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CHAPTER 15General Relativity

- 15.1 Tenets of General Relativity
- 15.2 Tests of General Relativity
- 15.3 Gravitational Waves
- 15.4 Black Holes
- 15.5 Frame Dragging

An excellent introductory General Relativity text

Albert Einstein (1879-1955)

There is nothing in the world except empty,

curved space. Matter, charge, electromagnetism,

and other fields are only manifestations of the

curvature. - John Archibald Wheeler

Inertial Mass and Gravitational Mass

- Recall from Newtons 2nd Law that an object

accelerates in reaction to a force according to

its inertial mass, mi - Inertial mass measures how strongly an object

resists a change in its motion. - Gravitational mass, mg, measures how strongly it

attracts and is attracted by other objects - Equating the forces, we get a ratio of masses
- We always take the inertial and gravitational

masses to be equal. Einstein considered this

equivalence fundamental.

where

15.1 Tenets of General Relativity

- General relativity is the extension of special

relativity to non-inertial (accelerating) frames. - And because the effects of gravity and

acceleration prove to be indistinguishable, it

will also be a theory of gravity. - Its based on two concepts
- (1) the principle of equivalencethe extension

of Einsteins first postulate

of special relativity to the case of

non-inertial reference frames. - (2) the modeling of these effects as the

curvature of space- time due to

matter.

Principle of Equivalence

- The principle of equivalence is an experiment in

non-inertial reference frames. - Consider an astronaut sitting in a confined space

on a rocket at rest on Earth. The astronaut sits

on a scale that indicates a mass M. The astronaut

also drops a book that falls to the floor in the

usual manner.

Principle of Equivalence

- Now let the rocket accelerate through space,

where grav-ity is negligible. If the

acceleration is g, then the scale indicates the

same mass M that it did on Earth, and the book

still falls with the same acceleration as

measured on earth. - The question is How can the astronaut tell

whether the rocket is at rest on earth or

accelerating in space? - Principle of equivalence There is no experiment

that can be done in a small confined space that

can detect the difference between a uniform

gravitational field and an equivalent uniform

acceleration.

Light Deflection

- Consider accelerating where gravity is

negligible, with a window to allow a beam of

starlight to enter the spacecraft. Since the

velocity of light is finite, it takes time for

the light to reach the spaceships opposite wall.

- During this time, the rocket has accelerated

upward. Inside the rocket, the light path bends

downward. - The principle of equivalence implies that, at

rest on earth, light must also bend downward.

Curvature of Space and Geodesics

- Lights constant velocity in special relativity

implies that it travels in a straight line. Light

bending for the Earth observer seems to violate

this premise. - Einstein recognized that we need to expand the

notion of a straight line. - The shortest distance between two points on a

flat surface differs from

the shortest distance between points on a

sphere, which is curved. We shall expand our

definition of a straight line to mean the

minimum distance between two pointswhich may be

curved. Such a minimal distance on a curved

surface iscalled a Geodesic.

New York

Paris

Note that all paths are curved!

Curvature of Space

Thus if the space-time near a massive body is not

flat, then the straight-line path of light (and

other objects) near that body will appear curved.

The Unification of Mass and Space-time

- Einstein postulated that the mass of the Earth

creates a dimple on the space-time surface. In

other words, the mass changes the geometry of

space-time. - The geometry of space-time then tells matter how

to move. Gravity is no longer a force, but,

instead, a property (curvature) of space-time. - Einsteins famous field equations sum up this

relationship as

Matter tells space-time how to curve. Space-time

curvature tells matter how to move.

Recall the Space-time Interval and Metric

Recall the space-time interval, Ds.

Ds2 Dx2 Dy2 Dz2 c2Dt2

This interval can be written in terms of the

space-time metric

The space-time metric

We can rewrite the expression for the space-time

interval

where

and

This matrix will be the key quantity in general

relativity. And itll become more complicated

soon!

Summation notation

Notice that the summed indices occur once as

subscripts and again as superscripts

When the same index appears as a superscript and

a subscript, summation is assumed, and we can

omit the summation symbols. This is called

Einstein Summation Notation

In General Relativity, space is curved, and the

space-time metric can be more complex.

The more general metric coefficients of general

relativity (which may not be -1s, 0s, and 1s)

are denoted by gmn

Example An expanding (flat) universe

Ds2 a(t)2 Dx2 Dy2 Dz2 c2Dt2 where

a(t) t q

Values of q range from 1/2 (in a

radiation-dominated universe) to 2/3 (in a

matter-dominated universe).

The Mathematics of General Relativity Tensors

A Tensor is a function of one or more vectors

that yields a real number.

gmn is a Tensor. It takes two input vectors and

yields a number, the interval

Because gmn operates on two vectors, we say its

a tensor of Rank 2. Example a vector can undergo

dot products with other vectors to yield a

number, so its a tensor of Rank 1. Scalars have

Rank zero. The rank is also the number of indices

on the tensor and the dimension of the matrix

necessary to write it down.

The Mathematics of General Relativity

General Relativity distinguishes between vectors

and tensors that are covariant (with lower

indices) and contravariant (with upper indices).

To raise or lower an index, simply multiply by

the metric

Ordinarily, we dont usually have to worry about

this because our metric is simple, and covariant

and contravariant tensors are essentially the

same.

To raise the indices of the metric gmn itself,

just take its inverse

The Geodesic Equation

In Newtonian space, geodesics are straight lines,

and one way of saying this is that acceleration

is zero

where t is proper time, and xa is the position

vs. t of a test particle. In curved space, this

expression generalizes to

where is called a Christoffel symbol and

is given by

The Ricci Curvature

The curvature of space-time is complicated

because there are several dimensions, and the

curvature at each point can be different in each

dimension (including time). Think of a saddle in

two dimensions

The curvature of space-time is given by the Ricci

Tensor

The Einstein tensor, Gmn is a related measure of

the curvature.

The Einstein tensor can be written in terms of

the Ricci tensor

where R is the trace of the Ricci tensor.

Matters effect on space-time occurs through the

stress-energy tensor, T.

Ttt is the mass-energy density Txt , Tyt and Tzt

are how fast the matter is movingits

momentum Txx , Tyy and Tzz are the pressures in

each of the three directions Txy , Txz and Tyz

are the stresses in the matter

Einsteins Field Equations

This set of coupled nonlinear partial

differential equations (one for each element)

relates the curvature of space, Gmn, to the

energy-momentum tensor, Tmn

Only six component equations are independent.

and where G is the usual gravitational constant.

The goal is to solve for gmn, for all values of

m and n. In free space, where Tmn 0, this

reduces to

The correspondence principle Einsteins Field

Equations reduce to Newton's law of gravity in

the weak-field and slow-motion limit. In fact,

the above constant (8p G / c4) is determined this

way.

The spherically symmetrical case the

Schwarzschild Solution

Using spherical coordinates, r, q, f, and

spherical symmetry, we can solve Einsteins Field

Equations for the metric to find

The other elements of gmn are zero, and Note

that, when r 2GM/c2 (called the Schwarzchild

radius), this becomes

Whats going on?

15.4 Black Holes

Escape velocity and black holes

An object will escape from a massive body when

its kinetic energy equals or exceeds its

gravitational potential energy

Consider the case that the escape velocity is the

speed of light

Yeah, we probably shouldnt use the

non-relativistic expression for the kinetic

energy here

This can only occur when the mass M is crammed

into a radius r

The Schwarzschild radius!

Black Holes and Event Horizons

- When a stars thermonuclear fuel is depleted, no

heat is left to counteract the force of gravity,

which becomes dominant. The stars mass collapses

into an incredibly dense ball that could warp

space-time enough to not allow light to escape.

The point at the center is called a singularity.

A collapsing star greater than 3 solar masses

will distort space-time in this way to create a

black hole. Karl Schwarzschild determined the

radius of a black hole, known as the event

horizon

Hawking Radiation

Event horizon

- Due to quantum fluctuations, particle-antiparticle

pairs are created near the event horizon. One

particle falls into the singularity as the other

escapes. Antiparticles that escape radiate as

they annihilate with matter.

Black hole

Hawking calculated the blackbody temperature of

the black hole to be

k is Boltzmanns constant

The power radiated is

s is the Stefan-Boltzmann constant from blackbody

theory.

Black Hole Evaporation

- Small primordial black holes (10-19 solar masses)

could be detected by their Hawking radiation, but

it is negligible for other black holes. - Energy expended to pair production at the event

horizon decreases the total mass-energy of the

black hole, causing the black hole to slowly

evaporate with a lifetime

Stephen Hawking

n is a constant 1 / 15,360p

The smaller the mass the shorter the lifetime.

Solar-mass black holes would live much longer

than the age of the universe, but small black

holes (lt1014 g, about the size of a mountain)

would explode.

Black Hole Detection

- Since light cant escape, black holes must be

detected indirectly.

Mass falling into a black hole would emit x rays

Black Hole Candidates

- Several plausible candidates
- Cygnus X-1 is an x-ray emitter and part of a

binary system. Its roughly 7 solar masses. - The galactic center of M87 is 3 billion solar

masses. - That of NGC 4261 is a billion solar masses.

15.2 Tests of General Relativity

- Bending of Light
- A total solar eclipse allowed viewing starlight

passing close to the sun in 1919. It was bent,

causing the star to appear displaced. - Einsteins theory predicted a deflection of 1.75

seconds of arc, and two measurements found 1.98

0.16 and 1.61 0.40 seconds. - Many more experiments, using starlight and radio

waves from quasars, have confirmed Einsteins

predictions about the bending of light with

increasing accuracy.

Gravitational Lensing

When light from a distant (here blue) object like

a quasar passes by a nearby object on its way to

us on Earth, the light can be bent to give

multiple images as it passes in different

directions around the nearby object.

The Cosmic Horseshoe

Gravitational Red-shift

- Another test of general relativity is the

predicted frequency change of light near a

massive object. - A light pulse traveling vertically upward from

the surface of a massive body will gain potential

energy and lose kinetic energy, as with a rock

thrown straight up. - Since a light pulses energy depends on its

frequency n through E hn, as the light pulse

travels vertically, its frequency decreases. - This phenomenon is called gravitational red-shift.

Gravitational Red-shift Experiments

- An experiment conducted in a tall tower measured

the blue shift in frequency of a light pulse sent

down the tower. The energy gained when traveling

downward a distance H is mgH. If n is the

frequency of light at the top and n is the

frequency at the bottom, energy conservation

gives hn hn mgH. - The effective mass of light is m E / c2

h n / c2. - This yields the relative frequency shift
- Or in general
- Using gamma rays, the frequency ratio was

observed to be

in agreement with that predicted for the relevant

tower height, H.

Gravitational Time Dilation

- The frequency of a clock decreases near a massive

body, so a clock in a gravitational field runs

more slowly. - A very accurate experiment was done by comparing

the frequency of an atomic clock flown on a Scout

D rocket to an altitude of 10,000 km with the

frequency of a similar clock on the ground. The

measurement agreed with Einsteins general

relativity theory to within 0.02.

Perihelion Shift of Mercury

- The orbits of the planets are ellipses, and the

point closest to the sun is called the

perihelion. It has been known for hundreds of

years that Mercurys orbit precesses about the

sun. After accounting for the perturbations of

the other planets, 43 seconds of arc per century

remained unexplained by classical physics.

The curvature of space-time explained by general

relativity accounted for the 43 seconds of arc

shift in the orbit of Mercury.

Light Retardation

- The path of the light passing by a massive object

is longer due to the space-time curvature,

causing a delay for a light pulse traveling close

to the sun. - This effect was measured by sending a radar wave

to Venus, where it was reflected back to Earth

near the sun. It saw a delay of about 200 ms, in

excellent agreement with the general theory.

15.3 Gravitational Waves

- When a charge accelerates, the electric field

surrounding the charge redistributes itself. This

change in the electric field produces an

electromagnetic wave, which is easily detected.

In much the same way, an accelerated mass should

also produce gravitational waves. - Gravitational waves carry energy and momentum,

travel at the speed of light, and are

characterized by frequency and wavelength.

Gravitational waves

As gravitational waves pass through space-time,

they cause small ripples. The stretching and

shrinking is on the order of 1 part in 1021 even

due to a strong gravitational wave source.

Due to their small magnitude, gravitational waves

would be difficult to detect. Large astronomical

events could create measurable space-time waves

such as the collapse of a neutron star, a black

hole or the Big Bang. This effect has been

likened to noticing a single grain of sand added

to all the beaches of Long Island, New York.

Gravitational Wave Experiments

- Taylor and Hulse discovered a binary system of

two neutron stars that lose energy due to

gravitational waves that agrees with the

predictions of general relativity. - LIGO is a large Michelson interferometer device

that uses four test masses on two arms of the

interferometer. The device will detect changes

in length of the arms due to a passing wave.

NASA and the European Space Agency (ESA) are

jointly developing a space-based probe called the

Laser Interferometer Space Antenna (LISA) which

will measure fluctuations in its triangular shape.

15.5 Frame Dragging

- Josef Lense and Hans Thirring proposedin 1918

that a rotat-ing body can literally drag

space-time around with it as it rotates. This

effect is called frame dragging. - It was observed in 1997 by noticing fluctuating x

rays from several rotating black holes. The

objects were precessing from the space-time

dragging along with it. - Also, planes of satellite orbits shift 2 m per

year in the direction of the Earths rotationin

agreement with the predictions of the theory. - Global Positioning Systems (GPS) have to utilize

relativistic corrections for the precise atomic

clocks on the satellites.

Machs Principle

So what causes inertial mass, and what determines

which frames are inertial and which are not? Mach

speculated that inertial frames are those that

have constant velocity with respect to the fixed

stars, that is, the rest of the universe.

Ernst Mach (1838-1916)

Gödels Rotating Universe

Einstein acknowledged Machs ideas in his

writings on GR, but GR actually violates it! In

Gödels rotating universe, GR yields different

trajectories with respect to the fixed stars than

it does in a stationary universe.