Why Lorentz transformations instead of general linear transformations?

Question

I am trying to understand why, in physics, we look specifically at Lorentz transformations, instead of the larger group of general linear transformations.

To fix the terminology: let spacetime be modeled as a four-dimensional Lorentzian manifold $(M, g)$. Then a Lorentz transformation at $p \in M$ is a linear map $$\Lambda: T_pM \rightarrow T_pM $$ on the tangent space at $p$ such that $g_p(\Lambda X, \Lambda Y) = g_p(X, Y)$ for all $X, Y \in T_pM$. Given an orthonormal frame $(e_i)$ $(i=0,...3)$ at $p$, i.e., a basis of $T_pM$ with $g_p(e_i, e_j) = \eta_{ij} = $ diag$(-1, 1, 1, 1)_{ij}$, a Lorentz transformation maps it to another orthonormal frame at $p$.

Question: Obviously, geometric objects like tensors on $M$ and equations involving geometric objects are invariant under Lorentz transformations, for the trivial reason that we can choose and switch bases as we wish without affecting these objects (in contrast, their components with respect to a basis change, of course). However, geometric objects are even invariant under general linear transformations $GL(4, \mathbb{R})$ of bases, for the same reason. What is so special about Lorentz transformations then? What makes them stand out from the larger group of general linear transformations?

EDIT

An illustrative example: Maxwell equations on a curved (or, in particular, a flat) spacetime, written geometrically as dF = 0, d * F = J (* is the Hodge star), are invariant under GL(4, $\mathbb R$) transformations on tangent spaces, I think, not just under SO(1, 3), simply because they do not even make use of a basis. So preservation of equations of motion does not seem to single out Lorentz transformations.

END EDIT

A tentative thought: Lorentz transformations allow us to split spacetime locally into space and time in a physically meaningful way, in the following sense: given an orthonormal frame $(e_i)_{i=0,...,3}$ with $e_0$ tangent to an observer worldline (the "time" direction for this observer) and $\{e_1, e_2, e_3 \}$ spanning the "spatial" subspace of $T_pM$ for this observer, the spatial part of a photon 4-velocity has the same absolute value $c$, the speed of light, in any frame related to the given one by a Lorentz transformation. This is not the case for general linear transformations, without surprise: the decomposition of a 4-vector into a temporal and a spatial part is generally basis-dependent, if it makes sense at all in general non-orthonormal bases. This shows that in contrast to general linear transformations, Lorentz transformations preserve even some "non-geometric", generally basis-dependent quantities. But is that all that sets them apart?

Answer 1

General linear transformations correspond to coordinate transformations. Differential geometry is formulated in a way that geometric statements are independent of coordinates.

Poincaire transformations (translations + Lorentz transformations) are isometries of the Minkowski metric. Isometries are more than generic coordinate transformations; they are distance preserving transformations of a metric space.

From a physical point of view (stated in differential geometry language), isometries are important because they are associated with Killing vectors (or generally Killing tensors), which are associated with conserved quantities. So, the fact that energy and momentum are conserved in special relativity, for example, follows from the fact that these isometries exist on a Minkowski background. And, the fact that energy and momentum are not conserved in general relativity, follows from the fact that a general spacetime does not have corresponding time and space translation symmetries.

Stated in a different way, coordinate transformations are a local or gauge symmetry, and there is no conserved quantity associated with them. Poincaire transformations are a global symmetry of relativistic field theories in Minkowski space, and come with associated conserved quantities via Noether's theorem.

Answer 2

Because observations of physical reality show that space-time vectors, and the conjugate momentum-energy vector, transform as 4-vectors under Lorentz transformations. This is just not true of a general linear transformation.

Think of time dilation and length contraction. It is specifically the Lorentz transformation that correctly gives the distance and time between two events in one inertial frame based on the corresponding observation in another. A general linear transformation will not.

Answer 3

Space, time, velocity & Galilean relativity are intimately related in Newtonian physics. Any change in the definition of one of these notions will affect the others.

Einstein, by postulating the constancy of light velocity in any inertial frame, falsified Galilean relativity. This means how space, time & velocity are related change. These are now related by the Lorentz transformations. Further, the constancy of light in any inertial frame means we now have Lorentz relativity aka Lorentz symmetries. This is how Lorentz symmetry arose in physics although they had been discovered earlier as part of the symmetry group of EM.