$GL(4,R)$ vs $T(3,1)\times O(3,1)$?

Question

I am told from math class that a general coordinate transformation on a vector is $GL(4,R)$, and in my physics class I am told it is $T(3,1)\times O(3,1)$. Can $GL(4,R)$ do everything $T(3,1)\times O(3,1)$ does, and more? Or is it $T(3,1)\times O(3,1)$ that is 'bigger'?

Answer 1

The general coordinate transformation group of a $4$-dimensional vector space is $\text{GL}(4)$.

However, in special relativity, the structure of space-time is that of a lorentzian affine space, in which case the group of general coordinate transformations is $T(4)\rtimes O(3,1)$. More explicitely, there is no origin of space, so transformations are not required to fix the origin like in the vector space case. This is the origin of the factor $T(4)$. But the difference between two points lies in a vector space equipped with a lorentzian bilinear form, so we require coordinate transformation to preserve this form. This is why we have $O(3,1)$, which is strictly smaller than $\text{GL}(4)$.

Neither group is included in the other. Instead, we have $\text{GL}(4)\cap (T(4)\rtimes O(3,1)) = O(3,1)$ which is strictly smaller than both.

To recap, the symmetry groups are :

	Vector space	Affine space
No bilinear form	$\text{GL}(n)$	$T(n)\rtimes \text{GL(n)}$
Euclidean form	$O(n)$	$T(n) \rtimes O(n)$
Lorentzian form	$O(n-1,1)$	$T(n) \rtimes O(n-1,1)$