Exploring Geometry in General Relativity
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Presentation Transcript
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.
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
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:
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: One can work out how all the other geometric properties.
General Relativity • We saw in special relativity: • 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 • 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.
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 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:
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.
