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General Relativity

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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 time.

- 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

- 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:

- 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 line.

- 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.

- 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.

- All the information about the curvature of the space is in how we add up distances:
- Given:
One can work out how all the other geometric properties.

- We saw in special relativity:
- We’ve seen that in curved spaces how you combined distances can change.
- Can they change in spacetime?

- 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.

- 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.

- 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.

- 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.

- 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.

- 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.

- 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.

- 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.

- 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.

- 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 have been observed:

- 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.

- Spacetime is one thing
- It can bend, its geometry can alter like the surface of a rubber sheet
- The bending is described mathematically by:

- 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.