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					General Relativity
    David Berman
 Queen Mary College
 University of London
   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

             s  d t
               2         2       2
   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.
   Polar coordinates

    d  r  r 
     2      2    2      2       r

   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
   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.
   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.
   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
   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
   Pi is the ratio of the circumference to
    the diameter of a circle.
   Parallel lines never meet.
   In curved space:
   Parallel lines may meet in curved
   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.
   All the information about the curvature
    of the space is in how we add up
   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
   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                      2

 h(t , x, y, z )y  k (t , x, y, z )z
                   2                       2
          General Relativity
   What makes spacetime curve?

   Mass and energy make spacetime
   The more mass and energy the more
    the geometry of spacetime curves
    and is affected.
           General Relativity
   How do objects more on curved a
   They move in straight Lines.
   That is they move so as to minimise
    the distance travelled. That is the
    shortest distance in between two
   This is like the straight lines we had
    on a sphere they bend when
    compared to flat space.
           General Relativity
   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
   This moving along geodesics explains
    how things move in a gravitational
   Mass bends spacetime.
   Objects in curved spaces move on
    bent trajectories.
   Therefore objects with mass cause
    other things to move on curved
   This is a lot like gravitation.
           General Relativity
   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
   Light also follows geodesics.
           General Relativity
   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
   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
   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
Bending of spacetime
Bending of spacetime
Bending of spacetime
              Black holes
   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
   Black holes have been observed:
   Curving spacetime is something we
    also see in more ordinary

   GPS satellite positioning system has
    to correct for general relativistic
    effects or else it would be wrong by
    200 meters per day.
   Spacetime is one thing
   It can bend, its geometry can alter like
    the surface of a rubber sheet
   The bending is described mathematically
      s 2   f (t , x, y, z )t 2  g (t , x, y, z )x 2
       h(t , x, y, z )y  k (t , x, y, z )z
                          2                      2
   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.
   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.