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					General Relativity
    David Berman
 Queen Mary College
 University of London
                   Geometry
   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
   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
   Polar coordinates

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


                            
                 Geometry
   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
   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
   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
   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
   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
   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
   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
    curve.
   The more mass and energy the more
    the geometry of spacetime curves
    and is affected.
           General Relativity
   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
   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
    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
   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
   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
    close.
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:
            Consequences
   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
   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 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
              Conclusions
   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
   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.