### The Schwarzschild Metric

The thing is, the

Einstein Field Equations (the trowsers part of

General Relativity) are quite hard to solve. They simplify a great deal in a vacuum, and as such explain

gravitational radiation, but as soon as you start to introduce matter, things rapidly become very complicated.

Schwardschild's solution, is in a sense the simplest solution to Einstein's equations where metter is present: a (point) mass M is localed at the centre of the 'universe', with vacuum everywhere else. As

Dystopian Autocrat points out, the solution is spherically symmetric, and static. The resulting

metric tensor when presented in r, θ, φ, t coordinantes is:

((1-2M/r)^{-1} 0 0 0 )
( 0 r^{2} 0 0 )
( 0 0 r^{2}sin^{2}θ 0 )
( 0 0 0 -(1-2M/r))

(as can easily be read off from the form in which

95% fat free expresses them).

Remarkably, despite the simplicity of this solution, it explains most of the 'interesting' phenomena in GR.

### Consequneces:

# Black Holes

Firstly, notice that things mess up a little when

r=2M: this is called the Schwarzschild Singularity, although is not a space-time

singularity in the strictest sense, as it goes away when you change coordinates. That is, the (r, θ, φ, t) doesn't work very well here, in a similar way in which

longitude is not a good coordinate at the North Pole.

Well, it is a little more serious than that in fact: a particle observed (from outside) to fall towards the Schwarzschild Singularity will appear to take an

*infinite* time to reach it, whereas the particle itself will think that it did reach r=2M in a finite time. Something odd is definitely happening here, and the conclusion to be drawn is that time (t) is not a sensible coordinate to be dealing with in this region (there is a better coordinate system - see below).

Secondly, things mess up again when r=0. This time it is a genuine space-time singularity (does not dissapear in

*any* coordinate system). This is a singularity at the centre of a

black hole, and the surface r=2M surrounding it is its

event horizon.

This would seem to imply that every particle with mass is a (possibly very small) black hole, since it will have a positive Schwarzchild radius. Here we need to question the assumption of being a point mass - it is only valid to say a black hole exists if the mass M is contained entirely within the Schwarzschild radius, which is (fortunately) small enough to prevent this from happening with most everyday objects (even for the Sun, it is only a few kilometres).

# Particle motion

You can show that for a massive particle (having mass, not necessarily enormous!) moving around the black hole in this metric, it experiences the following effective

potential (I won't, as the derivation is long, and I think most most people would rather I cut to the chase):

, + B
x# # ##
+# # #+ #=
+# # # #,
+# # #+ #
+# # # #
+# # # =#
+# # x# #=
+# # # #
+# # # #
+# # # x#
+# # .# #+
+# # #x #
=+ 0 2M X # # r
=#################-#-#################################################################XxxxX#
=+ X # # -x#####
+# # # -# -X##########=.
+# # # # .=###########+,
+# # =# ##+ .-X############+. D
+# # #+ =###x=,
+# # # A C
+# # #
+# # #
+# # #
+# # #
=# # #
x# # -#
,- x ,x

Radial distance is measured on the x-axis. The picture gives a qualitive description of how the particle behaves at different distances from the hole - all you need bear in mind is that the particle seeks to

*minimise* its potential, and will move accordingly.

**A**: near the hole, the potential is steeply sloped, directing all matter in towards it.
**B**: there is a position of unstable equilibrium here, allowing a partile to orbit here providing that it is at *precicely* the right distance from the hole. Any small change will send it off to **A** or **C** regions. It is called the last stable circular orbit.
**C**: in this region, the potential has a minimum, allowing for stable equilibrium. Here, the particles take elliptical orbits about the hole or star (at this distance the metric makes a good approximation for that around a star), which are observed to precess (ie. the axis of the ellipse rotates through some angle each time the particle orbits). This explains the phenomenon of the precession of the prerhelion of Mercury, which had been observed by astronomers prior to Einstein's theory and was not adequately explained by Newtonian Gravity.
**D**: at larger distances, the potential agrees with that predicted by the Newtonian theory.

For particles without mass, eg.

photons, the potential takes a different form:

, +
x# #
+# # B
+# # ########
+# # #. ###
+# # # ###
+# # #, ;##-
+# # -# ###
+# # # ### C
+# # -# ###
+# # # .###.
+# # # ####
+# # #, ######- D
=+ X # ,X##########++===-;;,;
=########################x############################################xxxxxxxX##############
=+ 0 2M X # r
+# # #
+# # x#
+# # #
+# # #
+# # A ##
+# # #
+# # #
+# # #+
+# # #
=# # +#
x# # #
,- x .#

It is like that for the massive particles, except that there is not minimum allowing for stable orbits. What is interesting is that there is still a(n unstable) circular orbit near the hole - this means that if you were situated here, you could look out in some direction and see that back of your head (that is, if the

x-rays and

spaghettification had not already killed you, which they almost certainly would have).

In region

**C**, the potential is still varies appreciably, giving rise to photn deflection and

gravitational lensing.

# Wow

So, despite being only a simple solution to Einstein's equations, it provides a theory of gravity richer than Newton's!

## Q & A:

# What's a better coordinate to use near the Schwarzschild singularity?

Change from (r, θ, φ, t) to (r, θ, φ, u) defined by:

These are the

*Ingoing Eddington-Frinkelstein Coorinates*, and the metric becomes:

ds^{2} = (1-2M/r)du^{2} + dudr + r^{2}(dθ^{2} + sin^{2}θdφ^{2})

with no coordinate singularity near r=2M. Then there is the

Kruskal extension, which deserves its own write up.

# What's a metric got to do with particle motion?

Well, that's quite a central question to General Relativity in fact. The key principle is this: given a metric, you can measure distances along curves and freely falling particles move along curves of minimal length, called

geodesics.

So, given a form of the metric, you can produce an expression for the length of a curve, then use the

Euler-Lagrange equations to find out what the paths are which minimise the length. Photons follow different curves to massive particles because there is an extra condition on the normal to the curve: its norm measures the res mass of the particle concerned, and this gives rise to a different set of geodesics.