General Relativity - Virginia Commonwealth Universityrgowdy/vcu.pdf · General Relativity is the...

General RelativityEinstein�s Theory of Gravitation

Robert H. Gowdy Virginia Commonwealth University

March 2007

R. H. Gowdy (VCU) General Relativity 03/06 1 / 26

What is General Relativity?

General Relativity is the currently accepted theory of gravity.

It has passed every experimental test so far.It is the low energy limit of most proposed �theories of everything�.

General Relativity is the currently accepted theory of space and time.

It includes Special Relativity as a limiting case.It provides a detailed theory of measurement.It predicts black holes and gravitational waves.It provides a framework for describing the history of the universe.It does not play well with others (quantum theory in particular).

General Relativity is best expressed in the language of di¤erentialgeometry.

But we will make do without that.

What is General Relativity?

General Relativity is the currently accepted theory of gravity.

It has passed every experimental test so far.It is the low energy limit of most proposed �theories of everything�.

General Relativity is the currently accepted theory of space and time.

It includes Special Relativity as a limiting case.It provides a detailed theory of measurement.It predicts black holes and gravitational waves.It provides a framework for describing the history of the universe.It does not play well with others (quantum theory in particular).

General Relativity is best expressed in the language of di¤erentialgeometry.

But we will make do without that.

The Plan

A Reminder of Special Relativity

The Equivalence Principal

Curved Spacetime

Why rubber-sheet pictures are misleading

Einstein�s Field Equations

In vacuum: Directly from Newton�s inverse square law of gravity.With matter: Pressure attracts.

Describing Spacetime

Moving reference frames and the metric tensor

Solving the Field Equations

The spacetime around a spherical objectThe large-scale geometry of the universeRipples in spacetime

Testing the predictions of General Relativity

The Plan

Curved Spacetime

The Plan

Curved Spacetime

The Plan

Curved Spacetime

The Plan

Curved Spacetime

The Plan

Curved Spacetime

In vacuum: Directly from Newton�s inverse square law of gravity.

With matter: Pressure attracts.

The Plan

Curved Spacetime

The Plan

Curved Spacetime

The Plan

Curved Spacetime

The Plan

Curved Spacetime

The Plan

Curved Spacetime

The spacetime around a spherical object

The large-scale geometry of the universeRipples in spacetime

The Plan

Curved Spacetime

The spacetime around a spherical objectThe large-scale geometry of the universe

Ripples in spacetime

The Plan

Curved Spacetime

The Plan

Curved Spacetime

Spacetime

We live in a four dimensional geometry.

Three space dimensions and one time dimension.

What we think of as an object is represented by its history orworldline.

What we think of as the speed of an object is the slope of itsworldline.

The worldline of a free object is always straight. Its graph in a plot ofits Minkowski coordinates is a straight line.

The Equivalence Principal: Fictitious Forces

Drop an object inside an accelerating rocket and it will appear toaccelerate toward the �oor as if acted upon by an invisible downward force.

An outside observer will claim that this force is ��ctitious�because it isreally the �oor of the laboratory that accelerates up toward the object.

The Equivalence Principal: Gravity = Acceleration.

An observer inside a (su¢ ciently small) rocket cannot tell whether therocket is accelerating through empty space or at rest in a gravitational�eld as here.

The inertial reference frames are freely falling.R. H. Gowdy (VCU) General Relativity 03/06 6 / 26

Curvature: How Straight Lines Can Be Curved

These airplanes start out on parallel headings and each �y straightaccording to local maps.

Any map large enough to show their whole trips will have their pathscurving toward each other.

Curved Spacetime: Dust in a Falling Elevator

A cloud of dust particles is initially at rest relative to a freely fallinglaboratory. Their world-lines start out parallel but curve toward (and alsoaway from) each other, like the airplanes on the curved surface of theEarth.

Curved Spacetime: A common misunderstanding

The analogy between the airplane paths and curved spacetime has aproblem.

The airplane paths are drawn at an instant of time and re�ect thespace curvature of the Earth�s surface.

The dust particle worldlines, initially at rest, start out in the timedirection.

Pictures showing gravity as a result of curved space are misleading.

Space is indeed curved and that has e¤ects, but only for objectsmoving at high speed.

The Newtonian gravity that we mostly experience at low speeds isdue to curvature in the time direction.

Einstein�s Field Equations: What Newton�s Theory Says

The acceleration of a freely falling particle is given by the gradient ofa potential.

ai = �∂φ

∂x i

For the acceleration of a freely falling particle relative to a freelyfalling laboratory, expand the potential in a Taylor series around thecenter of mass of the lab and subtract out the gradient term.

∆ai = �∂

∂∆x i ∑r

∂2φ

∂x r ∂x s∆x r∆x s

�= �∑

∂2φ

∂x i∂x j∆x j

∆~a = ��∂2φ�

In vacuum, Newton�s inverse square law of gravity is equivalent to

Tr�∂2φ�= r2φ = 0

or, the tidal force matrix,�∂2φ�is trace-free.

ai = �∂φ

∂x i

∆ai = �∂

∂∆x i ∑r

∂2φ

�= �∑

∂2φ

∂x i∂x j∆x j

∆~a = ��∂2φ�

Tr�∂2φ�= r2φ = 0

ai = �∂φ

∂x i

∆ai = �∂

∂∆x i ∑r

∂2φ

�= �∑

∂2φ

∂x i∂x j∆x j

∆~a = ��∂2φ�

Tr�∂2φ�= r2φ = 0

Einstein�s Field Equations: In vacuum

Require that spacetime be curved in such a way that every small,freely falling laboratory will see a trace-free tidal force matrix.

In each laboratory, particles that move slowly in that lab obeyNewtonian gravity.

In each moving laboratory, the trace of the tidal force matrix is acomponent of the Ricci curvature tensor of spacetime.

For all laboratories to see vanishing trace tidal forces, the spacetimemust obey

Ricci = 0

There are actually ten equations here. They make up Einstein�sVacuum Field equations and follow directly and simply fromNewtonian Gravity and the Theory of Relativity.

Ricci = 0

Einstein�s Field Equations: When Matter is Present

Where there is a mass-energy density ρ the Newtonian potential obeys

r2φ = 4πGρ

A �rst guess at the equation that governs the curvature of spacetimewould be

Ricci = something involving mass-energy

= 4πG (stress-energy tensor)

Unfortunately the stress-energy tensor must obey anenergy-momentum conservation law, while the Ricci tensor must obeythe Bianchi Identity.

The Bianchi Identity and the conservation law are not compatible, sothe �rst guess does not work.The stress-energy needs a correction term that mixes mass-energydensity and pressure.

The local content of Newton�s theory of gravity has to be modi�ed.

r2φ = 4πGρ

The Bianchi Identity and the conservation law are not compatible, sothe �rst guess does not work.

The stress-energy needs a correction term that mixes mass-energydensity and pressure.

r2φ = 4πGρ

The local content of Newton�s theory of gravity has to be modi�ed.R. H. Gowdy (VCU) General Relativity 03/06 12 / 26

Einstein�s Field Equations: Pressure attracts (but usuallynot very much)

The simplest consistent coupling of matter to spacetime curvature is

Ricci = 8πG (trace-reversed stress-energy tensor)

The corresponding Newtonian �eld equation is

r2φ = 4πG�

ρ+ 3pc2

For ordinary pressures, the correction is very small because c2 is verybig.However, if an object becomes massive enough for its central pressureto dominate its gravity, a paradox can result:

Instead of preventing the object from collapsing, increasing centralpressure can accelerate the collapse.

That is one way to understand how a star can collapse to form ablack hole. No physical "sti¤ening" process can stop the collapsebecause increasing the central pressure just makes things worse.

r2φ = 4πG�

ρ+ 3pc2

�For ordinary pressures, the correction is very small because c2 is verybig.

However, if an object becomes massive enough for its central pressureto dominate its gravity, a paradox can result:

r2φ = 4πG�

ρ+ 3pc2

�For ordinary pressures, the correction is very small because c2 is verybig.However, if an object becomes massive enough for its central pressureto dominate its gravity, a paradox can result:

r2φ = 4πG�

ρ+ 3pc2

r2φ = 4πG�

ρ+ 3pc2

That is one way to understand how a star can collapse to form ablack hole. No physical "sti¤ening" process can stop the collapsebecause increasing the central pressure just makes things worse.R. H. Gowdy (VCU) General Relativity 03/06 13 / 26

Describing Spacetime: Coordinates

Label events by arbitrary coordinates:

An event P is labeled by the four numbers

x0 (P) , x1 (P) , x2 (P) , x3 (P)

These coordinates can cover an extensive region of spacetime, but maynot be as regular as we would like.The worldlines of freely falling objects usually look curved in thesecoordinates.

x0 (P) , x1 (P) , x2 (P) , x3 (P)

These coordinates can cover an extensive region of spacetime, but maynot be as regular as we would like.

The worldlines of freely falling objects usually look curved in thesecoordinates.

x0 (P) , x1 (P) , x2 (P) , x3 (P)

Describing Spacetime: Local Frames

Use the procedures of Special Relativity to de�ne regular Minkowskicoordinates, t, x , y , z near each event.

Near each event, we have a choice of Minkowski coordinates, whichare regular there, but not elsewhere, and arbitrary coordinates, whichlabel an extensive region but may not be regular anywhere.

Freely falling objects have worldlines that look straight in these localMinkowski frames.

Describing Spacetime: Frame-�elds

At each event, relate small changes in the Minkowski coordinatesconstructed from that event to small changes in the arbitrarycoordinates.0BB@

∆t∆x∆y∆z

1CCA =

0BB@f (0)0 f (0)1 f (0)2 f (0)3f (1)0 f (1)1 f (1)2 f (1)3f (2)0 f (2)1 f (2)2 f (2)3f (3)0 f (3)1 f (3)2 f (3)3

1CCA0BB@

At each event labeled by the arbitrary coordinates, there is a matrixof frame coe¢ cients

[f ] =

1CCAthat o¤ers direct access to all of the local laws of physics as they arestated in a Minkowski reference frame.

Describing Spacetime: Frame-�elds

At each event, relate small changes in the Minkowski coordinatesconstructed from that event to small changes in the arbitrarycoordinates.0BB@

∆t∆x∆y∆z

1CCA =

1CCA0BB@

1CCAAt each event labeled by the arbitrary coordinates, there is a matrixof frame coe¢ cients

[f ] =

1CCAthat o¤ers direct access to all of the local laws of physics as they arestated in a Minkowski reference frame.R. H. Gowdy (VCU) General Relativity 03/06 16 / 26

Describing Spacetime: The Metric Tensor

In Special relativity, the proper time interval between two events isrelated to the di¤erences in Minkowski coordinates by

(∆τ)2 = (∆t)2 � (∆x)2 � (∆y)2 � (∆z)2

Now we can express the proper time interval in terms of arbitrarycoordinate di¤erences

(∆τ)2 =3

∑α=0

∑β=0

f (0)αf (0)k∆xα∆x β �3

∑i=1

∑α=0

∑β=0

f (i )αf (i )β∆xα∆x β

∑α=0

∑β=0

f (0)αf (0)β �

∑i=1f (i )αf (i )β

!∆xα∆x β

The combinations of frame coe¢ cients that appear here are thespacetime metric tensor components:

�gαβ = f(0)

αf (0)β �3

∑i=1f (i )αf (i )β

(∆τ)2 = (∆t)2 � (∆x)2 � (∆y)2 � (∆z)2

(∆τ)2 =3

∑α=0

∑β=0

f (0)αf (0)k∆xα∆x β �3

∑i=1

∑α=0

∑β=0

∑α=0

∑β=0

f (0)αf (0)β �

∑i=1f (i )αf (i )β

!∆xα∆x β

�gαβ = f(0)

αf (0)β �3

∑i=1f (i )αf (i )β

(∆τ)2 = (∆t)2 � (∆x)2 � (∆y)2 � (∆z)2

(∆τ)2 =3

∑α=0

∑β=0

f (0)αf (0)k∆xα∆x β �3

∑i=1

∑α=0

∑β=0

∑α=0

∑β=0

f (0)αf (0)β �

∑i=1f (i )αf (i )β

!∆xα∆x β

�gαβ = f(0)

αf (0)β �3

∑i=1f (i )αf (i )β

Describing Spacetime: Spherical Coordinates

Here is Minkowski spacetime with a frame-�eld adapted to sphericalcoordinates.

Small changes in the frame coordinates, t, x , y , z , are related tochanges in the general coordinates by

dt = dt, dz = dr , dx = rdθ, dy = r sin θdφ

Describing Spacetime: Spherical Coordinates

Here is Minkowski spacetime with a frame-�eld adapted to sphericalcoordinates.

Small changes in the frame coordinates, t, x , y , z , are related tochanges in the general coordinates by

dt = dt, dz = dr , dx = rdθ, dy = r sin θdφ

Solving the Field Equations: Near a star (or a black hole)

Einstein published the correct �eld equations for spacetime in 1915.

They are a system of partial di¤erential equations for the metric tensorcomponents (or equivalently the frame coe¢ cients) as functions ofarbitrary coordinates.

Karl Schwarzschild found the general spherically symmetricsolution of the equations in 1916. Here it is in terms of the localMinkowski coordinates t, x , y , z .

r1� 2Gm

c2rd t, dz =

1q1� 2Gm

dx = rdθ, dy = r sin θdφ

Notice that something odd happens when r = 2Gmc2 .

That value is called the Schwarzschild Radius. If m is the mass of theEarth, it is about a centimeter.It took almost 50 years for its signi�cance to become clear.

Einstein published the correct �eld equations for spacetime in 1915.They are a system of partial di¤erential equations for the metric tensorcomponents (or equivalently the frame coe¢ cients) as functions ofarbitrary coordinates.

r1� 2Gm

c2rd t, dz =

1q1� 2Gm

r1� 2Gm

c2rd t, dz =

1q1� 2Gm

r1� 2Gm

c2rd t, dz =

1q1� 2Gm

r1� 2Gm

c2rd t, dz =

1q1� 2Gm

That value is called the Schwarzschild Radius. If m is the mass of theEarth, it is about a centimeter.

It took almost 50 years for its signi�cance to become clear.

r1� 2Gm

c2rd t, dz =

1q1� 2Gm

Solving the Field Equations: For the universe

The simplest model of the universe assumes that, on the largestscales, spacetime is

The same everywhere and in all directions.Has constant-time surfaces that are Euclidean.

This type of universe can be described by a global time coordinate tand global Cartesian coordinates x , y , z .A local Minkowski coordinate system t, x , y , z in such a universecould be connected to the global coordinates by the relations

dt = dt, dx = a (t) dx , dy = a (t) dy , dz = a (t) dz

Plug these frame coe¢ cients into Einstein�s �eld equations along withthe equations that govern the density and pressure of the mattercontent of the universe.

Get an ordinary di¤erential equation for the function a (t) thatindicates how the universe is expanding.

There are more complicated models with non-Euclidean constant-timesurfaces, but it is this simple Euclidean one that best �ts the data.

The same everywhere and in all directions.

Has constant-time surfaces that are Euclidean.

This type of universe can be described by a global time coordinate tand global Cartesian coordinates x , y , z .

A local Minkowski coordinate system t, x , y , z in such a universecould be connected to the global coordinates by the relations

There are more complicated models with non-Euclidean constant-timesurfaces, but it is this simple Euclidean one that best �ts the data.R. H. Gowdy (VCU) General Relativity 03/06 20 / 26

Solving the Field Equations: For gravitational waves

To represent small ripples in spacetime, use a matrix of framecoe¢ cients

[f ] = [1] + ε [h]

where ε is a parameter that ranges from 0 to 1.

Expand Einstein�s Field Equations in powers of ε and just keep the�rst order terms. The result is called the linearized theory.

Impose the coordinate condition that each extended coordinatefunction should obey the wave equation in the curved spacetime:

�x = �y = �z = �t

and �nd that the frame coe¢ cients then obey a wave equation

� [h] = 0

as well as some constraints.

[f ] = [1] + ε [h]

�x = �y = �z = �t

� [h] = 0

[f ] = [1] + ε [h]

�x = �y = �z = �t

� [h] = 0

Solve the constraints and the wave equation the way we always do bysubstituting

[h] = Re�[a] e i(

~k �~r�ωt)�

For a wave propagating in the z direction, there are two independentsolutions:

[h]+ = Re�[a+] e ik (z�ct)

�, [h]� = Re

�[a�] e ik (z�ct)

�where

[a+] =

0BB@0 0 0 00 a 0 00 0 �a 00 0 0 0

1CCA , [a�] =

0BB@0 0 0 00 0 a 00 a 0 00 0 0 0

Solve the constraints and the wave equation the way we always do bysubstituting

[h] = Re�[a] e i(

~k �~r�ωt)�

For a wave propagating in the z direction, there are two independentsolutions:

[h]+ = Re�[a+] e ik (z�ct)

�, [h]� = Re

�[a�] e ik (z�ct)

�where

[a+] =

0BB@0 0 0 00 a 0 00 0 �a 00 0 0 0

1CCA , [a�] =

0BB@0 0 0 00 0 a 00 a 0 00 0 0 0

Here are the changes in coordinates.

For both polarizations, nothing changes in the direction of propagation:

dt = dt, dz = dz

For the + polarization:

dx = (1+ εa) dx ,dy = (1� εa) dy

For the � polarization:

dx = dx + εady ,dy = dy + εadx

Here are the changes in coordinates.

For both polarizations, nothing changes in the direction of propagation:

dt = dt, dz = dz

For the + polarization:

dx = (1+ εa) dx ,dy = (1� εa) dy

For the � polarization:

dx = dx + εady ,dy = dy + εadx

Testing the Predictions of General Relativity

Solar System Tests (Testing the Schwarzschild metric)

Include light bending near the Sun, the Perihelion Precession ofMercuryPhase shifting of quasar radio signals past the SunLaser ranging to retrore�ectors on the MoonRadar ranging experiments and space probes

The Binary Pulsar (Taylor and Hulse 1993 Nobel Prize)

An accurate clock (pulsar) in close orbit around a collapsed objectTests the Schwarzschild metric in the nonlinear regimeProvided �rst observational evidence of gravitational waves

The Microwave Background Radiation

Direct observational evidence of Big Bang CosmologyHigh Precision measurements of the average matter content of theuniverseSupports the In�ation model of the very early universe (another story)Indicates the universe is mostly "dark energy," which is matter withnegative pressure.

Solar System Tests (Testing the Schwarzschild metric)Include light bending near the Sun, the Perihelion Precession ofMercury

Phase shifting of quasar radio signals past the SunLaser ranging to retrore�ectors on the MoonRadar ranging experiments and space probes

Solar System Tests (Testing the Schwarzschild metric)Include light bending near the Sun, the Perihelion Precession ofMercuryPhase shifting of quasar radio signals past the Sun

Laser ranging to retrore�ectors on the MoonRadar ranging experiments and space probes

Solar System Tests (Testing the Schwarzschild metric)Include light bending near the Sun, the Perihelion Precession ofMercuryPhase shifting of quasar radio signals past the SunLaser ranging to retrore�ectors on the Moon

Radar ranging experiments and space probes

Solar System Tests (Testing the Schwarzschild metric)Include light bending near the Sun, the Perihelion Precession ofMercuryPhase shifting of quasar radio signals past the SunLaser ranging to retrore�ectors on the MoonRadar ranging experiments and space probes

The Binary Pulsar (Taylor and Hulse 1993 Nobel Prize)An accurate clock (pulsar) in close orbit around a collapsed object

Tests the Schwarzschild metric in the nonlinear regimeProvided �rst observational evidence of gravitational waves

The Binary Pulsar (Taylor and Hulse 1993 Nobel Prize)An accurate clock (pulsar) in close orbit around a collapsed objectTests the Schwarzschild metric in the nonlinear regime

Provided �rst observational evidence of gravitational waves

The Binary Pulsar (Taylor and Hulse 1993 Nobel Prize)An accurate clock (pulsar) in close orbit around a collapsed objectTests the Schwarzschild metric in the nonlinear regimeProvided �rst observational evidence of gravitational waves

The Microwave Background RadiationDirect observational evidence of Big Bang Cosmology

High Precision measurements of the average matter content of theuniverseSupports the In�ation model of the very early universe (another story)Indicates the universe is mostly "dark energy," which is matter withnegative pressure.

The Microwave Background RadiationDirect observational evidence of Big Bang CosmologyHigh Precision measurements of the average matter content of theuniverse

Supports the In�ation model of the very early universe (another story)Indicates the universe is mostly "dark energy," which is matter withnegative pressure.

The Microwave Background RadiationDirect observational evidence of Big Bang CosmologyHigh Precision measurements of the average matter content of theuniverseSupports the In�ation model of the very early universe (another story)

Indicates the universe is mostly "dark energy," which is matter withnegative pressure.

The Microwave Background RadiationDirect observational evidence of Big Bang CosmologyHigh Precision measurements of the average matter content of theuniverseSupports the In�ation model of the very early universe (another story)Indicates the universe is mostly "dark energy," which is matter withnegative pressure.

Astronomical Phenomena

Hubble Expansion of the UniversePhenomena only explainable as black holesThe relative abundances of elementsModels of Neutron Stars and Supernovas

Hubble Expansion of the Universe

Phenomena only explainable as black holesThe relative abundances of elementsModels of Neutron Stars and Supernovas

Hubble Expansion of the UniversePhenomena only explainable as black holes

The relative abundances of elementsModels of Neutron Stars and Supernovas

Hubble Expansion of the UniversePhenomena only explainable as black holesThe relative abundances of elements

Models of Neutron Stars and Supernovas

Hubble Expansion of the UniversePhenomena only explainable as black holesThe relative abundances of elementsModels of Neutron Stars and Supernovas

The Search for Gravitational Radiation

Hanford, WA Livingston, LA GEO600, Hannover

VIRGO on the Arno AIGO in West Australia TAMA, Tokyo

General Relativity - Virginia Commonwealth Universityrgowdy/vcu.pdf · General Relativity is the...

Documents

Transcript of General Relativity - Virginia Commonwealth Universityrgowdy/vcu.pdf · General Relativity is the...