Sciences
What is General Relativity?
What's the basic idea?

Both space and spacetime can either be curved
or flat. 
We've described the
Euclidean
(or EuclideanMesopotamian
:) metric
in two space dimensions:
dL^{2} = dX^{2}
+ dY^{2}
and we've discussed at some length the complications that arise with
the
addition of time to space to give the
Minkowski metric
(shown here in
just one space and one time dimension):
dS^{2} = c^{2}
dT^{2}  dX^{2}
What else can we do to our
spacetime distance function to make life more interesting (and
hopefully solve
the problem with Newtonian gravity discussed in the last section)?
What if we play around
with the form of the Minkowski metric? It turns out that if the
spacetime
metric is arranged in the right manner, we can get something
called
spacetime
curvature. And that is what the
General Theory of
Relativity is all
about.
For example, suppose we
add some extra space and time dependence to the Minkowski metric to
make a new
spacetime distance function
dS^{2} = g_{TT}(T,X)
c^{2} dT^{2}  g_{XX}(T,X)
dX^{2}
Using
differential
geometry, taking the right combination of first and second
derivatives of
g_{TT}(T,X)
and
g_{XX}(T,X), we could
calculate the what is called the
curvature
tensor R_{uv} for this choice of
spacetime distance function. The
subscripts on
R_{uv} are called
tensor indices and refer
back to the coordinates used in the above metric. The Minkowski metric
corresponds to the choice
g_{TT} = g_{XX}
= 1 and it has
R_{uv}
= 0 for all values of the tensor indices. This is why the
Minkowski metric
is known also as
flat spacetime  because the
spacetime curvature
calculated from this distance function is zero.
In Einstein's time they
were already learning about differential geometry, but Einstein
motivated this
field of mathematics even more when he came up with
an
equation relating the
curvature tensor of the spacetime distance function to the
distribution
of matter and energy in spacetime, encoded in a tensor
T_{uv}
called the stressenergy tensor.
This equation is now
called the
Einstein equation:
R_{uv}  (1/2) g_{uv}
R = (8 Pi G/c^{4}) T_{uv}

This equation (or
actually, set of equations, for there is an equation for every
combination of
tensor indices
u and
v) models
a lot of phenomena in the Universe
that was impossible to describe with mathematics just using Newton's
law of
gravity. For example, observations of the bending of light by gravity,
gravitational radiation emitted by pulsars, new observations of black
holes and
the observed expansion of the visible Universe can all be modelled
rather
successfully using this elegant formalism uncovered by Einstein.
What happens to light cones?
Most people have heard the
phrase
A straight line is the shortest distance
between two points.
But in
differential
geometry, they say this same thing in a different language.
They say instead
Geodesics for the Euclidean metric are
straight lines.
A
geodesic is a curve that represents the extreme
value of a distance
function in some space or spacetime.
Geodesics
are important in
the relativistic description of gravity. Einstein's
Principle
of Equivalence,
part of the General Theory of Relativity, tells us that
geodesics
represent
the paths of freelyfalling particles in a given spacetime.
(Freelyfalling
in this context means moving only under the influence of gravity, with
no other
forces involved.)

The shortest path between the two red
points
in the Poincare upper half plane is on a semicircle, not a straight
line. 
Space geodesics
When our
distance function
is the
Pythagorean Rule
dL^{2} = dX^{2}
+ dY^{2},
also known as the
Euclidean metric, straight lines
are the curves that
give the minimum Euclidean distance between two points.
In two
space dimensions
there are many metrics one can dream up in addition to the Euclidean
metric. For
example, take a class of metrics of the form:
dL^{2} = (k^{2}/Y^{2})
(dX^{2} + dY^{2})
For
Y>0,
this
distance function or metric is called the
Poincare
upper half plane.
The geodesics for this metric are described by the formula:
(X  X_{0})^{2}
+Y^{2} = k^{2}/h^{2}
The geodesics consist of
two types of curves: a)
semicricles of radius
k/h
centered at
X = X_{0} and in
the limit
h = 0,
vertical lines with
X = X_{0}.
The
shortest path between
any two points on the
Poincare upper half plane
is along one of
those two types of curves,
not along the straight line that
connects the two
points. This is shown in the figure above.

In flat spacetime in two space and
one time
dimensions, the light cones really do look like cones.
If we add the right kind of curvature, we can twist the light cones so
that they overlap as shown below. 

Spacetime geodesics
Things get
more
complicated when we graduate from
space to
spacetime.
Remember
that the
Minkowski metric has a spacetime distance
function
dS^{2}
that can be negative, positive or zero, whereas the distance functions
in space
dL^{2}
can only be positive.
The means
we have to
separate our geodesics on the basis of whether the distance function
dS^{2}
is positive, negative or zero. Goedesics with
dS^{2}
< 0 are
called
spacelike geodesics. Goedesics with
dS^{2}
= 0 are
called
null geodesics. Goedesics with
dS^{2}
> 0 are
called
timelike geodesics. The behavior of timelike
and null geodesics
are the most important for understanding time travel.
Timelike
geodesics behave
the opposite from geodesics in space. They actually represent the
longest
spacetime distance between two spacetime events.
In
Minkowski
spacetime,
all of the geodesics all straight lines, whether timelike, spacelike or
null.
The light cones are made of the null geodesics, and they rigidly
separate the
past from the future. In flat spacetime in two space and one time
dimensions,
the light cones really do look like cones, as shown in the top figure
to the
right.
But in
a
generic
curved
spacetime, the null geodesics won't as a rule be straight
lines, sometimes
they can be more interesting. The light cones made from null geodesics
from a
spacetime metric in a curved spacetime can even be made to have the
past and
future light cones overlap. An example of a spacetime satisfying the
Einstein
equations in three spacetime dimensions (two space and one time) where
the past
and future light cones overlap is shown in the figure on the bottom
right. We'll
see more of this spacetime later. See how it's twisted? Angular
momentum can
twist light cones and even make time travel possible in theory if not
in
practice.
How does that relate to Newton's model?
General
Relativity
didn't make
Newtonian gravity false. Newton's model
is still a good model
for most measurements and calculations of the motion of planets and
rockets and
so forth, and General Relativity must agree with Newton's model in that
limit.
Here is proof that these two very different models agree on the motion
of the
Earth around the Sun but disagree in the region where a new
relativistic
phenomenon  the black hole  is present in Einsetin's model but not in
Newton's.

Newtonian and relativistic
potentials are
almost the same at large distance scales like the radius of the Earth's
orbit R = 1.5 10^{13} cm. 
The Earth and Sun in the Newtonian model
In
Newton's model of
nature based on the math he invented, the
differential
calculus, the
primary equation by which motion of objects is calculated is
Force =
mass
x
acceleration
This gives a
set of secondorder differential equations
and when we solve
that system, we get the motion of the object in question when subjected
to the
force in question.
If the
object in question
is the
Earth and the force in question is the
gravitational
force of
the Sun on the Earth, then after writing the above equation
in spherical
coordinates, we get the following equation for the change in time of
the radial
coordinate
r for the position of the Earth:
m (dr/dt)^{2}
= 2 (E 
V_{N}(r))
V_{N}(r) =  M m G/r + m (L/m)^{2}/(2
r^{2})
G is Newton's gravitational
constant,
M
stands for the mass of the
Sun and
m is the (approximately) the mass of the
Earth. Because the
Newtonian force law for gravity depends only on radial distance, not on
time or
angle, there are two
constants of motion called
angular momentum
L
and energy
E.
The
function
V(r)
is plotted above for the Earth/Sun system. Notice that whenever
V(r)
= E,
the radial coordinate stops changing because
dr/dt = 0.
These are called
turning
points. The
turning points classify the
type
of orbit, as
shown in the above figure.
Notice
that it is the
L^{2}/r^{2}
term in the potential that causes the turning points for smaller
r.
Angular
momentum acts almost like a force of repulsion to counter gravitational
attraction. That will be important later as an agent of
causality violation
in the relativistic model.

Newtonian and relativistic
potentials begin to
disagree very close to the center of attraction. 
The Earth and Sun in the relativistic model
In
General
Relativity,
the first ingredient in modelmaking is the
spacetime metric.
This metric
must solve the
Einstein equation that relate the
spacetime curvature to
the matter and energy present.
If the
matter and energy
present is idealized by a
pointlike object of mass M,
then the spacetime
metric that solves the Einstein equation is called the
Schwarzschild
metric.
We won't
display that
metric here for the sake of brevity. The
timelike geodesics
are easy to
calculate for the Schwarzschild metric. We wind up with an equation
that looks a
lot like the Newtonian formula above:
m (dr/dt)^{2}
= 2 (E 
V_{R}(r))
V_{R}(r) =  M m G/r + m (L/m)^{2}/(2
r^{2})  M G m (L/m)^{2}/(2 c^{2}
r^{3})
At large
distances from
the Sun, the last term in the potential
V_{R}(r)
can be
neglected, and the classification of orbits is the same as for the
Newtonian
case.
But if the
Sun really were
a pointlike mass, the last term in the potential
V_{R}(r)
would
give us trouble. Notice in the bottom figure the red line, which
represents the
relativistic potential, veers down to become infinitely negative,
instead of
infinitely positive as with the Newtonian potential. This behavior is
the signal
of a
black hole. We can't go into that in depth
right here, but you can
read about it in books specializing in black holes.
The Sun is
not pointlike,
the mass of the sun is spread within a radius of about
700,000
kilometers.
The black hole behavior would not set in unless the mass of the Sun
were
confined to a within radius of about
three kilometers.