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Oxymoron
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Im trying to understand more about the foundations upon which general relativity lies.
The first thing I need to clarify is the space in which GR works. I've read that GR merges time and space into spacetime, which to my understanding means that, when considered separately, time and space are invariant.
This spacetime must then be much like any other mathematical space, such as Euclidean space or Hilbert space, in that notions of distance must be defined.
In Euclidean space, [tex]\mathbb{R}^3[/tex], you have the Euclidean metric,
[tex]\Delta s^2 = x^2 - y^2[/tex]
which obeys all the axioms of a metric and defines the distance between two points in the Euclidean space.
However, I have read, that the metric in spacetime, the spacetime metric, is defined as
[tex]\Delta s^2 = \Delta x^2 + \Delta y^2 + \Delta z^2 - c^2 \Delta t^2[/tex]
This spacetime metric raises some issues with me. If this is the metric of spacetime then points in spacetime must be represented as a 4-vector. Are these points actually called events by physicists?
The Euclidean plane has structure, right, in the form of the Euclidean metric:
1. [tex]d(x,y) \geq 0[/tex]
2. [tex]d(x,y) = 0 \Leftrightarrow x = y[/tex]
3. [tex]d(x,y) = d(y,x)[/tex]
4. [tex]d(x,z) \leq d(x,y) + d(y,z)[/tex]
In a similar fashion I am interested to know what the structure on spacetime is? Are there similar axioms for the spacetime metric?
The first thing I need to clarify is the space in which GR works. I've read that GR merges time and space into spacetime, which to my understanding means that, when considered separately, time and space are invariant.
This spacetime must then be much like any other mathematical space, such as Euclidean space or Hilbert space, in that notions of distance must be defined.
In Euclidean space, [tex]\mathbb{R}^3[/tex], you have the Euclidean metric,
[tex]\Delta s^2 = x^2 - y^2[/tex]
which obeys all the axioms of a metric and defines the distance between two points in the Euclidean space.
However, I have read, that the metric in spacetime, the spacetime metric, is defined as
[tex]\Delta s^2 = \Delta x^2 + \Delta y^2 + \Delta z^2 - c^2 \Delta t^2[/tex]
This spacetime metric raises some issues with me. If this is the metric of spacetime then points in spacetime must be represented as a 4-vector. Are these points actually called events by physicists?
The Euclidean plane has structure, right, in the form of the Euclidean metric:
1. [tex]d(x,y) \geq 0[/tex]
2. [tex]d(x,y) = 0 \Leftrightarrow x = y[/tex]
3. [tex]d(x,y) = d(y,x)[/tex]
4. [tex]d(x,z) \leq d(x,y) + d(y,z)[/tex]
In a similar fashion I am interested to know what the structure on spacetime is? Are there similar axioms for the spacetime metric?
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