General Relativity
David Berman
Queen Mary College
University of London
Geometry
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In the previous lecture we saw that the
important thing was to have an invariant
quantity (the distance in spacetime).
Remarkably the distance in spacetime
involves changing how we add up the
distance in space with the distance in
time.
s  d t
2
2
2
Geometry
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Actually there are many ways we can
add distance depending on the
coordinates that we use.
Consider using polar coordinates
r- radial distance from the origin and
an angle say theta.
Geometry
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Polar coordinates
d  r  r 
2
2
2
2
r

r
Geometry
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
Suppose we restrict ourselves to the circle.
Distances on the circle would be given by
theta only but the actual distance would be
given by:
d  r 
2
2
2
Geometry
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The point is, on a curved surface how
you measure distance may not be as
simple as we’ve seen so far.
There are many things that change
once we are on a curved space.
Imagine the surface of the earth.
Geometry

Changes in geometry:
To understand geometry we need to
understand what makes a straight
line on a curved space.
A straight line between two points is
given by the shortest distance
between those two points along the
curved surface.
Geometry
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See how this can work on a curved
surface. On the surface of a sphere
the shortest distance between two
points always lies on a great circle.
This is what we mean by a straight
line.
Geometry
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How does geometry change when we
are on a curved surface?
The things we are used to:
Angles of a triangle add up to 180
degrees.
Pi is the ratio of the circumference to
the diameter of a circle.
Parallel lines never meet.
Geometry
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In curved space:
Parallel lines may meet in curved
space
The angles of a triangle do not add
up to 180 degrees.
The ratio of the circumference of a
circle to its diameter is not Pi.
Geometry
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All the information about the curvature
of the space is in how we add up
distances:
Given:
d  f ( x, y)x  g ( x, y)y
2
2
2
One can work out how all the other
geometric properties.
General Relativity

We saw in special relativity:
s  d t
2


2
2
We’ve seen that in curved spaces how
you combined distances can change.
Can they change in spacetime?
General Relativity
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Spacetime can curve.
It can bend and its geometry can
change just as on a curved surface.
Spacetime distance will no longer be
given by our favourite formula but by
something more general.
General Relativity
s   f (t , x, y, z )t  g (t , x, y, z )x
2
2
 h(t , x, y, z )y  k (t , x, y, z )z
2
2
2
General Relativity
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What makes spacetime curve?
Mass and energy make spacetime
curve.
The more mass and energy the more
the geometry of spacetime curves
and is affected.
General Relativity
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How do objects more on curved a
space.
They move in straight Lines.
That is they move so as to minimise
the distance travelled. That is the
shortest distance in between two
points.
This is like the straight lines we had
on a sphere they bend when
compared to flat space.
General Relativity
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How do we interpret this physically?
The shortest path between two
points is how any particle will move.
This is called a geodesic.
Anything moving will follow a
geodesic path.
General Relativity
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This moving along geodesics explains
how things move in a gravitational
field.
Mass bends spacetime.
Objects in curved spaces move on
bent trajectories.
Therefore objects with mass cause
other things to move on curved
trajectories.
This is a lot like gravitation.
General Relativity
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In fact it is gravitation.
Einstein realised in 1915 that this is
what gravity is.
Mass bends spacetime and objects
move in spacetime along geodesics.
Thus mass effects how objects move
though bending spacetime. That is
gravity.
Light also follows geodesics.
General Relativity
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Just like we had with special
relativity where most the speeds we
are used to are small, most
spacetime curvatures are also small.
There are places where spacetime
curvatures are large, near very
massive objects.
These are black holes.
Black holes
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We have learned that light itself
follows geodesics. It bends according
to the curvature of the spacetime.
There are regions where spacetime is
so heavily bent that light itself
cannot escape that is a black hole.
Bending of spacetime
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Space and time distort near very
heavy objects. The following
animations show how this happens.
Speeds appear slower far away since
time appears to slow down far away
when compared to when you are
close.
Bending of spacetime
Bending of spacetime
Bending of spacetime
Black holes
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Black holes form when there is
enough mass to collapse spacetime
and prevent light from escaping.
This shows the spacetime bending as
a star collapses creating a
gravitational field strong enough to
trap light.
Black holes
Black holes
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Black holes have been observed:
Consequences
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Curving spacetime is something we
also see in more ordinary
circumstances.
GPS satellite positioning system has
to correct for general relativistic
effects or else it would be wrong by
200 meters per day.
Conclusions
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Spacetime is one thing
It can bend, its geometry can alter like
the surface of a rubber sheet
The bending is described mathematically
by:
s   f (t , x, y, z )t  g (t , x, y, z )x
2
2
 h(t , x, y, z )y  k (t , x, y, z )z
2
2
2
Conclusions
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Once spacetime can bend we have to
consider new geometries.
Objects travel on geodesics in
spacetime that is the shortest path.
That is gravity.
This can lead to things like black
holes where spacetime bends so
much light can’t escape.
Conclusions
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Spacetime is a rich varied place
where time and space bend in
beautiful and miraculous ways.
We must be amazed that we can
imagine so much that is distant from
our usual everyday view of the
world; and it exists in the universe
around us.