General Relativity

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General Relativity
Topics Philosophical originsA new view of
gravityGravity as a tidal forceThe field
equationsConsequences of general relativityThe
twin paradox truly resolvedGeneral relativity
and cosmology
Motivation Learn about general relativity.
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Leading to general relativity

• In 1907, Einstein was asked to publish a summary
  of his theory of invariance (special relativity).
  He felt special relativity was consistent with
  all of physics except for gravity.
• Some examples of how gravity is not consistent
  with special relativity
• If we made the Sun disappear, Newtonian gravity
  predicts the Earth would instantly go careening
  into spacethis is information conveyed faster
  than light.Ultimately, this is due to the
  mysteries of action at a distance, which even
  Newton did not like.
• Gravity is an inverse square law force. But
  according to special relativity, spatial distance
  is a function of velocity (i.e., Lorentz
  contraction). How can gravity be a sensible force
  if it relied upon a frame-reliant parameter such
  as distance?

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Leading to general relativity

• It took Einstein six weeks of focused work to
  develop special relativity. He based it on a
  single postulate
• Postulate of special relativity
• Physical laws are the same for all observers in
  inertial reference frames
• It took him nine years to figure out how to
  incorporate gravity when he realized his key
  idea, der gluecklichste Gedanke meines Lebens
  (it was the happiest thought of my life)
• Einsteins happy thought
• If a person fell off the roof of a house, they
  would not be able to feel the effects of gravity
  during the fall.

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Leading to general relativity

• Einsteins gluecklichste Gedanke led him to build
  general relativity upon the equivalence
  principle, which results in a new way of thinking
  about gravity, and in turn, spacetime
• Equivalence Principle
• A uniformly accelerating reference frame is
  indistinguishable from a gravitational field.
• Let us see the equivalence principle in practice,
  with three examples

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Three examples of the equivalence principle

• Example 1 free fall is equivalent to
  weightlessness
• Suppose you followed Einsteins happy idea, and
  jumped off a roof.
• For a few seconds you would plummet downwards, as
  you offered no resistance to gravity.
• If you were holding a ball, you could let it go
  and it would float weightlessly.
• Accelerating in free fall conditions zero
  gravity

A free-fall condition is not a simulation of
weightlessness. It is exactly equivalent to
weightlessness.
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Three examples of the equivalence principle

• Example 2 acceleration gravity
• Suppose you were in a windowless rocket.
• You measure yourself with a scale and find that
  your weight is normal.
• There is no way for you to determine whether you
  are accelerating in deep space, or sitting on the
  surface of the Earth in 1g.
• Acceleration gravity

Acceleration does not simulate the affects of
gravity. It is exactly equivalent to gravity.
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Three examples of the equivalence principle

• Example 3 equivalent effects on a beam of light
• You are in a room and a horizontal beam of light
  enters through a little window. Instead of
  following a straight line, the light bends
  towards the ground before it strikes the far
  wall. There are two ways to explain this.
• A) Your room is accelerating upwards. As you
  accelerate upwards, the light beam bends
  downwards. The deflection is due to your
  acceleration.

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Three examples of the equivalence principle

• Example 3 equivalent effects on a beam of light
• B) Your room is stationary with respect to the
  beam, but a gravitational field is bending the
  light. The deflection is due to gravity.
• These two explanations are equivalent, and lead
  to a prediction of general relativity

Gravity bends beams of light.
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New interpretation of gravity

• In studying special relativity, we learned that
  if you enter a different constant-velocity
  reference frame, you can make an objects
  velocity disappear.
• Hence, velocity does not exist as something
  everyone can agree upon. It is a consequence of
  shifting reference frames.
• In general relativity, if you enter an
  accelerating reference frame, you can similarly
  affect gravityyou can even make gravitys
  affects disappear.
• Therefore, gravity is not an element of objective
  reality.
• Gravity can be considered to be a fictitious
  force.

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New interpretation of gravity

• Consider a car turning on a curved path. It would
  love to take the easy path and go in a straight
  line. Forcing it to follow the curved path puts
  the car in a strange frame of reference. In this
  frame of reference, a fictitious force appears.
• You call this fiction the centrifugal force.
• Similarly, the structure of space is not flat,
  like a carefully constructed framework.
• Space is warped. Objects moving in warped space
  follow warped paths. If you insist that objects
  should travel in straight lines, you will be
  surprised when they follow warped paths.
• You must create a fictitious force to account for
  this strange behavior.
• You call this fiction the gravitational force.

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New interpretation of gravity

• Why does an object traveling in a gravitational
  field follow a curved path?
• It is not following a curved pathit is moving in
  a straight line. The effect of gravity is to
  curve space.
• A straight line in curved space looks curved!

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Geodesics in non-flat space

• Straight lines in curved space are called
  geodesicsthe path that takes the least amount
  of time to traverse from point A to point B.
• Geodesics are not always easy to figure out if
  the curvature of space iscomplicated.

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New interpretation of gravity

• One can use Einsteins perspective of space to
  rewrite Newtons first law

LAW 1 (Newton) An object at rest stays at rest,
unless acted upon by an external, unbalanced
force. Similarly, an object in motion continues
in motion, unless acted upon by an external,
unbalanced force.
LAW 1 (Einstein) An object in warped space,
allowed to fall freely, will stay at rest as
viewed from the perspective within that freely
falling reference frame, unless acted upon by an
external, unbalanced force. Similarly, an object
in motion in warped space continues in motion in
a geodesic path, unless acted upon by an
external, unbalanced force.
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Letting gravity have its way

• Consider the advantages of this perspective.
• Imagine looking out a window, studying the
  trajectory of a ball shot out of a rocket.
• Now imagine you are in free fall, but are (oddly
  enough) holding a window frame, and looking
  through it as you fall.
• What do you observe?

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Letting gravity have its way

• In this downwards-accelerating reference frame,
  the ball follows a straight line trajectory, as
  if we were in an inertial frame. Consider five
  snapshots taken during the balls trajectory.

What you might think of as a force, i.e.,
gravity
Can be made to go away in a free-fall reference
frame.
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Effects of gravity v. 2.0

• Einstein then reasonedif gravity as we normally
  conceive it is fictitious, are there any effects
  from gravity that cannot be frame-shifted into
  non-existence?
• Consider two balls traveling in parallel paths in
  space. As they approach a planet, the two balls
  accelerate towards the planet because of gravity.
  
• Next, gravity pulls both towards the center of
  the planet, to what will eventually be a
  collision with each other (and the planet)
• Fine for Newton, but WRONG for Enstein. Try
  again, but from the perspective of general
  relativity!

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Effects of gravity v. 2.0

• Consider two balls traveling in parallel paths in
  space. As they approach a planet, the two balls
  accelerate towards the planet because of gravity.
  
• Next, gravity pulls both towards the center of
  the planet, to what will eventually be a
  collision with each other
• Next, their geodesic paths both lead towards the
  center of the planet, to what would eventually be
  a collision with each other
• The objects, moving at uniform speeds on geodesic
  paths (through warped space), merely happened to
  collide.

travel at constant speeds, following geodesic
paths in warped space.
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Effects of gravity v. 2.0

• What would the two balls see, in their natural
  free-fall frame of reference?
• They would observe that they slowly drift towards
  each other and then collide.
• Collisions are undeniable events, and are not
  merely a consequence of weird reference frames.
• Einstein interprets this is due to travellingon
  geodesics in curved space. No forcesneed be
  invoked.
• But any student of geometry will tell you that
  straight lines on a sphere will intersect.
• Balls moving on a sphere would collide.

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Effects of gravity v. 2.0

• Einstein proceeded by modeling whatever might
  cause the balls to come sliding together. His
  solution was to model the differential aspects of
  gravityhow strong gravity was at one place in
  space, compared to how strong it was at another,
  nearby location.
• We are familiar with differential aspect of
  gravity, which we call tidal forces.
• It was by treating tidal forces, which cannot be
  frame-shifted away, that Einstein developed
  general relativity.

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Minkowski and Einstein

• As an aside, there is an interesting historical
  storyline
• involving Einstein and Hermann Minkowski.
• In the 1890s (when Einstein was 20), he was
  aninfuriating student. Minkowski, his
  mathematics instructor at the ETH (Zurich),
  called him a lazy dog.
• In 1908, two years after Einstein published
  special relativity, Minkowski rewrote the theory
  in amathematically elegant and beautiful form.
• Einstein disliked Minkowskis formulation and he
  frequently made fun of it, saying it made
  relativity so difficult even physicists couldnt
  understand it.
• Four years later, Einstein ate his wordshe
  realized Minkowskis spacetime was an essential
  and fundamental component of building general
  relativity.
• He spent many painful years formulating his
  physics in curved spacetime.

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Einsteins basic field equationsthe math

• In their simplest form, the field equations of
• general relativity can be written as
• Gµ? is called the Einstein tensor, and describes
  the various curvature elements of space.
• Tµ? is the stress-energy tensor and describes the
  distribution of energy and matter.
• Tensors are matrices, so the equation above is
  actually a system of equations in this case, 10
  coupled nonlinear partial differential
  equations.
• Each separate equation is identified by the µ, ?
  index numbers.
• (Note often you will see systems where G and c
  are set to 1.)

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Einsteins basic field equationsthe math

• Let us step away from the details for a moment.
  The key revelation of this equation is that
  spacetime tells matter how to move matter tells
  spacetime how to curve (Wheeler, 1990).
• Changes in the distribution of matter and energy
  (Tµ?) change the structure of spacetime (Gµ?)
  those changes in spacetime ripple through
  spacetime at the speed of light.
• Matter responds to the spacetime that the matter
  sits in. Matter gets its marching orders
  locally.

Spacetime warpage (Gµ?) forces matter and energy
in the Universe to move on geodesics.
Matter and energy (Tµ?) in the Universe warps
spacetime.
There is no such thing as action at a
distance!Gravity is exposed as the fiction it is!
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Einsteins basic field equationsthe math

• In the next level of complexity, the Einstein
  tensor can be expanded
• Rµ? is the Ricci curvature tensor
• It describes how spacetime is different from
  Euclidean spacetime.
• gµ? is the metric tensor
• This is the key feature you are studying in
  general relativity. It determines the nature of
  space, and defines quantities such as distance,
  volume, curvature, angles, and how rapidly time
  flows.
• R is the scalar curvature
• This is a spatially varying value, similar in
  information to Rµ?.
• ? is the cosmological constant
• This is an arbitrary value that can be added to
  the equations.

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Einsteins basic field equationsthe math

• Einsteins equations are so complex, they can be
  solved only for certain highly simplified cases.
• For example, the first solution of Einsteins
  equations were made by Karl Schwarzschild for a
  perfectly symmetric, non-rotating star with a
  uniform interior.
• His second solution was for the stars interior
  only much later was it realized that this
  solution was for a non-rotating black hole.
• Even the case of gravity around an axially
  symmetric object becomes horribly complex.
• In the current era, numerical methods are used to
  solve equations that could never have been solved
  in the past.
• I vote we avoid the math, and keep the discussion
  generalOK?

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Four new implications of general relativity

• 1) Light escaping from a gravitational field
  becomes redshifted!
• Imagine you release a mass (m) from a tower on
  Earth. Starting at zero speed, the masss initial
  energy is E mc2.
• It falls a distance (h) in the gravitational
  field to a receiver on the ground. At the bottom,
  its energy is E mc2 mgh.
• The receiver converts the mass into a photon with
  the same total energy E(?) mc2 mgh.
• The photon is fired upwards, back to the top of
  the tower.
• The receiver at the top converts the photon back
  into matter. The blobs new mass is m E(?)/c2
  m mgh/c2.
• This is more massive than it was before!
• The only way around this gain of mass is if the
  photon loses energy on the flight upwards, out of
  the gravitational well!

m
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Four new implications of general relativity

• 2) Time runs slower in a gravitational field!
• It is easy to imagine that the structure of space
  is warped by the stress-energy tensor.
• But, as Minkowski demonstrated, space and time
  are intermixed. As space is warped, so is time.
• 10,000 km above the Earth, clocks run slightly
  faster than those at the Earths surface, by the
  difference of 4.5 parts per 1010.
• ? Earth clocks run slow by 1 sec every 70 years.
• In 1976, the Smithsonian Astrophysical
  Observatory shot a Scout rocket to 10,000 km
  during two hours of free fall it transmitted
  pulsed signals to the ground, the lengths of the
  time intervals (after corrected for special
  relativity) confirmed general relativity to
  within 0.01.

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Four new implications of general relativity

• 2) Time runs slower in a gravitational field!
• Global Positioning Systems must correct for
  relativity!
• GPS depends upon receiving signals from
  satellites the timing of the signals must be
  known to within 25 nanoseconds.
• But satellites are moving rapidly (14,000 km/hr)
  and are far from the Earth (about 27,000 km from
  the Earths center).
• In a single day, compared to a person on the
  ground, special relativity says that a satellite
  clock (moving rapidly) would lose 7 microseconds
  general relativity says a satellite clock (in
  comparatively low spacetime warpage) would gain
  45 microseconds.
• The difference (38 microseconds/day) means a GPS
  fix would be accurate to only a few kilometers if
  it did not correct for both special and general
  relativity.

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Four new implications of general relativity

• 3) Gravity bends the path of light
• For neophytes who do not know that light travels
  on geodesics, this is a surprise. We know better,
  and regularly use gravitational lenses as a tool
  in astronomy.
• 4) Rotating masses drag spacetime with them
• This effect is called frame-dragging. This means
  an object falling straight towards a rotating
  mass will start moving sideways before it reaches
  the surface. The resulting lateral fictitious
  force is called gravitomagnetism or
  gravitoelectromagnetism. It has been verified by
  a space mission called Gravity Probe B.

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Remember the Twin Paradox?

• In summary
• Rocket-Ronda and Spacestation-Sally are both 20
  years old. Rhonda flies at 0.8c to a point 8 LY
  away and then returns.
• Spacestation-Sallys PerspectiveRocket-Rhonda
  took 10 years (t 0.8 LY/0.8c) for each leg of
  the trip.
• Spacestation-Sally is therefore 40 years old when
  Rocket-Rhonda returns.
• Because of time dilation, Rocket-Rhonda will age
  only 20/? years (12 y) during her high velocity
  trip. When the two twins reunite, ones age is
  40, and the others is 32.

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Remember the Twin Paradox?

• Rocket-Rhondas Perspective
• Traveling at 0.8c, the distance to the
  turn-around point is Lorentz-contracted to 4.8 LY
  (i.e., 8/? LY). It takes 6 years (4.8 LY/0.8c)
  for each leg of her trip, so she is 32 years old
  (2066) upon her return (this agrees with
  Spacestation-Sallys perspective).
• But.in Rocket-Rhondas frame, Spacestation-Sally
  was the one who was moving.
• During Rocket-Rhonda 12-year trip, she observes
  Spacestation-Sallys to age only 7.2 years (12/?
  years).
• Upon their reunion, 32-year-old Rocket-Rhonda
  expects to find her sister to be only 207.2
  27.2 years old. Imagine her surprise, when she
  meets a woman who is 40!

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Remember the Twin Paradox?

• In our study of the paradox, I pointed out that
  Rocket-Rhonda accelerates at the turn-around
  point of her travel. Accelerating reference
  frames are not treated by special relativity.
• If Rocket-Rhonda filmed Spacestation-Sally for
  the entire trip, what would she see?
• 1) During the 6-yr outward bound trip,
  Spacestation-Sally ages only 3.6 yrs (6 yr/?).
• 2) At the turn-around point, Rocket-Rhonda
  decelerates hard, then she turns around and
  accelerates hard, again. Although this takes her
  but a few instants, her accelerating reference
  frame is equivalent to being in a huge
  gravitational field.Spacestation-Sally (and the
  rest of the Universe) ages by 12.8 yrs in the
  tiny amount of time it takes for Rocket-Rhonda to
  reverse direction!
• 3) During the 6-yr return trip,
  Spacestation-Sally ages only 3.6 yrs. When the
  two reunite, Spacestation-Sally is 40, and
  Rocket-Rhonda is 32.
• And that is the real resolution to the Twin
  Paradox!

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Proofs of general relativity

• Mercurys orbit
• Mercurys perihelion precesses over time. After
  all known sources of gravitational perturbations
  are removed, there is a tiny, residual
  0.431/year, taking 3 million years for the
  orbit to precess once. After a number of
  attempts, Einstein was able to explain this with
  general relativity in 1915.
• Deflection of light
• In 1915, Einstein predicted that starlight would
  be deflected as it passed by the Sun. In 1919,
  during a total solar eclipse of the sun, Sir
  Arthur Eddington was able to photograph the Sun
  and nearby stars. He observed a deflection of
  about 1.7.
• General relativity is tested on a daily basis
  gravitational time dilation, redshifts, lensing
  are all readily measured.

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General relativity and cosmology

• The Universe is highly uniform, with a simple
  stress-energy tensor permeating all of space. The
  field equations have been solved for the
  Universe, with different simplifying
  approximations.
• A few examples
• FriedmannLemaîtreRobertsonWalker (FLRW)
  solutions
• The standard model, developed in 1920-30. This
  can lead to the Friedmann equation
• G gravitational constant
• ? density of matter and energy
• H Hubble constant
• K curvature of space
• De Sitter
• A Universe empty of matter, and only driven to
  expand by dark energy (or its equivalent). Our
  Universe is heading in this direction.

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The curvature of the Universe

• The amount of energy and matter of the Universe,
  compared to how fast it is expanding, will
  determine if its density exceeds the critical
  density ?crit.
• The value of ?crit 10-29 g/cm3, or about 1 H
  atom/m3.
• Positive curvature (K gt 1), ? gt ?crit
• The Universe is finite
• The Universe will collapse upon itself.
• Negative curvature (K lt 1), ? lt ?crit
• The Universe is infinite
• The Universe will expand forever.
• Zero curvature (K 1), ? ?crit
• The Universe is infinite
• The Universe will expand forever, but just barely!

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The curvature of the Universe

• Positive curvature, ? gt ?crit
• The sum of interior angles in a triangle gt180º
• Parallel lines cross.
• Negative curvature, ? lt ?crit
• The sum of interior angles in a triangle lt180º
• Parallel lines diverge.
• Zero curvature, ? ?crit
• The sum of interior angles in a triangle 180º
• Parallel lines stay parallel.

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The cosmological constant

• Those who are familiar with calculus know that
  when integrating equations, you develop a
  constant of integration.
• The value of this constant must be determined by
  comparing your result to measurements.
• When Einstein solved his equations for simple
  cases, his equations predicted the Universe was
  expanding. He set a cosmological constant to
  counter this expansion.
• Cosmological redshifts were discovered by Hubble
  (1929), and Einstein concluded that ?0.
• We shall see, with dark energy, that the
  cosmological constant was revived.

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Extreme general relativity

• General relativity predicts some very weird
  things
• Black holes
• Gravitational radiation
• Extreme relativistic effects with binary pulsars
• Wormholes and warp bubbles
• More about these at the end of the semester!

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Appendix Measuring the curvature of the Universe

• We can measure the spatial curvature of the
  Universe directly by using a standard ruler. A
  standard ruler is an object of a calibrated size.
• Standard rulers are analogous to standard
  candles, such as Type Ia supernovae.
• Standard rulers are hard to come by, since
  everything large in the Universe tends to come in
  a range of physical sizes.
• Furthermore, since the curvature of the Universe
  is subtle (at most), you need a very large shape
  at a very large distance to detect the tiny
  amounts of curvature.

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Measuring the curvature of the Universe

• Suppose an object of size (L) is a distance (D)
  away its apparent angular size (?) would depend
  upon the curvature of space.
• If the Universe is flat, the geometry is simple
  tan? L/D.
• But if space is positively curved, tan? gt L/D.
• If space is negatively curved, tan? lt L/D.

L
?
D
?
?
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Measuring the curvature of the Universe

• Consider collapsing lumps of gas in the early Big
  Bang (i.e., before the recoupling, t380,000
  years).
• Very large lumps did not have time to collapse,
  meanwhile small lumps would have collapsed, then
  rebounded as they heated.
• Lumps of size L ct 380,000 LY would show the
  maximum amount of density enhancements. These
  would appear as the most common, most intense
  variations in the cosmic microwave background.
• This is a very good standard ruler, and at this
  distance, such lumps should be about 1º (flat),
  gt1º (positive curvature), lt1º (negative
  curvature).
• The lumps in the cosmic microwave background are
  1º.
• The overall warpage of the Universes spacetime
  is flat.

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