Re: Gravitational Confusion
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Posted by ronron on December 21, 19100 at 11:21:50:
In Reply to: Re: Gravitational Confusion posted by Brian on December 21, 19100 at 08:37:52:
i share your wish to know WHY gravitation exists. we
might have to rest content being able to predict. you
spoke above about nuclear reactions, and this involves
energy and mass, so it might be useful to look at the
equations involved in gravitation. we could compare the Newtonian gravitational force
law: F = (G*m1*m2)/(|r1 - r2|^2) with the Einstein field equation: G = 8*pi*T in the Newtonian force law, the right hand side
of the equation contains the masses and the inverse
square of their separation distance. the left hand
side of this equation is the value of the force
produced by those masses and separation distance. so
we could loosely say that the "masses CAUSE the
gravitational attraction", but we still wouldn't have
answered your 'WHY?' question. but we at least relate
gravity to mass. now look at the Einstein equation: G = 8pi T
here the right hand side of the equation represents
the stress-energy tensor. this would loosely
correspond to the mass on the right hand side of
the Newtonian equation. it's also an energy thing
since we know from SR about the equivalence of mass
and energy. the left hand side of this equation, the
G, is the Einstein tensor. G depends only on the
Riemann curvature tensor and the metric (the "tool"
for calculating intervals between observers in GR).
G represents the curvature of spacetime. so the
relation G = 8pi T relates mass/energy to curvature,
it tells you how the curvature G is determined by
the stress-energy ("mass") tensor T, and you have
somewhat of a parallel with the Newtonian theory
to the extent that you have "mass" causing gravitation
regardless of how you explain the gravitational
effects. in fact, the 8pi constant is a result of
the requirement that the Einstein theory agree with
the Newtonian theory under certain conditions. it's
a constant that's pretty easy to calculate. some notes: the Newtonian force law above is just that - a way
of calculating the force given the masses and
separation distance. Newtons second law, F = ma
still holds as a definition, so you can substitute
the force law into Newton's second law to get ma = (Gm1m2)/r^2 this differential equation you can solve to
get the equation of motion. but you are working
in the domain of an invariant Euclidean space. that is,
any solutions you get involving trajectories will
assume that space is R^3 and will not change. in the Einstein equation, G = 8pi T is a tensor
equation and you have to expand it. when you do
this you get that set of non-linear, coupled, partial
differential equations that we were talking about
yesterday. a mess. but what is significant here
is this: in the Newtonian case, you are solving for
a trajectory, x(t), or you are computing a force, F.
but you are *not* solving for the global topology
or geometry. in the Einstein case, you ARE solving
for the actual spacetime curvature itself, and the
spacetime you get depends on the stress-energy tensor
T, which we can think of as the "mass" for now. but if you want to know whether G = 8pi T tells you
WHY mass/energy causes a curvature of spacetime, the answer
is no. a lot of people would probably answer your
question "What causes gravity?" by saying "mass",
and that's reasonable. recall too that any endeavor to explain the "why"
will usually get you in a pickle of using GR itself
to work out the "why". you mentioned nuclear reactions.
you suggested that a "dense reaction" might cause a
bending of spacetime. but what you are really saying
is that with a sufficiently "large" stress-energy
tensor T, you might get a curvature that is reflected
in the Einstein tensor G. so you are here using GR
to explain "why" GR works, and you can't do that.
you could solve the Einstein field eqn to get a solution
for a given T (that involved star masses, energies, etc)
but the solution would only give you the
corresponding spacetime. it wouldn't tell you why
the mass/energy caused the bending. the power of Newtonian action-at-a-distance gravitation
or Einstein's geometric formulation is the ability to
predict more than to truly explain. i wish i had a
better answer, but the best response i can think of
to the question "what causes gravitational effects?"
is "mass" or "stress-energy tensor". à bientôt, ronron
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