General RelativityGeneral Relativity

David Berman David Berman

Queen Mary CollegeQueen Mary College

University of LondonUniversity of London

GeometryGeometry

In the previous lecture we saw that the In the previous lecture we saw that the important thing was to have an invariant important thing was to have an invariant quantity (the distance in spacetime).quantity (the distance in spacetime).

Remarkably the distance in spacetime Remarkably the distance in spacetime involves changing how we add up the involves changing how we add up the distance in space with the distance in distance in space with the distance in time.time.

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GeometryGeometry

Actually there are many ways we can Actually there are many ways we can add distance depending on the add distance depending on the coordinates that we use.coordinates that we use.

Consider using polar coordinatesConsider using polar coordinates r- radial distance from the origin and r- radial distance from the origin and

an angle say theta.an angle say theta.

GeometryGeometry

Polar coordinatesPolar coordinates

r

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GeometryGeometry

Suppose we restrict ourselves to the circle.Suppose we restrict ourselves to the circle. Distances on the circle would be given by Distances on the circle would be given by

theta only but the actual distance would be theta only but the actual distance would be given by:given by:

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GeometryGeometry

The point is, on a curved surface how The point is, on a curved surface how you measure distance may not be as you measure distance may not be as simple as we’ve seen so far.simple as we’ve seen so far.

There are many things that change There are many things that change once we are on a curved space.once we are on a curved space.

Imagine the surface of the earth. Imagine the surface of the earth.

GeometryGeometry

Changes in geometry:Changes in geometry:

To understand geometry we need to To understand geometry we need to understand what makes a straight line understand what makes a straight line on a curved space.on a curved space.

A straight line between two points is A straight line between two points is given by the shortest distance given by the shortest distance between those two points along the between those two points along the curved surface.curved surface.

GeometryGeometry

See how this can work on a curved See how this can work on a curved surface. On the surface of a sphere surface. On the surface of a sphere the shortest distance between two the shortest distance between two points always lies on a great circle.points always lies on a great circle.

This is what we mean by a straight This is what we mean by a straight line.line.

GeometryGeometry

How does geometry change when we How does geometry change when we are on a curved surface?are on a curved surface?

The things we are used to:The things we are used to: Angles of a triangle add up to 180 Angles of a triangle add up to 180

degrees.degrees. Pi is the ratio of the circumference to Pi is the ratio of the circumference to

the diameter of a circle.the diameter of a circle. Parallel lines never meet.Parallel lines never meet.

GeometryGeometry

In curved space:In curved space: Parallel lines may meet in curved Parallel lines may meet in curved

spacespace The angles of a triangle do not add The angles of a triangle do not add

up to 180 degrees.up to 180 degrees. The ratio of the circumference of a The ratio of the circumference of a

circle to its diameter is not Pi.circle to its diameter is not Pi.

GeometryGeometry

All the information about the curvature of All the information about the curvature of the space is in how we add up distances:the space is in how we add up distances:

Given:Given:

One can work out how all the other One can work out how all the other geometric properties.geometric properties.

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General RelativityGeneral Relativity

We saw in special relativity:We saw in special relativity:

We’ve seen that in curved spaces how We’ve seen that in curved spaces how you combined distances can change.you combined distances can change.

Can they change in spacetime?Can they change in spacetime?

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General RelativityGeneral Relativity

Spacetime can curve.Spacetime can curve.

It can bend and its geometry can It can bend and its geometry can change just as on a curved surface.change just as on a curved surface.

Spacetime distance will no longer be Spacetime distance will no longer be given by our favourite formula but by given by our favourite formula but by something more general.something more general.

General RelativityGeneral Relativity

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General RelativityGeneral Relativity

What makes spacetime curve?What makes spacetime curve?

Mass and energy make spacetime Mass and energy make spacetime curve.curve.

The more mass and energy the more The more mass and energy the more the geometry of spacetime curves the geometry of spacetime curves and is affected.and is affected.

General RelativityGeneral Relativity

How do objects more on curved a space.How do objects more on curved a space. They move in They move in straight Lines.straight Lines. That is they move so as to minimise the That is they move so as to minimise the

distance travelled. That is the shortest distance travelled. That is the shortest distance in between two points.distance in between two points.

This is like the This is like the straight linesstraight lines we had on a we had on a sphere they bend when compared to flat sphere they bend when compared to flat space.space.

General RelativityGeneral Relativity

How do we interpret this physically?How do we interpret this physically?

The shortest path between two The shortest path between two points is how any particle will move. points is how any particle will move. This is called a geodesic.This is called a geodesic.

Anything moving will follow a Anything moving will follow a geodesic path.geodesic path.

General RelativityGeneral Relativity

This moving along geodesics explains This moving along geodesics explains how things move in a gravitational field.how things move in a gravitational field.

Mass bends spacetime.Mass bends spacetime. Objects in curved spaces move on bent Objects in curved spaces move on bent

trajectories.trajectories. Therefore objects with mass cause other Therefore objects with mass cause other

things to move on curved trajectories.things to move on curved trajectories. This is a lot like gravitation. This is a lot like gravitation.

General RelativityGeneral Relativity

In fact it is gravitation.In fact it is gravitation. Einstein realised in 1915 that this is Einstein realised in 1915 that this is

what gravity is.what gravity is. Mass bends spacetime and objects Mass bends spacetime and objects

move in spacetime along geodesics.move in spacetime along geodesics. Thus mass effects how objects move Thus mass effects how objects move

though bending spacetime. That is though bending spacetime. That is gravity.gravity.

Light also follows geodesics.Light also follows geodesics.

General RelativityGeneral Relativity

Just like we had with special relativity Just like we had with special relativity where most the speeds we are used where most the speeds we are used to are small, most spacetime to are small, most spacetime curvatures are also small.curvatures are also small.

There are places where spacetime There are places where spacetime curvatures are large, near very curvatures are large, near very massive objects.massive objects.

These are black holes.These are black holes.

Black holesBlack holes

We have learned that light itself We have learned that light itself follows geodesics. It bends according follows geodesics. It bends according to the curvature of the spacetime.to the curvature of the spacetime.

There are regions where spacetime is There are regions where spacetime is so heavily bent that light itself so heavily bent that light itself cannot escape that is a black hole.cannot escape that is a black hole.

Bending of spacetimeBending of spacetime

Space and time distort near very Space and time distort near very heavy objects. The following heavy objects. The following animations show how this happens.animations show how this happens.

Speeds appear slower far away since Speeds appear slower far away since time appears to slow down far away time appears to slow down far away when compared to when you are when compared to when you are close. close.

Bending of spacetimeBending of spacetime

Bending of spacetimeBending of spacetime

Bending of spacetimeBending of spacetime

Black holesBlack holes

Black holes form when there is Black holes form when there is enough mass to collapse spacetime enough mass to collapse spacetime and prevent light from escaping.and prevent light from escaping.

This shows the spacetime bending as This shows the spacetime bending as a star collapses creating a a star collapses creating a gravitational field strong enough to gravitational field strong enough to trap light. trap light.

Black holesBlack holes

Black holesBlack holes

Black holes have been observed:Black holes have been observed:

ConsequencesConsequences

Curving spacetime is something we Curving spacetime is something we also see in more ordinary also see in more ordinary circumstances.circumstances.

GPS satellite positioning system has GPS satellite positioning system has to correct for general relativistic to correct for general relativistic effects or else it would be wrong by effects or else it would be wrong by 200 meters per day.200 meters per day.

ConclusionsConclusions

Spacetime is one thingSpacetime is one thing It can bend, its geometry can alter like It can bend, its geometry can alter like

the surface of a rubber sheetthe surface of a rubber sheet The bending is described mathematically The bending is described mathematically

by:by:

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ConclusionsConclusions

Once spacetime can bend we have to Once spacetime can bend we have to consider new geometries.consider new geometries.

Objects travel on geodesics in Objects travel on geodesics in spacetime that is the shortest path.spacetime that is the shortest path.

That is gravity.That is gravity. This can lead to things like black This can lead to things like black

holes where spacetime bends so holes where spacetime bends so much light can’t escape.much light can’t escape.

ConclusionsConclusions

Spacetime is a rich varied place Spacetime is a rich varied place where time and space bend in where time and space bend in beautiful and miraculous ways.beautiful and miraculous ways.

We must be amazed that we can We must be amazed that we can imagine so much that is distant from imagine so much that is distant from our usual everyday view of the world; our usual everyday view of the world; and it exists in the universe around and it exists in the universe around us.us.