In Landau's classical field theory, chapter 1, I'm getting mad with an argument used to derive the $ds^2$invariance from Einstein's postulate of the invariance of the speed of light. Landau says that if

$ds^2= c^2dt^2-dx^2-dy^2-dz^2=0$ then we have to have $ds'^2=c^2dt'^2-dx'^2-dy'^2-dz'^2=0$

That implies, according to Landau, that $ds^2 = A(v) ds^2$

I had a differential geometry exam, succeeding in it, but I can't understand what the object $ds^2$ is. If we want to interpret it as a metric, the scalar product $x^T I x$ is mapped to $x'^TI x'$

How can be it proven that there exists a function of velocity $A(v)$ such that

$$x'^T I x' = A(v) x^T I x?$$

At this moment I can't use that the transformation is linear since we don't know that for sure (if it is, please explain why).

But also if we assume linearity, it's not true that if $x^T I x$ and $x^T M^T I M x$ have the same zeros, then $I= M^T IM$...

How can I write Landau's argument in a rigorous way? What are the precise mathematical tools and assumptions made?

In Landau & Lifshitz's "classical field theory", chapter 1, I'm getting mad with an argument used to derive the $ds^2$invariance from Einstein's postulate of the invariance of the speed of light. Landau & Lifshitz say that if

$$ds^2= c^2dt^2-dx^2-dy^2-dz^2=0$$ then we have to have $$ds'^2=c^2dt'^2-dx'^2-dy'^2-dz'^2=0$$

That implies, according to Landau, that $$ds^2 = A(v) ds^2.$$

I had a differential geometry exam, succeeding in it, but I can't understand what the object $ds^2$ is. If we want to interpret it as a metric, the scalar product $x^T I x$ is mapped to $x'^TI x'$

How can be it proven that there exists a function of velocity $A(v)$ such that

$$x'^T I x' = A(v) x^T I x?$$

At this moment I can't use that the transformation is linear since we don't know that for sure (if it is, please explain why).

But also if we assume linearity, it's not true that if $x^T I x$ and $x^T M^T I M x$ have the same zeros, then $I= M^T IM$...

How can I write Landau's argument in a rigorous way? What are the precise mathematical tools and assumptions made?

In the Landau's classical field theory, chapter 1, I'm getting mad with an argument used to derive the $ds^2$invariance from the Einstein's postulate of the invariance of the speed of light. Landau says that if

$ds²= c²dt²-dx²-dy²-dz²=0$ Then we have to be $ds'²=c²dt'²-dx'²-dy'²-dz'²=0$

That implies, according to Landau, that $ds² = A(v) ds²$

I had a differential geometry's exam, succeeding in it, but I can't realize what the object $ds²$ are. If we want to interpret that as a metric, the scalar product

 $x^T I x$

Is mapped in

$x'^TI x'$

How can be proven that exists a function of velocity $A(v)$ s.t

$x'^T I x' = A(v) x^T I x$?

At this moment I can t use that the transformation is linear, i don't now that (if it is, explain me why)

But also if we assume linearity, it's not true that if $x^T I x$ and $x^T M^T I M x$ have the same zeros, then $I= M^T IM$...

How can I write Landau argument in a rigorous way? What are the precise mathematically tools and assumptions made?

In Landau's classical field theory, chapter 1, I'm getting mad with an argument used to derive the $ds^2$invariance from Einstein's postulate of the invariance of the speed of light. Landau says that if

$ds^2= c^2dt^2-dx^2-dy^2-dz^2=0$ then we have to have $ds'^2=c^2dt'^2-dx'^2-dy'^2-dz'^2=0$

That implies, according to Landau, that $ds^2 = A(v) ds^2$

I had a differential geometry exam, succeeding in it, but I can't understand what the object $ds^2$ is. If we want to interpret it as a metric, the scalar product  $x^T I x$ is mapped to $x'^TI x'$

How can be it proven that there exists a function of velocity $A(v)$ such that

$$x'^T I x' = A(v) x^T I x?$$

At this moment I can't use that the transformation is linear since we don't know that for sure (if it is, please explain why).

But also if we assume linearity, it's not true that if $x^T I x$ and $x^T M^T I M x$ have the same zeros, then $I= M^T IM$...

How can I write Landau's argument in a rigorous way? What are the precise mathematical tools and assumptions made?

In the Landau's classical field theory, chapter 1, I'm getting mad with an argument used to derive the $ds^2$invariance from the Einstein's postulate of the invariance of the speed of light. Landau says that if

$ds²= c²dt²-dx²-dy²-dz²=0$ Then we have to be $ds'²=c²dt'²-dx'²-dy'²-dz'²=0$

That implies, according to Landau, that $ds² = A(v) ds²$

I had a differential geometry's exam, succeeding in it, but I can't realize what the object $ds²$ are. If we want to interpret that as a metric, the scalar product

$x^T A x$

Is mapped in

$x'^TA x'$

How can be proven that exists a function of velocity $A(v)$ s.t

$x'^T A x' = A(v) x^T A x$?

At this moment I can t use that the transformation is linear, i don't now that (if it is, explain me why)

But also if we assume linearity, it's not true that if $x^T A x$ and $x^T M^T A M x$ have the same zeros, then $A = M^T AM$...

How can I write Landau argument in a rigorous way? What are the precise mathematically tools and assumptions made?

In the Landau's classical field theory, chapter 1, I'm getting mad with an argument used to derive the $ds^2$invariance from the Einstein's postulate of the invariance of the speed of light. Landau says that if

$ds²= c²dt²-dx²-dy²-dz²=0$ Then we have to be $ds'²=c²dt'²-dx'²-dy'²-dz'²=0$

That implies, according to Landau, that $ds² = A(v) ds²$

I had a differential geometry's exam, succeeding in it, but I can't realize what the object $ds²$ are. If we want to interpret that as a metric, the scalar product

$x^T I x$

Is mapped in

$x'^TI x'$

How can be proven that exists a function of velocity $A(v)$ s.t

$x'^T I x' = A(v) x^T I x$?

At this moment I can t use that the transformation is linear, i don't now that (if it is, explain me why)

But also if we assume linearity, it's not true that if $x^T I x$ and $x^T M^T I M x$ have the same zeros, then $I= M^T IM$...

How can I write Landau argument in a rigorous way? What are the precise mathematically tools and assumptions made?