In SR are Lorentz transformations the only transformations that leave invariant the spacetime interval / proper time?

Question

If yes:

• Then why are coordinate transformations such as Cartesian $\to$ spherical not included in this set, despite the fact that the spacetime interval / proper time is unchanged (since the space coordinates $x$, $y$, $z$ are simply being re-labeled, while time stays the same)?

If no:

• Then what is the name of the full group of transformation that leaves invariant the spacetime interval (including Lorentz transformations and Cartesian $\to$ spherical coordinate transformations) ?
• Then what makes the Lorentz transformations different/special with respect to this group (outlined in the previous question), meaning, why do we only talk about them in SR, disregarding the others members of the group?

Answer 1

Spacetime interval, defined as $g_{\mu\nu}dx^\mu dx^\nu$ (where $g_{\mu\nu}$ are the components of the metric tensor in a given set of coordinates, and $dx^\mu$ are the components of the infinitesimal displacement in a given set of coordinates), is, by construction, invariant under general coordinate transformations -- assuming that you are transforming the $g_{\mu\nu}$ and $dx^\mu$ as tensor components. This follows simply from the mathematical fact about tensors that transformations acting on upper indices "cancel out" the transformations acting on lower indices. So, the spacetime interval is invariant under any general coordinate transformation.

What is special about Lorentz transformations is that they leave the Minkowski metric invariant! In particular, under a general coordinate transformation, the metric components $g_{\mu\nu}$ will change. For example, you see various terms such as $r^2$ or $\sin^2\theta$ in the flat-space metric when you write it down in polar coordinates. On the contrary, under a Lorentz transformation, the flat-space metric remains Minkowskian, i.e., $\mathrm{diag}(1,-1,-1,-1)$. Lorentz transformations are not the only transformations that have this property, the exhaustive group of transformations that leave the Minkowskian metric invariant is called the Poincare group. The Lorentz group is the homogeneous part of the Poincare group (i.e., such transformations that map the origin to the origin).

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PS: It should also be noted that if you define the spacetime interval to be $\eta_{\mu\nu}dx^\mu dx^\nu$ where you take $\eta_{\mu\nu}$ to be a constant matrix given by $\mathrm{diag}(1,-1,-1,-1)$ then the Lorentz transformations (or, Poincare transformations if you include non-homogeneous transformations) are the only transformations that leave the spacetime interval invariant. See this old answer of mine for worked-out details of the proof.

Answer 2

(Note I am considering every transformation here as a passive transformation, i.e. as transformations of globally defined coordinates rather than transformations of the actual spacetime. This is very important, because the reasoning here applies to passive transformations only!)

If you're asking what passive transformations preserve what the spacetime interval actually is, then all possible coordinate transformations preserve it, because the spacetime distance between any two events is a coordinate-independent notion.

However, if you're asking about what passive transformations preserve the way the spacetime interval is written, then the answer is that it is the Poincare group (reinterpreted not as group of isometries but as a group of passive transformations). The transformations from the Poincare group preserve the form of the spacetime interval as $$ ds^{2} = -c^{2}dt^{2} + dx^{2} + dy^{2} + dz^{2} \qquad (*) $$ in Cartesian inertial coordinates.

The Poincare group is the group of all (1) Lorentz boosts, (2) rotations, (3) translations, and (4) reflections, and these are the only passive transformations that preserve the way the spacetime interval is written.

Again, I emphasize that there is a difference in asking what changes the spacetime metric vs asking what changes the way the spacetime metric is written. It is extremely important to understand what is coordinate-independent and what is not.

This gets us to another question you posed.

Then why are coordinate transformations such as Cartesian → spherical not included in this set, despite the fact that the spacetime interval / proper time is unchanged (since the space coordinates x, y, z are simply being re-labeled, while time stays the same)?

The Cartesian → spherical transformations are not included, because they change the way the spacetime metric is written. In spherical coordinates, the spacetime interval is $$ ds^{2} = -c^{2}dt^{2} + dr^{2} + r^{2}d\theta^{2} + r^{2}\sin^{2}\theta d\phi^{2}. $$ Clearly this is not the same form as in $(*)$, so this transformation is not in the Poincare group.

So to summarize,

• All coordinate transformations of any kind (no matter how crazy and weird) preserve the spacetime distance between two events, because the spacetime distance between two events is coordinate-independent.
• Only transformations of the Poincare group preserve the way the spacetime distance between two events is written and calculated.

Answer 3

We use the special coordinate system and the (indefinite) quadratic form (Minkowski metric) to define Minkowski space (as a manifold, if it suits you). Then we can attach to it any other coordinate system (including local or coordinate charts) to express the same manifold in any other coordinate system ("atlas"). Then, of course, we can say that the tensor of the Minkowski metric is the same in all coordinates, therefore the spacetime interval (proper time) remains the same in any coordinate system.

It is important, however, to not forget that the original choice of coordinates -- and the quadratic form expressed in those coordinates -- has a very privileged role because that's how we defined the manifold (Minkowski space in this case). It could have been done in any other coordinate system, to begin with, but this appears simplest and easiest to correlate with observations.

With this, the inhomogeneous Lorentz transformations (the Poincaré group) should be the only ones that preserve the Minkowski metric (I am not sure if there are some pathological transformations, mathematical artifacts without physical meaning). Any such transformation can be expressed in any coordinate system. The form ("how it is written") will, in general, differ from one coordinate system to another, so the representation in the spherical coordinates will differ from that in the Cartesian.