5.8.4 Second Example: Stars and Black Holes (General Relativity)
Description
This article is from the Relativity and FTL Travel FAQ, by Jason W. Hinson jason@physicsguy.com with numerous contributions by others.
5.8.4 Second Example: Stars and Black Holes (General Relativity)
In this second example, we will briefly look at the description GR
gives us for the gravitational field of certain stars. We will also take a
look at one of the most widely publicized consequences of GR--black holes.
To make our discussion simpler, the types of stars we will be
considering will be spherically symmetric. What does that mean? Well,
consider an imaginary sphere with some radius. Place the center of that
sphere at the center of the star. If the star is spherically symmetric, then
the strength of the gravitational field everywhere on the surface of our
imaginary sphere will be exactly the same. For example, a star who's density
is spherically symmetric and which is not spinning would work.
Now, it will be helpful for us to discuss the space around the star in
terms of spherical coordinates; therefore, I should make sure the reader
knows what these coordinates are. Rather than using x, y, and z coordinates
for the three dimensional space around the star, we will use r, a, and b
coordinates, which I will define here. In Diagram 5-11 I have tried to draw
(in three dimensions) an z-y-z coordinate system, and I have marked a point
in space, *. There is a line segment drawn from the origin (o) to that
point, and the lengths of the x, y, and z components of the line segment are
the values for the x, y, and z coordinates of the point, *. These components
have been indicated on the diagram using "dotted" lines. Now, note that
there is one other dotted line which is not labeled. If you imagine a light
shining down on our line segment, then the unlabeled dotted line would be
the shadow that light produced on the x-y plane. It is called the projection
of the line segment on the x-y plane, but let's just call it "the x-y
component" for convenience.
Diagram 5-11
z
|
|
| *
| /'
| / '
|_a / 'z-comp
| \/r '
| / '
|/ '
o------'----- y
/ '. ' '
/__b/'. ' ' x-comp
/ '.''
/'''''''''''
x y-comp
Now we can define the r-a-b coordinates for the point, *. First, the
distance from the origin to the point (the length of the line segment) is
the "r" coordinate as indicated on the diagram. Next, the angle between the
z axes and the line segment is our "a" coordinate (though it is usually
denoted by the Greek letter "theta"). It too is indicated on the diagram.
Finally, there is the angle between x and the x-y component of the line
segment. That angle is our "b" coordinate (though it is usually denoted by
the Greek letter "phi"), and it is indicated on the diagram as well. Thus,
with r-a-b coordinates as defined here, we can specify any point in three
dimensional space.
As a final note about this coordinate system, we should look at the
metric of a flat 3-space using these coordinates. For an x-y-z system, the
metric is (of course) given by this invariant interval:
(Eq 5:52)
ds^2 = dx^2 + dy^2 + dz^2.
However, for our new coordinate system in the same flat 3-space, it is given
by the following:
(Eq 5:53)
ds^2 = dr^2 + r^2*da^2 + r^2*[sin(a)]^2*db^2.
For convenience, a new infinitesimal (call it du) is sometimes defined such
that:
(Eq 5:54)
du^2 = da^2 + [sin(a)]^2*db^2.
Then we can rewrite Equation 5:53 as
(Eq 5:55)
ds^2 = dr^2 + r^2*du^2.
We will therefore continue to use du throughout this discussion, but
remember it is just a convenient way to write the a and b components of the
invariant interval.
Next, let's look at some properties of the star we will be considering.
Basically, we will say it has a total mass of m(star) and a radius R. The
center of the star will be centered at the origin, o. Finally, we will only
be considering the gravitational field outside of the star itself. In
general, physicists are interested in the gravitational field inside the
star as well, but we will not worry about it that much.
We also want to define a new variable for mass using the Newtonian
gravitational constant G. In Newtonian gravitation, the force between two
objects of mass m1 and m2 which are a distance r apart is given by
(Eq 5:56)
F(Newtonian Gravity) = G * m1 * m2 / r^2
(where G = 6.672*10^-11 m^3/(s^2*kg) and we note that kg is the symbol for
kilogram). We will use G to define a new variable, M, such that
(Eq 5:57)
M = G*m(star)/c^2
Notice that M has the units of meters, and so M gives us a way of specifying
the mass of an object in units of meters (similar to the way w allows us to
specify time in units of meters). It is called the "geometrized" mass. So,
using M we can say that an object has a mass of 1 meter, and one can
decipher what mass we are talking about in terms of conventional units by
using Equation 5:57. As a note, a mass of M = 1 meter corresponds to
m(conventional) = 1.35E27 kg, the mass of the sun is M(sun) = 1477 meters
(1.989E30 kg), and the mass of the earth is M(earth) = 0.004435 meter
(5.973E24 kg).
Now, with this information in mind, the next step is to figure out what
the metric of the space-time around the star would be because of the
stress-energy tensor of the star. Generally, one uses the fact that we are
considering spherically symmetric stars in order to make some assumptions
about the form of the metric. One then uses this general form to calculate
the general form the stress-energy tensor would have. Finally, one uses what
we know physically about the star compared to the form of the stress-energy
tensor, and one can decipher what equations must have made up the metric in
the first place. In the end, one finds a metric for the space-time around
this type of star, and for our purposes, we will simply state that end
result. Thus, the metric is as follows (expressed in terms of the invariant
interval):
(Eq 5:58)
ds^2 = -(1 - 2*M/r)*dw^2 + [1/(1 - 2M/r)]*dr^2 + r^2 du^2
= g *dw^2 + g *dr^2 + g *du^2
ww rr uu
Note that we are using du as defined earlier, and we are using dw = c*dt as
our time component as discussed in the previous section. Also, we are using
M (as defined in Equation 5:57 ) to denote the mass of the star rather than
m(star). This metric is known as the Schwarzschild metric.
The next step, then, is to show that we can get useful physics by
considering this metric. We will again (as we did with the Special
Relativity discussion earlier) be looking at a particle of mass m, and here
we will be interested in its motion in the space-time around the star.
Because of the spherical symmetry of the space-time, the motion of such a
particle will remain within a plane, and we can orient our coordinate system
so that the plane is one where the angle "a" = 90 degrees (and sin(a) = 1).
Since the particle doesn't move out of that plane, there is never a change
in the angle "a" (da = 0). Thus, for this particle, we can consider the
metric as follows (putting sin(a) = 1 and da = 0 into Equation 5:58):
(Eq 5:59)
ds^2(particle's path)
= -(1 - 2*M/r)*dw^2 + [1/(1 - 2*M/r)]*dr^2 + r^2*db^2
= g *dw^2 + g *dr^2 + g *db^2
ww rr bb
In the interest of time (because we simply haven't been able to cover
everything we need to know about tensor analyses in this text), I will have
to simply state a couple of facts which we will use to produce the physics
we will look at. Namely, we notice that the form of the metric depends on
your particular position in r (because g_ww, g_rr, and g_bb are all
functions of r). However, none of the metric's components are functions of
w. Because of that, as it turns out, p_w (the covariant form of the time
component of the momentum four-vector) is constant throughout the motion of
the particle. The metric is also independent of the angle b. This, as it
turns out, implies that p_b is a constant. We can therefore define two
constants, E and L such that
(Eq 5:60) p = -E*m*c w
and
(Eq 5:61) p = L*m*c b
where m is the mass of the particle. These definitions will simplify the
equations we will produce below (and they are related to our usual concepts
of energy and angular momentum, so the fact that they are constant basically
say that energy and angular momentum are conserved as the particle moves).
Now, so far we have only defined the contravariant form of the
momentum, p^a. However, when we discussed the metric tensor we learned how
to use it to "raise" and "lower" indices. So, we can write the following
from Equation 5:22:
(Eq 5:62)
w ww wr wb wa
p = g *p + g *p + g *p + g *p
w r b a
Note that we are considering the case where the angle "a" is a constant so
that p^a = 0 in Equation 5:62. Also recall that in Equation 5:23 we noted
how to go from contravariant to covariant forms of the metric. For the
metrics we are discussing we thus have (note that the metric components come
from Equation 5:59).
(Eq 5:63)
ww 1 -1
g = --- = ---------
g 1 - 2*M/r
ww
rr 1
g = --- = 1 - 2M/r
g
rr
bb 1 1
g = --- = ---
g r^2
bb
all other covariant metric components = 0
Thus, only the p_w part remains in Equation 5:62 giving us the following
(note that I substitute using Equation 5:60):
(Eq 5:64)
w -1 1
p = ----------- * p = ----------- * E*m*c
(1 - 2*M/r) w (1 - 2*M/r)
Similarly we can find the equation for p^b:
(Eq 5:65)
b bb 1 1
p = g *p = --- * p = --- * L*m*c
b r^2 b r^2
Now, recall that in the last section we found that p(dot)p was a
constant, -(m*c)^2. That remains true here, so we find the following:
(Eq 5:66)
w w r r b b
p (dot) p = g *p *p + g *p *p + g *p *p = -(m*c)^2
ww rr bb
We can express each of the parts for that equation by substituting in the
metric components from Equation 5:59, using the above equations for p^w and
p^b, and writing p^r as m*c*dr/dW to get the following:
(Eq 5:67)
w w [ (E*m*c)^2 ]
g *p *p = -(1 - 2*M/r) * [-------------]
ww [(1 - 2*M/r)^2]
-E^2*(m*c)^2
= ------------
(1 - 2*M/r)
r r 1 [ dr] 2
g *p *p = ----------- * [m*--] (NOTE: dr/dW = c*dr/dT)
rr (1 - 2*M/r) [ dW]
(dr/dT)^2*(m*c)^2
= -----------------
(1 - 2*M/r)
b b (L*m*c)^2
g *p *p = r^2 * ---------
bb r^4
L^2*(m*c)^2
= -----------
r^2
Substitute this into Equation 5:66 and the (m*c)^2 portions will cancel out
on both sides giving this:
(Eq 5:68)
-E^2 (dr/dT)^2 L^2
-1 = ----------- + ----------- + -----
(1 - 2*M/r) (1 - 2*M/r) r^2
From this, we can find the following equation which describes the orbits the
particle can take. It is the equation of motion of the particle:
(Eq 5:69) (dr/dT)^2 = E^2 - (1 - 2*M/r)*(1 + L^2/r^2)
Now, it turns out that if one examine this equation for the case of a
circular orbit (where r is a constant and dr = 0) and for the case where the
mass is small or the orbit is large, we find things to be quite similar to
what Newtonian physics predicts. However, it is interesting to note that for
orbits for which r can change (elliptical orbits in Newtonian physics) GR
predicts something a bit different from Newtonian physics. Basically, in
Newtonian physics, the path of the particle in space is a true, closed
ellipse. However, with the above equation one finds that the "elliptical"
orbit in GR does not close in on itself. Instead, it's as if the ellipse
changes position as the particle's orbit goes on. We thus see a difference
in the predictions of the two theories, and we will mention this again in
the next section.
With this quick look at the physics one can derive using the metric for
such a star, we now want to go on and look at a very special case where this
metric comes into play. Consider for a moment what would happen if the
star's radius were to somehow become smaller than 2*M. Such a thing can
theoretically happen for certain stars at the end of their life cycle,
(though we won't get into how in our discussion).
So, consider the case where the radius of the star is smaller than 2*M.
We can then consider a point above the star for which r < 2*M. Now look back
at the metric of the star. If r < 2*M then g_tt becomes positive, while g_rr
becomes negative. That is to say that the time component of the invariant
interval will contribute to the interval in the same way that a space-like
coordinate did when r was greater than 2*M, and the radial component will
contribute in the same way as a time-like coordinate did when r was greater
than 2*M. Further, when r was greater than 2*M, we understood that all
particles followed a space-time path which took them "forward" in time.
Similarly, when g_rr becomes negative and d_tt becomes positive, (when r <
2*M) we find that all particles must continue along a space-time path for
which r continually decreases. In other words, the point r = 0 becomes part
of the "future" of every particle/observer for which r is less than 2*M.
Thus, such a particle will be doomed to fall in toward the center of the
star. One can then imagine that the star itself would be doomed to fall in
upon itself completely, becoming nothingness at r = 0.
This is known as a black hole (specifically, for the metric we are
considering, it is a spherically symmetric black hole), and the radius r=2*M
is called the Schwarzschild radius or the event horizon. Any observer with
an r coordinate less than 2*M must fall into the point r = 0. Note that at r
= 0 our metric becomes truly infinite, and as it turns out, that would be a
point where physical laws break down. Such a point is called a singularity.
We should also note that any signal (even a light signal) which the observer
tries to send outside of the event horizon must also fall into the
singularity (because all space-time geodesics for r < 2*M fall into the
singularity). Thus, there is no way to get any information from the
singularity to the "outside universe". There is no way for one to "see" the
singularity and its destruction of physical laws. In that sense, the
singularity's existence isn't a problem for our physical laws outside of the
event horizon.
As a last consideration about black holes, one might ask what would
happen to an observer who starts where his r coordinate is greater than 2*M
and then falls toward the event horizon. I won't go through the math, but
one finds that in our coordinates, the observer will take an infinite amount
of time to reach r = 2*M. However, if we ask about how much time the
observer himself reads on his watch as he falls (the proper time) we find
that in his coordinates, the time it takes for him to reach the event
horizon is finite. To try and understand how this can be, we will start by
considering the equation for p^w (the time component of the momentum
four-vector) as defined in Equation 5:46:
(Eq 5:70)
w dw
p = m*c*--
dW
However, if we look back at Equation 5:64, we can combine it with Equation
5:70 to find the following:
(Eq 5:71) dw E -- = ----------- dW (1 - 2*M/r)
Rewriting this, one finds that
(Eq 5:72)
(1 - 2*M/r)
dW = ----------- * dw.
E
So what does that tell us? Well, consider an observer at the coordinate
position r. If a small time ticks in our coordinate w = c*t, then the amount
of time which ticks on the observer's clock (dW = c*dT, where dT is the
proper time) depends on the r position of the observer. The smaller his r
position (as long as he is above the event horizon) the smaller dW will be
for a given dw. This is similar to time dilation in SR, but here it is
caused by the gravitational field and not by the relative motion of two
observers.
Applying this to our discussion of the observer falling towards the
event horizon, we find the following: In our coordinates (w) the clock of
the infalling observer (who is constantly falling to smaller and smaller r
values) takes longer and longer to tick its next tick. For example, let's
say that for the observer's clock, it ticks 10 ticks before it reaches the
event horizon. As we mentioned earlier, the coordinate time (w) will have to
become infinitely large before the observer will reach the horizon. However,
as the observer gets closer and closer to the event horizon, his clock takes
longer and longer to tick its next tick. Essentially, in our coordinate
system, the observer's clock will never be able to tick the 10th tick.
Meanwhile, for the observer, time goes on as usual. For him, therefore, the
10th tick will come, and he will enter the event horizon. However, once in
the horizon, he will not be able to send any signals out of the r = 2*M
event horizon (in our coordinates). Thus, no one with r greater than 2*M in
our coordinates will ever be able to see the infalling observer go into the
event horizon. This then explains how we can say that the infalling observer
never reaches the horizon according to our coordinate system.
As it was in SR, there are different explanations for how certain
outcomes come to be. The explanation depends on what coordinate system you
use to explain the occurrences (which means that it depends on your frame of
reference). The important point is that the end result of the explanations
agree with the each other as far as any physical laws can be applied. In the
twin paradox of SR, when the two twins come back together and stand next to
one another at the end of the trip, each explanation must agree as to which
twin is actually, physically older. For the question of whether an infalling
observer reaches the event horizon, regardless of which coordinate system we
use, we must agree that the observer is never seen to enter the horizon by
any observer outside of the event horizon. The fact that the infalling
observer "sees" himself enter the horizon has no physical consequences to
the outside world.
Thus, with spherically symmetric stars and black holes, we have found
the following: the metric of the surrounding space-time is given by the
following (using variables we have defined earlier):
(Eq 5:73--Copy of Eq 5:58)
ds^2 = -(1 - 2*M/r)*dw^2 + [1/(1 - 2M/r)]*dr^2 + r^2 du^2
= g *dw^2 + g *dr^2 + g *du^2
ww rr uu
Symmetries in this metric can be used along with the metric itself to find
the equations of motion for a particle which moves within this space-time.
Finally, the space-time has interesting consequences for the measurement of
space and time for observers at different points in the curved space-time
surrounding such stars and black holes.
That ends our look at some examples of the application of GR. The only
thing left in our discussion of this theory is to show some experimental
evidence for its existence, as we will do in the following section.
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