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Re: law of conservation



Ninduk wrote:

Mr. Hugh Logan wrote:

In the first place, what do you mean by the total energy of the
universe?

i refer to Hawking in brief history "the TOTAL ENERGY of the universe is exactly zero"-p.136.

I haven't gotten around to reading Hawking's book. That should be my
next reading project. I suspect this result depends very much on how the
total energy of the universe has been defined. The only thing I have
seen about this is the comment in the Weiss and Baez article (
http://math.ucr.edu/home/baez/physics/energy_gr.html ) stating that the
total energy of a closed universe is zero if the Hamiltonian definition
of energy is used.

In particular, how do you sum the infinitesimal contributions over the entire universe?

may i ask where do all these infinitesimal components come from?

I was thinking of the universe as a continuum. One does this in
classical mechanics, even though one knows that matter is composed of
atoms. I believe this is the usual procedure for classical GR. (I
haven't read much about quantum gravity).

if these are of quantum mechanical origin then we can safely disregard
the component because there is an allowed uncertainty of delE for a >very small time delT such that delE*delT~h.

another point should be
cleared that if stress-energy tensor in GR is conserved as T^jk ;j=o >so why it should not hold for everywhere in the universe?

The fact that the covariant divergence T^jk ;j is zero everywhere in the
universe doesn't mean that the stress-energy tensor T^jk is zero
everywhere. (The semicolon in the covariant divergence refers to
covariant differentiation which takes into account the curvature of the
4-d hyperspace. The ordinary (coordinate) divergence uses a comma rather
than a semicolon. It must be generalized to the covariant divergence for
curved spacetime, which involves the concept of parallel displacement.)

as in
tensor we know that if any tensor vanishes in any coordinate system, >so does it in every coordinate system.

More generally, if you know the elements of a tensor in any coordinate
system, you know the elements of the tensor in any other coordinate
system, since you know how the tensor transforms. But this does not say
that, if all the elements of a tensor are zero at one point in spacetime
that they are zero at all points in spacetime using the same coordinate
system or other coordinate systems.

I presume that T^jk ;j = 0 is a statement that the covariant divergence
of the stress-energy tensor is zero everywhere in the universe. It
follows from taking the covariant divergence of Einstein's field
equations: G_{mu,nu} = 8pi T_{mu,nu}, since the covariant derivative of
the Einstein curvature tensor G can be shown to be zero using the Bianci
identities, according to Weiss and Baez. T^jk ;j = 0 implies that the
flux of the energy-momentum vector across a surface bounding an
infinitesimal volume surroundin the point of 4-dimensional hyperspace at
which T^jk ;j is evaluated is zero. Thus it is a very local statement.
The stress-energy tensor T^jk may be defined throughout the
4-dimensional hyperspace, but T^jk is a function of position in
hyperspace. T^jk ;j = 0 is a differential statement of energy-momentum
conservation.

The components of the stress energy tensor T^jk (j,k = 0,1,2,3) are
defined as the flux of j momentum across a surface of constant
x^k. The 0 direction is the time direction, and the time or 0 component
of 4-momentum is energy. (Actually, it is customary to use Greek letter
indices for indices that range over all four dimensions including time,
but Latin letters for those that range over only the three spacial
coordinates, but Greek letters are difficult to type).

T^00 is interpreted as energy density. One might think that T^00 could
be integrated over the entire universe to get the total energy, but
this cannot be done in curved space. (Conservation of energy, where
locally valid, would be expressed T^0j ; j = 0). Apparently, a somewhat
better approach is to work with the stress-energy tensor T^jk, not just
the energy density term. Weiss and Baez point out the
difficlty here:

To compute a divergence, we need to compare quantities (here vectors) on opposite faces. Using parallel transport for this
leads to the covariant divergence. This is well-defined, because we're dealing with an infinitesimal hypervolume. But to add up
fluxes all over a finite-sized hypervolume (as in the contemplated extension of Gauss's theorem) runs smack into the
dependence on transportation path. So the flux integral is not well-defined, and we have no analogue for Gauss's theorem.

The use of pseudotensors has been mentioned, but this is not completely
satisfactory or mathematically consistent, although useful results have
been obtained -- by Einstein in particular (the quadrupole formula - see
the article by Weiss and Baez). A good reference on the stress-energy
tensor is _A first course in general relativity_ by Bernard Schutz,
Chapter 4.

I hope that those more expert than I will clarify these matters more
fully.

Hugh Logan






How would this be possible if energy is lost by the universe? The only
thing like creation of matter out of nothing that I have heard of is the
continuous creation theory of Bondi,Gold and Hoyle in which particles
are spontanously created so as to keep the particle density of the
universe constant.

yes, they had devised a so-called c-field theory of such kind.