lriuui0x0
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I'd like to get some help on checking my understanding of special relativity, specifically I'm trying to clarify the idea of coordinates. Any comment is really appreciated!
The spacetime is an affine space ##M^4##, which is associated with a 4 dimensional real vector space ##\mathbb{R}^4##. This vector space is abstract, and no basis is prechosen, so there's no canonical way to define what the coordinate might be.
There's a metirc g defined on the vector space ##\mathbb{R}^4##. This inner product has the property that for a particular set of basis, it has ##(+,−,−,−)## signature. Such a basis is a standard Cartesian basis, which is not unique.
The linear maps ##\mathbb{R}^4 \to \mathbb{R}^4## between the sets of standard basis form the Lorentz group. The affine maps ##M^4 \to M^4## between the set of standard basis form the Poincare group. All such maps have the metric signature ##(+, -, -, -)##.
Coordinates is a map ##M^4 \to \mathbb{R}^4## (here ##\mathbb{R}^4## means a four real number tuple, not an abstrct vector space). Any Cartesian basis at a point defines a Cartesian coordinates by defining the coordinates to be the components of the vector. The standard Cartesian coordinates defined as above is the same as the coordinates being inertial. Other coordinates are non-inertial, in which the metric components don't have ##(+, -, -, -)## signature.
The spacetime is an affine space ##M^4##, which is associated with a 4 dimensional real vector space ##\mathbb{R}^4##. This vector space is abstract, and no basis is prechosen, so there's no canonical way to define what the coordinate might be.
There's a metirc g defined on the vector space ##\mathbb{R}^4##. This inner product has the property that for a particular set of basis, it has ##(+,−,−,−)## signature. Such a basis is a standard Cartesian basis, which is not unique.
The linear maps ##\mathbb{R}^4 \to \mathbb{R}^4## between the sets of standard basis form the Lorentz group. The affine maps ##M^4 \to M^4## between the set of standard basis form the Poincare group. All such maps have the metric signature ##(+, -, -, -)##.
Coordinates is a map ##M^4 \to \mathbb{R}^4## (here ##\mathbb{R}^4## means a four real number tuple, not an abstrct vector space). Any Cartesian basis at a point defines a Cartesian coordinates by defining the coordinates to be the components of the vector. The standard Cartesian coordinates defined as above is the same as the coordinates being inertial. Other coordinates are non-inertial, in which the metric components don't have ##(+, -, -, -)## signature.