15.1Tenets of General Relativity

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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
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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
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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.

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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.

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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.

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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.

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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!
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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.
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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.
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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
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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!
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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
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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).
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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.
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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
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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
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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
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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.
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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
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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.
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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?
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15.4 Black Holes
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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!
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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
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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.
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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.
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Black Hole Detection

• Since light cant escape, black holes must be
  detected indirectly.

Mass falling into a black hole would emit x rays
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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.

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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.

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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
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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.

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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.
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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.

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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.
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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.

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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.

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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.
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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.
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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.

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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)
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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.