Invariance of $ds^2$ from invariance of all null intervals

Is this linear algebra statement true?

Let $$\eta= \begin{pmatrix}1&0&0&0\\0&-1&0&0\\0&0&-1&0\\0&0&0&-1\end{pmatrix}$$.

If $$x^T (\Lambda^T\eta\Lambda) x$$=0 for all $$x$$ such that $$x^T \eta x=0$$, then $$\Lambda^T\eta\Lambda=a\eta$$ for some $$a \in \mathbb{R}$$.

If so how does one prove it? Is a stronger statement true (e.g. $$a>0$$)?

Motivation: I can't figure out why this statement from the Wikipedia article is true using the above mathematical language:

Since if $$ds^{2}=0$$, then the interval will be null in any other system (second postulate), and since $$ds^{2}$$ and $$ds'^{2}$$ are infinitesimals of the same order, they must be proportional to each other,

$$ds^{2}=ads'^{2}$$.

The translation of the above is that $$x^T (\Lambda^T\eta\Lambda) x$$ must be proportional to $$x^T \eta x$$ for all $$x$$. Why?

• For the $\Lambda$ you gave $\Lambda^T \eta \Lambda=\begin{pmatrix}0&-2\\-2&0\end{pmatrix}$ and $x^T\Lambda^T \eta \Lambda x$ doesn't equal zero for any null vector (i.e. an $x$ where $x^T \eta x=0$) so I don't see how this is a counterexample. Also, what is your definition of "a set of Minkowski coordinates"? The mathematical condition I gave is equivalent to saying that the new coordinates are a linear transformation of inertial coordinates that preserves the speed of light. Is that not a suitable definition of Minkowski coordinates? – Alex Mar 12 at 15:53
• I've converted my comment into an answer, and replied to your comment as part of the answer. I've also given a simpler counterexample than the one you were replying to above. – Ben Crowell Mar 12 at 19:08
• FWIW, the statement is taken pretty much verbatim from Landau & Lifshitz, who don't provide a proof either. – Javier Mar 12 at 20:15

Prelude: 1+2D

In 1+2D we have a matrix $$\Lambda = \begin{bmatrix}a&b&c\\ d&e&f\\ g&h&i \end{bmatrix}$$ which is being used to generate a symmetric matrix $$\eta' = \Lambda^T \eta \Lambda$$. We can rotate the two spatial dimensions into each other to find a family of null vectors $$v(\theta) = [1, \cos\theta,\sin\theta]^T$$ such that $$v^T \eta v = 0$$ and you want to consider only the $$\Lambda$$ such that $$v^T \eta' v =0$$ too, for all $$\theta$$. This would mean that $$(a + b\cos\theta + c\sin\theta)^2 = (d+e\cos\theta + f \sin\theta)^2 + (g + h\cos\theta + i\sin\theta)^2.$$ So we have 6 degrees of freedom (symmetric 3x3 matrix $$\eta'$$) but presumably we have 5 equations here: terms in $$\theta$$ proportional to $$1,$$ $$\cos\theta,$$ $$\sin\theta,$$ $$\cos(2\theta),$$ and $$\sin(2\theta):$$ \begin{align} 2a^2 + b^2 + c^2 &= 2d^2 + e^2 + f^2 + 2g^2 + h^2 + i^2\\ ab &= de + gh\\ ac &= df + gi\\ b^2 - c^2 &= e^2 - f^2 + h^2 - i^2\\ bc &= ef + hi \end{align}The three "small" equations above set all of the off-diagonal elements to be 0 in the resulting matrix. The first "big" equation can be reduced to $$a^2 + b^2 = d^2 + e^2 + g^2 + h^2$$ in light of the second, thus we would have \begin{align}a^2 - d^2 - g^2 &= - b^2 + e^2 + h^2 &:= K\\ b^2 - e^2 - h^2 &= c^2 - f^2 - i^2 &= -K \end{align} proving the form $$\begin{bmatrix}K&0&0\\0&-K&0\\0&0&-K\end{bmatrix}$$ for some $$K$$ which likewise does not have to be positive.

Extending to 1+N dimensions

Now let's just do the same thing as before, but probe $$\eta'$$ in 1+N dimensions with some basic null vectors that comprise the unit vector in the time direction $$\hat w$$ and the unit vector in some arbitrary space dimension $$\hat x$$, e.g. $$(\hat w \pm \hat x)^T\eta'(\hat w \pm \hat x) = 0.$$Since $$\eta'$$ is symmetric one gets results like $$\hat w^T \eta' \hat w ~\pm~ 2 \hat w^T\eta'\hat x ~+~ \hat x^T\eta'\hat x ~=~ 0$$ and this then argues that these off-diagonal elements $$\hat w^T\eta'\hat x = 0$$ directly.

The above rotational argument from 1+2D gives the same for the $$\hat x^T\eta'\hat y$$ terms if we just do a rotation from any spatial coordinate into any other, call them $$\hat x$$ and $$\hat y$$: we have even that $$(\hat w + \hat x \cos\theta + \hat y \sin\theta)^T \eta' (\hat w + \hat x \cos\theta + \hat y \sin\theta) = 0$$ and the $$\sin(2\theta)$$ component of that equation comes exclusively from $$2 \hat x^T \eta' \hat y \cos\theta \sin\theta$$ and this can only be zero if $$\hat x^T \eta' \hat y = 0.$$

So we've proven that all off-diagonal elements must be zero and then we can just probe with those first null vectors again, so if we use $$\hat w + \hat x$$ then we determine that the $$(w, w)$$ diagonal element must be the negative of the $$(x, x)$$ diagonal element, but since we chose $$\hat x$$ arbitrarily this must apply to all of the other diagonal elements: it must have the form $$\operatorname{diag}(K, -K, -K, \dots)$$.

Proposition. In a space $$V\cong \mathbb{R}^n$$ of dimension $$n=p+q$$ let there be given an indefinite metric tensor $$\eta~=~\begin{pmatrix} \mathbf{1}_{p\times p}& \mathbf{0}_{p\times q} \cr \mathbf{0}_{q\times p} & -\mathbf{1}_{q\times q} \end{pmatrix}_{n\times n}~=~{\rm diag}(\underbrace{+1,\ldots,+1}_{p\text{ times}},\underbrace{-1,\ldots,-1}_{q\text{ times}})\tag{1}$$ of signature $$(p,q)$$, and a (possibly degenerate & indefinite) metric tensor $$g$$. Assume that all null-vectors for $$\eta$$ are also null-vectors for $$g$$: $$\forall v\in V :~~v^t\eta v~=~0~~\Rightarrow~~v^tg v~=~0.\tag{2}$$ Then $$g$$ is proportional to $$\eta$$: $$\exists \lambda\in \mathbb{R}:~~g~=~\lambda \eta.\tag{3}$$

Sketched proof of proposition:

1. Let $$e_i=(0,\ldots, 0,1,0,\ldots, 0)^t$$ be the $$i$$th unit-vector (of $$\eta$$-length-square $$\pm 1$$). Write the metric tensor $$g~=~ \begin{pmatrix} a& b^t \cr b & c \end{pmatrix} \tag{4}$$ in terms of a symmetric $$p\times p$$ matrix $$a$$, a symmetric $$q\times q$$ matrix $$c$$, and a rectangular $$q\times p$$ matrix $$b$$.

2. Use the following "polarization trick" to show that the $$b$$-block vanishes: $$b~=~0.\tag{5}$$ If $$g_{ij}=e_i^tge_j$$ corresponds to a matrix element in the $$b$$-block, then $$v_{\pm}:=e_i\pm e_j$$ are null-vectors, so $$4g_{ij}~=~4e_i^tge_j~=~(v_++v_-)^tg(v_+-v_-)~=~v_+^tgv_+ -v_-^tgv_-~\stackrel{(2)}{=}~0+0~=~0. \tag{6}$$

3. We can diagonalize the symmetric $$a$$ and $$c$$ blocks by orthogonal matrices while keeping $$\eta$$ invariant. In other words, we may assume w.l.o.g. that $$g\text{ is diagonal}.\tag{7}$$

4. Finally, by considering null-vectors of the form $$v:=e_i+ e_j$$, it becomes clear that $$g_{ii}+g_{jj}~=~e_i^tge_i+e_j^tge_j~\stackrel{(7)}{=}~v^tgv~\stackrel{(2)}{=}~0.\tag{8}$$ This implies that both $$a$$ and $$c$$ are proportional to an identity matrix. The sought-for eq. (3) follows. $$\Box$$

[converted from a comment into an answer]

The WP article isn't claiming that it's true on mathematical grounds, it's just saying that it needs to be true on physical grounds. It also isn't claiming that it holds for any Λ, but only for a Λ that represents a change of coordinates to a new set of Minkowski coordinates.

I don't think your claim, without any conditions on Λ, holds. Let $$\Lambda=\operatorname{diag}(2,2)$$. This is just rescaling the coordinates by a factor of 2. For this $$\Lambda$$, we have $$\Lambda^T\eta\Lambda=4\eta$$. This is a counterexample to your conjecture that "If $$x^T (\Lambda^T\eta\Lambda) x$$=0 for all $$x$$ such that $$x^T \eta x=0$$, then $$\Lambda^T\eta\Lambda=\eta$$. "

Also, what is your definition of "a set of Minkowski coordinates"?

Minkowski coordinates are coordinates in which the metric has the form $$\operatorname{diag}(1,-1,-1,-1)$$.

So I think what you were interpreting in the WP article as a mathematical theorem is actually a combination of a physical argument (second postulate) with a definition (defining Minkowski coordinates as above).

A straightforward way to see that your conjecture shouldn't be expected to hold without any condtions on $$\Lambda$$ is that it's written in a way that assumes that we can do a certain coordinate transformation and the corresponding inverse transformation using the matrix $$\Lambda$$ and its transpose. This is not generally true for coordinate transformations. It's a mor special property that happens to hold for Lorentz boosts from one set of Minkowski coordinates to another.

• (1) Your counterexample is good and leads me to think that, in accordance with the wikipedia argument (originating with Landau & Lifshiftz?), that my statement needs to be changed to $\Lambda^T\eta\Lambda=\mathrm{const}\eta$. I'll edit the question. (2) I think the definition you give of Minkowski coordinates puts the cart before the horse in this case. I'm trying to show that the transformation $\Lambda$ preserves the Minkowski metric $\eta$, not start from that point. (3) There is no assumption that $\Lambda^T$ is the inverse transformation. $(\Lambda x)^T$ is a forward transformation of $x$ – Alex Mar 12 at 20:01
• Your counterexample is good and leads me to think that, in accordance with the wikipedia argument (originating with Landau & Lifshiftz?), that my statement needs to be changed to ΛTηΛ=constη No, that doesn't work. This counterexample is just a special case of the fact that all of these ideas only work for certain special transformations, not for all changes of coordinates. – Ben Crowell Mar 12 at 21:58
• I think the definition you give of Minkowski coordinates puts the cart before the horse in this case. I'm trying to show that the transformation Λ preserves the Minkowski metric η, not start from that point. Then you have to decide what premise you're going to start from. Your basic problem here is that you haven't clearly identified any particular set of assumptions that you want to start from, so you can't prove anything. There is more than one way to set up the logic here, but it won't work if you don't start with any set of assumptions at all. – Ben Crowell Mar 12 at 22:00
• Assumptions should be clear: 1. transformation is linear (i.e. no preferred origin of space and time). 2. something traveling at the speed of light in the first frame also travels at the speed of light in the other frame. Goal: prove that such a transformation preserves $\eta$ (maybe up to a scalar multiple). Can you explain why you are so sure this statement is false / give a counterexample? – Alex Mar 12 at 23:26