tessel@um.bot
Mar6-06, 04:00 AM
On Fri, 3 Mar 2006, zephir wrote:
> Let's say we have empty space, so that metric is flat. And we put some
> matter in it . So now we have some nonflat metric.
Right, although I'd prefer to say "nonflat Lorentzian spacetime", and you
should definitely say "spacetime" rather than "space".
> Question is:is that matter in flat or nonflat space? My difficulty is in
> understanding that if that matter is in curved space then we had no
> right to use GR formula which says how space is curved when we put
> matter is a flat space (cause that matter is NOT in flat space). And if
> we say that that matter is in flat space then it is wrong to say space
> is curved when we add matter in it.
By "GR formula which says how space[time] is curved when we put matter
[in] a flat space[time]", do you mean the Einstein field equation (EFE)?
If so, compare the spacetime curvature of the Lorentzian manifold part of
of your spacetime model (Lorentian manifold plus appropriate tensor
fields) with the stress-energy tensor. You can think of the EFE as giving
a consistency condition between the geometry and the physics of your
spacetime model. If you wrote down any old metric and then wrote down any
old stress-energy tensor field, the EFE will almost always insist that
that the geometric part of your spacetime model (the Lorentzian manifold
structure) is incompatible with the physical part (the stress-energy
tensor contributions from matter and any nongravitational fields). Thus,
the EFE provides a very stringent compatibility condition between the
geometry of a spacetime and any "physical fields" which have been placed
in that spacetime. Exact solutions of the EFE, OTH, are spacetime models
which satisfy this consistency condition.
The EFE does NOT directly answer questions like "if I take an exact
solution (M,g) and place a new stress-energy tensor T on M, how does this
change the metric tensor g?" (This way of thinking appears to be related
to the "hole argument" which held up Einstein for three long years on the
path to writing down the final field equations for gtr!)
Does this help?
"T. Essel"
> Let's say we have empty space, so that metric is flat. And we put some
> matter in it . So now we have some nonflat metric.
Right, although I'd prefer to say "nonflat Lorentzian spacetime", and you
should definitely say "spacetime" rather than "space".
> Question is:is that matter in flat or nonflat space? My difficulty is in
> understanding that if that matter is in curved space then we had no
> right to use GR formula which says how space is curved when we put
> matter is a flat space (cause that matter is NOT in flat space). And if
> we say that that matter is in flat space then it is wrong to say space
> is curved when we add matter in it.
By "GR formula which says how space[time] is curved when we put matter
[in] a flat space[time]", do you mean the Einstein field equation (EFE)?
If so, compare the spacetime curvature of the Lorentzian manifold part of
of your spacetime model (Lorentian manifold plus appropriate tensor
fields) with the stress-energy tensor. You can think of the EFE as giving
a consistency condition between the geometry and the physics of your
spacetime model. If you wrote down any old metric and then wrote down any
old stress-energy tensor field, the EFE will almost always insist that
that the geometric part of your spacetime model (the Lorentzian manifold
structure) is incompatible with the physical part (the stress-energy
tensor contributions from matter and any nongravitational fields). Thus,
the EFE provides a very stringent compatibility condition between the
geometry of a spacetime and any "physical fields" which have been placed
in that spacetime. Exact solutions of the EFE, OTH, are spacetime models
which satisfy this consistency condition.
The EFE does NOT directly answer questions like "if I take an exact
solution (M,g) and place a new stress-energy tensor T on M, how does this
change the metric tensor g?" (This way of thinking appears to be related
to the "hole argument" which held up Einstein for three long years on the
path to writing down the final field equations for gtr!)
Does this help?
"T. Essel"