Science > Physics > Energy In The Knowledge Equation 5.2 Bose and Dadhich 1999
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Science > Physics |
| User: |
"OsherD" |
| Date: |
27 Nov 2005 09:42:36 PM |
| Object: |
Energy In The Knowledge Equation 5.2 Bose and Dadhich 1999 |
From Osher Doctorow
Continuing with Bose and Dadhich (1999), introduced in discussing
Carlip, consider:
1) E_H - E_infinity = M_H
where M_H is the value of the gravitational charge at the horizon (M_c
is the general gravitational charge value), E_H is the QLE at the
horizon, E_infinity is the total energy of the spacetime. This is
equation (4) of p. 3 of Bose and Dadhich (1999).
This identity (1) only holds for specific foliations of spacetimes.
The BY quasilocal energy in the integral form of last time depends on
the choice of the foliation and B itself, but not on the choice of
coordinates on the quasilocal two-surface B.
M__H defines the strength of the leading Newtonian potential and so in
Newtonian terms it measures how strong the gravitational pull exerted
by a body is.
The field energy E_inifinty - E_H measures the amount of curvature of
the space due to a non-rotating matter distribution since it is related
to the spatial conformal factor arising in higher post post-Newtonian
orders in expansion of the spacetime metric around such a matter
distribution. The spatial components of the Riemann curvature tensor
up to such orders are determined by E_infinity - E_H.
The gravitational field energy appears to be related to the curvature
of space for the specific case of Schwarzschild spacetime, so (1)
implies that the horizon is a surface where gravitational field energy
equals gravitational charge in magnitude.
Osher Doctorow
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| User: "OsherD" |
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| Title: Re: Energy In The Knowledge Equation 5.2 Bose and Dadhich 1999 |
27 Nov 2005 11:57:26 PM |
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From Osher Doctorow
I should add that the system under consideration is a spatial 3-surface
SIGMA bounded by a 2-surface B in a region of spacetime that is
decomposable as a product of a spatial 3-surface and a real line
interval that represents time.
Bose and Dadhich also point out that for describing non-gravitational
interactions one uses separate measures for mass/energy of a particle
and its charge, where the latter determines its strength of coupling to
the field; but for GR one uses a single measure for both and the
gravitational field contributes to its gravitational charge since it
contains energy which has led to major ambiguity except for
asymptotically flat static spacetimes for example.
The expressions for M_c are:
1) M_c = (1/(4pi))I[g.ds]
2) M_c = -(1/(8pi))I[e_abcd DELTA^c t^b
where the integrals I... are surface integrals over closed 2-surface B
and g is given by:
3) g = -N DELTA(ln N), N = sqrt(-t^a t_a)
N plays the role of lapse function by suitably choosing a time
coordinate for a suitable choice of foliation of the region of interest
with spacelike hypersurfaces.
Here e_abcd is the volume element on the spacetime.
Equation (2) turns out to be the equation of the Komar mass and defines
a conserved gravitational chage when the spacetime is vacuum and admits
a timelike Killing vectors, while if one of these doesn't hold then M_c
depends on the location of B and describes the "quasilocal" charge of
the spatial volume bounded by B.
Osher Doctorow
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| User: "Androcles" |
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| Title: Re: Energy In The Knowledge Equation 5.2 Bose and Dadhich 1999 |
28 Nov 2005 04:55:59 AM |
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Yawn...
*plonk*
Androcles.
"OsherD" <mdoctorow@comcast.net> wrote in message
news:1133157446.251396.150520@z14g2000cwz.googlegroups.com...
From Osher Doctorow
I should add that the system under consideration is a spatial 3-surface
SIGMA bounded by a 2-surface B in a region of spacetime that is
decomposable as a product of a spatial 3-surface and a real line
interval that represents time.
Bose and Dadhich also point out that for describing non-gravitational
interactions one uses separate measures for mass/energy of a particle
and its charge, where the latter determines its strength of coupling to
the field; but for GR one uses a single measure for both and the
gravitational field contributes to its gravitational charge since it
contains energy which has led to major ambiguity except for
asymptotically flat static spacetimes for example.
The expressions for M_c are:
1) M_c = (1/(4pi))I[g.ds]
2) M_c = -(1/(8pi))I[e_abcd DELTA^c t^b
where the integrals I... are surface integrals over closed 2-surface B
and g is given by:
3) g = -N DELTA(ln N), N = sqrt(-t^a t_a)
N plays the role of lapse function by suitably choosing a time
coordinate for a suitable choice of foliation of the region of interest
with spacelike hypersurfaces.
Here e_abcd is the volume element on the spacetime.
Equation (2) turns out to be the equation of the Komar mass and defines
a conserved gravitational chage when the spacetime is vacuum and admits
a timelike Killing vectors, while if one of these doesn't hold then M_c
depends on the location of B and describes the "quasilocal" charge of
the spatial volume bounded by B.
Osher Doctorow
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