- #1
ecce.monkey
- 21
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Could someone please help me understand the Einstein hole argument (as outlined by Norton, see below). In particular the step that says that the second solution within the hole is a valid solution to the generally covariant field equation. I think my understanding of general covariance is at fault here.
I'll summarise the argument as described by Norton:
1) g(x) is a solution in the hole in one coordinate system...ok
2) g'(x') is the same solution in another coordinate system...fine
3) g'(x), gained by using the function from 2 with the first coord system args, is a different gravitational field...fine
4) g'(x) is a solution of the field equations (what!?)
How can he just say that g'(x) is a solution to the field equations? I can understand that the field equations are generally covariant and therefore take the same form in different coordinate systems. But I don't understand that a solution explicitly expressed in terms of one coordinate system can take the same form and be a solution in a different coord system.
This is a rough paraphrase of my question...
A generally covariant defintion of the circle is a curve equidistant from some point.
1)A solution in one coord system is x^2 + y^2 = 25
2)The same solution in another coord system is r=5
3)The equation x=5 is a different curve to 1)
4) The equation x=5 is a solution of the definition of a circle !?
How can 4) be stated?
I'll summarise the argument as described by Norton:
1) g(x) is a solution in the hole in one coordinate system...ok
2) g'(x') is the same solution in another coordinate system...fine
3) g'(x), gained by using the function from 2 with the first coord system args, is a different gravitational field...fine
4) g'(x) is a solution of the field equations (what!?)
How can he just say that g'(x) is a solution to the field equations? I can understand that the field equations are generally covariant and therefore take the same form in different coordinate systems. But I don't understand that a solution explicitly expressed in terms of one coordinate system can take the same form and be a solution in a different coord system.
This is a rough paraphrase of my question...
A generally covariant defintion of the circle is a curve equidistant from some point.
1)A solution in one coord system is x^2 + y^2 = 25
2)The same solution in another coord system is r=5
3)The equation x=5 is a different curve to 1)
4) The equation x=5 is a solution of the definition of a circle !?
How can 4) be stated?