mu} \eta^{\mu\nu}a^{(1)\sigma}_{\nu}~=~\eta^{\rho\sigma},$$

which not surprisingly says that the matrix $a^{(1)\rho}_{\mu}$ is a Lorentz (or an orthogonal) matrix, respectively. By possibly performing an appropriate "rotation", we will from now on assume without loss of generality that the constant matrix

$$a^{(1)\rho}_{\mu}~=~\delta^{\rho}_{\mu}$$

is the unit matrix.

4) In the following, it will be convenient to lower the index of the $y^{\sigma}$ coordinate as

$$y_{\rho}~:=~\eta_{\rho\sigma}y^{\sigma}.$$

Then the local coordinate transformation becomes

$$y_{\rho} ~=~ \eta_{\rho\mu} x^{\mu} + \frac{1}{2} a^{(2)}_{\rho,\mu\nu}x^{\mu}x^{\nu} + \frac{1}{3!} a^{(3)}_{\rho,\mu\nu\lambda}x^{\mu} x^{\nu} x^{\lambda}+ \ldots$$ $$+\frac{1}{n!} a^{(n)}_{\rho,\mu_1\ldots\mu_n}x^{\mu_1} \cdots x^{\mu_n}+ \ldots.$$

5) To the first order in $x$, the equation $(1)$ reads

$$a^{(2)}_{\rho,\sigma\mu}+a^{(2)}_{\sigma,\rho\mu}~=~0.$$

That is, $a^{(2)}_{\rho,\mu\nu}$ is symmetric in $\mu\leftrightarrow \nu$, but antisymmetric in $\rho\leftrightarrow \mu$. It is not hard to see (by applying the symmetry and the antisymmetry property in alternating order three times each), that the second order coefficients $a^{(2)}_{\rho,\mu\nu}=0$ must vanish.

6) To the second order in $x$, the equation $(1)$ reads

$$a^{(3)}_{\rho,\sigma\mu\nu}+a^{(3)}_{\sigma,\rho\mu\nu}~=~0.$$

That is, $a^{(3)}_{\rho,\mu\nu\lambda}$ is symmetric in $\mu\leftrightarrow \nu\leftrightarrow \lambda$, but antisymmetric in $\rho\leftrightarrow \mu$. For fixed $\lambda$, we can again reach the conclusion $a^{(3)}_{\rho,\mu\nu\lambda}=0$.

7) Similarly, we conclude inductively that the higher order coefficients $a^{(n)}_{\rho,\mu_1\ldots\mu_n}=0$ must vanish as well. So $y^{\mu}= x^{\mu}$. Q.E.D.

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Very nice. I also concluded that my approach needs induction on Taylor order (I was trying to figure out a direct way but wasn't able to find it). This makes we wonder whether the direct attack is at all possible. After all, perhaps we need to exploit the antisymmetry and consequently the transformation being polynomial depends crucially on the fact that we preserve a quadratic form and not a cubic (say), or even something nonhomogenous. Indeed, it's probably easy to construct polynomial invariants preserved by non-polynomial transformations. –  Marek Jul 26 '11 at 17:32
@Qmechanic: I'm not clear on the 3rd step. Specifically, the part that says "By possibly performing an appropriate "rotation", we will from now on assume without loss of generality that the constant matrix ..." Can you elaborate? Thanks. –  becko Jul 27 '11 at 17:13
@becko: Right, I'm being a bit brief. I'm trying to say that by introducing a third coordinate system $y'$, which is related by a standard linear Lorentz transformation (=a sort of "rotation") to the $y$ coordinate system, we may assume that the combined coordinate transformation $x\to y'$ is of the form claimed in the third step. The idea is that we can prove that the coordinate transformation $x\to y$ is linear, if we somehow can prove that that $x\to y'$ is linear. In the rest of the proof we remove the prime from the notation for convenience. Does that make sense? –  Qmechanic Jul 27 '11 at 17:40
@Qmechanic: How does one prove that a tensor $a_{\mu }^{\rho }$ that satisfies $\eta ^{\mu \nu }a_{\mu }^{\rho }a_{\nu }^{\sigma }=\eta ^{\rho \sigma }$ can be put in the form $\delta _{\mu }^{\rho }$ through Lorentz transformations? If this is supposed to be elementary, at least point me to some website or something that explains it, since I'm self-studying this stuff. Thanks. –  becko Jul 27 '11 at 18:16
@becko: The eq. you just wrote is the abstract/descriptive definition of a Lorentz matrix $a^{\rho}_{\mu}$. Does that help? –  Qmechanic Jul 27 '11 at 19:23

I had the feeling that a direct proof would be possible using only the relation $\eta _{ij}\frac{\partial y^i}{\partial x^p}\frac{\partial y^j}{\partial x^q}=\eta _{pq}$, assuming simple smoothness properties of the transformation and then using some algebra maneuvers. I found the following lovely argument on the book Gravitation and Cosmology by Steven Weinberg.

We start from the relation

$$\eta _{ij}\frac{\partial y^i}{\partial x^p}\frac{\partial y^j}{\partial x^q}=\eta _{pq}$$

Differentiating with respect to $x^k$ we obtain

$$\eta _{ij}\frac{\partial ^2y^i}{\partial x^p\partial x^k}\frac{\partial y^j}{\partial x^q}+\eta _{ij}\frac{\partial y^i}{\partial x^p}\frac{\partial ^2y^j}{\partial x^q\partial x^k}=0$$

We add to this the same equation with $p$ and $k$ interchanged, and subtract the same with $q$ and $k$ interchanged; that is,

$$\eta _{ij}\left(\frac{\partial ^2y^i}{\partial x^p\partial x^k}\frac{\partial y^j}{\partial x^q}+\frac{\partial y^i}{\partial x^p}\frac{\partial ^2y^j}{\partial x^q\partial x^k}+\frac{\partial ^2y^i}{\partial x^k\partial x^p}\frac{\partial y^j}{\partial x^q}+\frac{\partial y^i}{\partial x^k}\frac{\partial ^2y^j}{\partial x^q\partial x^p}-\frac{\partial ^2y^i}{\partial x^p\partial x^q}\frac{\partial y^j}{\partial x^k}-\frac{\partial y^i}{\partial x^p}\frac{\partial ^2y^j}{\partial x^k\partial x^q}\right)=0$$

This simplifies to

$$2\eta _{ij}\frac{\partial ^2y^i}{\partial x^p\partial x^k}\frac{\partial y^j}{\partial x^q}=0$$

Since the tensors $\frac{\partial y^i}{\partial x^j}$ and $\eta _{ij}$ are invertible, this implies that

$$\frac{\partial ^2y^i}{\partial x^p\partial x^k}=0$$

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