On Feb 17, 1:19 pm, Erik Max Francis <m...@[EMAIL PROTECTED]
> wrote:
> George W Harris wrote:
> > Global conservation of mass/energy requires a
> > simply connected space-time topology.
>
> Actually, it's worse than that; even in a simply-connected spacetime
> topology, the concept of the total amount of mass, energy, momentum,
> angular momentum of the Universe is problematic in general relativity.
> You can talk about one, but those familiar with general relativity know
> that they're basically cheating.
>
> But more im****tantly, global conservation of energy is not in any
> meaningful sense true in general relativity. Indeed, our models of the
> Universe require that it be violated (namely, with cosmological red****ft
> -- photons traversing an expanding spacetime are losing energy, but the
> energy isn't going anywhere).
>
> In general relativity, one can only talk about _local_ (that is, in the
> vicinity of a point) conservation laws. There are no global
> conservation laws in our best theories that deal with the large-scale
> structure and evolution of the Universe, as strange as that sounds.
Just to go into a little more detail - the conservation of dynamical
quantities (energy, momentum, angular momentum, and some weird stuff
without convenient Newtonian analogues) must hold in any space-time
that is asymptotically flat. That is, if you go far enough away and
you reach areas where gravity reaches its weak field (Newtonian) limit
and space-time is approximately given by the Minkowski metric (that
is, it is flat and not warped, curved, spindled, or mutilated), and
this is true no matter what direction you choose to go far away in,
then you are guaranteed that energy and momentum and stuff will be
conserved (although the conservation might not be local while within
the region where space-time is not flat). If this condition is not
true then you can probably find ways of violating these conservation
laws.
Note that this holds even for non-simply connected topologies. By
now, most readers of this newsgroup are probably familiar with the way
a wormhole mouth gains the conserved quantities of things that go
through them, and lose said quantities of things that come out of
them. This is an example of a non-simply connected topology where the
conservation of dynamical variables (as well as electric charge) still
holds (assuming that if you go far enough away from the wormhole you
reach asymptotically flat space-time, of course - but that is usually
part of the definition of a wormhole).
Also note that the space-time surrounding Earth, or solar system, and
our galaxy is, to a very good approximation, flat. Thus time
travelers to and from Earth will need to worry about their mass being
conserved.
I think the conservation of electric charge holds in any space-time
geometry, since it arises from gauge symmetries rather than
geometrical symmetries. If someone knows more detail about this,
though, feel free to correct me (and explain why, of course - this
stuff is interesting to learn about).
Luke
|
