5.7 The Metric Tensor and the Stress-Energy Tensor (General Relativity)
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This article is from the Relativity and FTL Travel FAQ, by Jason W. Hinson jason@physicsguy.com with numerous contributions by others.
5.7 The Metric Tensor and the Stress-Energy Tensor (General Relativity)
Now that we have had a glimpse at tensors, let's consider a couple that
will be important to us. The first is called the metric tensor. I mentioned
a couple of sections ago that this tensor is related to the invariant
interval for a certain coordinate system on a given manifold. So, let's go
back and look at a the two specific invariant intervals which we introduced.
First, in normal, x-y, Cartesian coordinates, we have Equation 5:1
duplicated here:
(Eq 5:15--Copy of Eq 5:1)
ds^2 = dx^2 + dy^2
Second, on the surface of a sphere, using the L-H coordinate system
which we defined, we have Equation 5:2 duplicated here:
(Eq 5:16--Copy of Eq 5:2)
ds^2 = dH^2 + [cos(H/R)]^2*dL^2
Now, let's make this more general by considering an arbitrary, two
dimensional manifold and an arbitrary coordinate system on that manifold.
Let's call the coordinates "a" and "b". Now, in general, the invariant
interval on this manifold is defined in terms of the square of that interval
ds^2. The equation for ds^2 involves the infinitesimal distances da and db
in second order combinations. By second order combinations, I mean, for
example, da^2 or da*db. Thus, in general, the invariant interval will have
the following form (note: the g components are generally formulas of "a" and
"b"):
(Eq 5:17)
ds^2 = g *da^2 + g *da*db + g *db*da + g *db^2
aa ab ba bb
In that equation you see the four components of the metric tensor in
this two dimensional, a-b coordinate system. They are the "g's" in the
equation. For our x-y coordinate system, we have
(Eq 5:18) g = 1, g = 0, g = 0, g = 1 xx xy yx yy
For our L-H coordinate system, we have
(Eq 5:19) g = 1, g = 0, g = 0, g = [cos(H/R)]^2 HH HL LH LL
So, we can construct the invariant interval if we know the metric
tensor for a coordinate system on a manifold. Now, remember that we said
that the form of the invariant interval for a particular coordinate system
tells us everything there is to know about the manifold for which those
coordinates are valid. So, now we see that all we need to know is the form
of the metric tensor. Once we know g, we know the geometry of the manifold.
Using tensor analysis, we can take the metric tensor and find an equation
for geodesics on the manifold. We can use it to find out all about the
curvature of the manifold. We can even use it to find the dot product (we
will discuss this a bit later) of two vectors in a particular coordinate
system.
Another thing the metric allows us to do is something generally called
"raising" or "lowering" indices. Basically, if you consider a tensor with a
contravariant index (which transforms in a particular way as discussed
earlier), then there is another way to express the tensor as one which has a
covariant index (and vice versa). That is to say that the geometric entity
represented by the tensor with the contravariant index has another
representation which involves a covariant index. For example, consider the
tensor A^a, which has a contravariant index, a. There is a corresponding
covariant tensor, A_a, which can be found using the metric of the space (and
coordinate system) we are dealing with. Here is an example of how you find
it (finding A_x when you know A^x) for a coordinate system with some
arbitrary coordinates, x and y:
(Eq 5:20)
x y
A = g A + g A
x xx xy
For a general space and coordinate system, you can write this rule as
follows (remember, "a" can be any one dimension in the space, so this
represents a number of equations):
(Eq 5:21)
b
A = SUM(b varies over all dimensions) g A
a ab
Similarly, if you know the covariant form of A (A_a) you can find the
contravariant form by using the following:
(Eq 5:22)
a ab
A = SUM(b varies over all dimensions) g A
b
But that equation involves the contravariant form of the metric g^ab. In the
invariant interval, the metric is expressed in its covariant form g_ab. It
is therefore important for the reader to remember as we discuss various
metrics below, that for all of them we have
(Eq 5:23)
ab 1
g = --- if a = b
g
ab
and
ab
g = 0 if a doesn't = b
Thus, using the metric tensor, one can "raise" or "lower" any index of
a tensor. Remember, what one is really doing is finding a form of that
tensor which transforms in a different way.
With this example of how the metric can be used, we will end our
discussion of this tensor. To sum up, the metric tensor on a manifold is a
very important entity which not only tells us all about the manifold's
geometry, but which also provides a very powerful tool which allows us to
deal with that geometry mathematically.
The second tensor we want to mention is the stress-energy tensor. I
don't want to get too deep into a discussion of the stress-energy tensor,
but the reader should know a couple of key points. With the stress-energy
tensor, we see our first example of a tensor explicitly defined in four
dimensional space-time (though later we will look at the metric tensor
defined in 4-d space-time). The stress-energy tensor (T) is also a tensor of
rank 2 (like the metric tensor), which gives it 16 components in 4
dimensions. Sometimes we express such a tensor in the form of a matrix as
follows:
(Eq 5:24)
+- -+
| tt tx ty tz |
| T T T T |
| |
| xt xx xy xz |
ab | T T T T |
T = | |
| yt yx yy yz |
| T T T T |
| |
| zt zx zy zz |
| T T T T |
+- -+
There you can see the 16 different components. Now, each of these components
tell us something about the distribution and "flow" of energy and momentum
in a region. More precisely, T contains information about all the stresses
and pressures and momenta in a region. For example, The "tt" component of
the stress-energy tensor would be the density of the energy in the region
(the amount of energy--including mass energy--per unit volume).
As to why the stress-energy tensor is important to us, that will be
discussed further in a bit. However, here we can note the following in order
to pull us back towards our discussion of relativity and gravity: In
Newtonian physics, gravity was caused by the density of mass in an area.
However, in SR we find that mass is just a form of energy, and so we might
think that the "tt" component of the stress-energy tensor would be the right
thing to look at when it comes to gravity. However, if we write a rule using
one component of a tensor, then because the value of that component will
depend on your coordinate system (or frame of reference in space-time) then
the rule will also be frame-dependent. In short gravity would not be an
invariant theory, and it would require a preferred frame if we based it only
on the "tt" component of T. However, if we use all the components of a
tensor to form our theory, then (as it turns out) the theory can be made
frame-independent. Einstein thus considered the possibility that the whole
stress-energy tensor would need to play a part as the source of gravity. Add
to this some insight on curved manifolds and you end up with general
relativity, as we will see.
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