Chapter 19.

Relativity and Black Holes

19.1. Problems With Newtonian Gravity

• Newtonian cosmology failed to develop a grand (encompassing) theory of gravity
• Serious defect in that gravity acted instantaneously everywhere, i.e., when apple falls to ground every place in the Universe receives information simultaneously
• Newtonian gravity ignores fact that speed of light is finite and information does not travel with infinite speed, i.e., inconsistent with special relativity
• Newtonian gravity overturns law of causality and allows effects to precede causes because of infinite speed of light
• Geometry of Newtonian space-time: Euclidean
  • Euclidean geometry, reasonable assumptions about nature of space
  • Parallel postulate: given straight line and point not on straight line, one and only one straight line equidistant from given straight line may be drawn through given point
  • Experience seems to suggest that these two lines remain equidistant (parallel) across all space even though we can not verify it
  • Plane triangle: sum of angles = 180^o
  • Pythagorean theorem: c² = a² + b²

19.2. Michelson-Morley Experiment, 1887

• Ether
  • Gravitational force (action-at-a-distance) transmitted by Descartes material medium, called ether, extending through out space
  • Maxwell's theory predicted light waves should travel at certain fixed speed, but Newton's theory of motion dispensed with concept of absolute standard of rest
  • Thus, light must travel at fixed speed relative to ether
  • Consequence: different observers, moving relative to ether, should see light coming toward them at different speeds
  • Consequence: speed of light measured in direction of Earth's motion should be larger than when measured in direction at right angles to Earth's motion
• Experiment
  • Split light beam so one part moved parallel Earth's motion and second part moved perpendicular
  • Paths for two beams same length
  • Measure difference in time over path
• Result
  • Times exactly the same
  • Did not depend on season of year
  • Bewildering to scientific community
• Einstein's resolution of problem was to abandon ether as unnecessary and also to abandon Newton's absolute time; theory of special relativity 1905
• Henri Poincaré (1854-1912); French mathematician; arrived at same solution a few weeks after and independent of Einstein

19.3. Theory of Special Relativity

• First postulates of special relativity
  • Laws of physics are the same (invariant mathematical form) for all observers in uniformly moving frames of reference no matter where in Universe.
• Second postulate of special relativity
  • Velocity of light is the same in all reference frames regardless of motion relative to each other, i.e., all observers no matter where in Universe measure same velocity for light.

19.4. Consequences of Special Relativity

• All motion is relative, i.e., no absolute standard of rest exists in Universe
• Nothing may travel faster than speed of light
• Mass is just one more manifestation of energy, i.e., E = mc² ,where E = energy, m = mass, c = velocity of light
• Space and time are determined both locally and globally by the Universe, in addition they are not independent of each other, but are combined together in concept of spacetime

  Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality. Hermann Minkowski, 1908

• Local consequence
  • Different observers (different reference frames) must agree on how fast light travels
  • Different observers must disagree on distance light traveled
  • Different observers must disagree on time taken for light travel
• Lorentz contraction
  • Consider reference frame moving relative to mine at large fraction of speed of light
  • Lengths appear contracted in direction of motion
  • Lengths appear unchanged at right angles to direction of motion
• Time dilation
  • In reference frame moving relative to mine at large fraction of speed of light
  • Time intervals seem shorter or time appears to slow down
• Spacetime
  • Intervals of space and intervals of time between events are no longer common properties
  • Intervals of spacetime and the speed of light are the things on which we all agree
  • Length of spacetime interval between any two events is the same for everybody
  • Two new invariants replacing Newtonian invariants of space and time are speed of light and the spacetime interval
  • All realms of spacetime are not accessible either to have come from there in the past or to go there in the future
• Geometry of spacetime
  • Euclidean geometry is geometry of Newtonian space and time
  • Pythagorean theorem for Newtonian space and time says that square of hypotenuse of a right triangle is equal to sum of squares of its two sides, i.e., c² = a² + b²
  • Geometry of spacetime: consider a right triangle made up of a space interval and a time interval whose hypotenuse is the spacetime interval; spacetime interval is distance between two events in spacetime
  • Pythagorean theorem in spacetime says that square of spacetime interval is equal to square of time interval minus square of space interval, even in different realms of spacetime, i.e., s² = t² - d²
• Consequence of spacetime form of Pythagorean theorem
  • What is distance in spacetime connect by a light ray?
  • Suppose light travels for 1 second in time, and therefore distance in space is 1 light-second
  • Hence spacetime interval equals: 0² = 1² - 1²
  • Thus spacetime interval for light is always zero
  • In spacetime curved world lines are always shorter than straight world lines
• Twin Paradox
  • Albert stays home, while identical twin Betty goes on a long journey and eventually returns to Earth
  • When Betty leaves, pulse of light is emitted from Earth, is reflected from distant mirror, and returns to Earth at moment of Betty's arrival
  • Spacetime interval for light pulse is zero, Betty's is intermediate between that for light pulse and Albert's
  • Time is measured along world line; amount of time that elapses between two events on a world line is length of world line between two events
  • Betty's age is same as Albert's when she leaves and is less than Albert's when she returns
  • Faster she goes relative to Albert, younger she is when she returns compared to Albert
  • If she could go at speed of light, she would not have aged at all compared to when she departed
• Einsteinian spacetime in the Universe
  • There is no space beyond the Universe, and there is neither time before nor after the Universe. Spacetime and its local and global features are properties of the Universe.

  The relativity theory arose from necessity, from serious and deep contradictions in the old theory [Newtonian theory] from which there seemed no escape. The strength of the new theory lies in the consistency and simplicity with which it solves all these difficulties, using only a few assumptions.... The old mechanics is valid for small velocities and forms the limiting case of the new one. Albert Einstein and Leopold Infeld

19.5. Geometry of Curved Spaces

• 1826, Karl Friedrich Gauss, German mathematician
  • Mathematics of curved surfaces
  • Showed that non-Euclidean spaces exist whose geometries do not conform to parallel postulate of Euclidean space in that an infinite number of lines can be drawn through given point parallel to given line
• 1854, Bernhard Riemann, German mathematician
  • Showed than non-Euclidean spaces exist whose geometries also do not admit parallel postulate in that no lines can be drawn through given point parallel to given line
  • Riemann foresaw intimate relationship between geometry and physics
  • "Only the genius of Riemann, solitary and uncomprehended, had won its way by the middle of the last century to a new concept of science." Albert Einstein
• 1885, William Clifford, English mathematician
  • Argued geometry and physics are interconnected
  • "We may conceive our space to have everywhere a nearly uniform curvature, but that slight variations of curvature may occur from point to point, and themselves to vary with time. These variations of the curvature with time may produce effects which we not unnaturally attribute to physical causes independent of the geometry of our space. We might even go so far as to assign to this variation of the curvature of space 'what really happens to that phenomenon which we term the motion of matter."
  • Predicted curvature waves (gravity waves) stating, "this property of being curved or distorted is continually being passed on from one region of space to another after the manner of a wave."
• Two-dimensional curved spaces
  • Flat space: example, flat two-dimensional plane
    • Possesses Euclidean geometry
    • Geodesic: segment of straight line
    • Obeys parallel postulate: equidistant straight lines remain equidistant toward infinity
    • Pythagorean theorem: c² = a² + b²
    • Sum of angles in triangle = 180^o
    • Circumference of circle = 2(pi)r
    • Area of circle = (pi)r²
  • Spherical space: example, surface of sphere
    • Possesses non-Euclidean spherical geometry
    • Geodesic: arc of great circle
    • Does not obey parallel postulate: equidistant
    • Small portion of surface may appear flat locally
    • Sum of angles in triangle > 180^o
    • Circumference of circle < 2(pi)r
    • Area of circle < (pi)r²
  • Hyperbolic space: example, saddle surface
    • Possesses non-Euclidean hyperbolic geometry
    • Geodesic: arc of hyperbola
    • Does not obey parallel postulate: equidistant lines diverge toward infinity
    • Small portion of surface may appear flat locally
    • Sum of angles in triangle < 180^o
    • Circumference of circle > 2(pi)r
    • Area of circle > (pi)r²

19.6. Theory of General Relativity

• General relativity devised by Einstein in 1916
• Applies to reference frames moving nonuniformly relative to each other, where special relative only applies to uniform motion
• Inertia and weight
  • Inertial mass: resistance body shows to change in its state of motion due to inertial force
  • Gravitational mass: resistance body shows to change in its state of motion due to gravitational force
  • Equality of inertial and gravitational mass long taken for granted
  • 1889, Baron von Eötvös, Hungarian physicist, showed experimentally inertial mass equals gravitational mass
  • Einstein argued that equality must mean that "the same quality of a body manifests itself according to circumstances as 'inertia' or as 'weight.' "
  • Consequence: impossible to distinguish between effects of inertial force and gravitational force on accelerated motion (principle of equivalence)
  • Inertial force: fictitious force resulting from nonuniform motion by observer's reference frame; example, merry-go-round, centrifugal force, apparent force that tends to move one toward rim
• Thus is Newtonian gravity also a "fictitious force?"
• General relativity is alternative theory of gravity since Newtonian gravity may be replaced by accelerations arising from curvature of spacetime produced by matter
• Moving bodies in spacetime are not attracted by action-at-a-distance force nor field of force, but follow shortest available path (geodesic) in response to local structure of curved spacetime
• Curved spacetime continuum is not permeated by a Newtonian gravitational field for it is itself a field and as such has field properties, such as curvature
• Flat rubber sheet model
  • Represents uncurved space in absence of gravity
  • Massive ball in middle produces depression which represents curvature of space by matter, such as star
  • Matter effects local as well as distant space curvature
  • Large curvature is where gravity is strong; small curvature is where gravity is weak
  • Small ball moving in vicinity of massive ball follows equivalent of straight line path (geodesic) for curved surface
• Einstein's equation of general relativity
  • Mathematical form is R_ij - 1/2g_ijR = kT_ij
  • 10 equations expressed in compact "tensor" notation
  • Left side deals with curvature of spacetime
  • Right side deals mainly with matter and energy
  • Simplistic interpretation: strain (curvature) of spacetime is proportional to stress of matter or energy
  • Principle of general relativity
  • Matter influences the curvature of spacetime, and curvature of spacetime influences the motion of matter.

19.7. Tests of General Relativity

• Precession of Mercury's orbit
  • Rotation of major axis of Mercury's orbit in direction of planet's revolution about Sun
  • Expected because Mercury is closest to Sun
  • Also most eccentric of planetary orbits
  • Primary effect due to gravitational attraction of other planets and predictable by Newtonian gravitational theory
  • Additional 43 seconds of arc per century observed but not predicted by Newtonian gravitational theory
  • Einstein's equations also account for effect by other planets as does Newton's
  • Einstein's equations contain an additional term not in Newton's that amounts to 43 seconds of arc per century
• Deflection of starlight
  • Ray of starlight passing close to edge of Sun follows curvature in near vicinity of Sun making ray appear to bend
  • Deflection is small fraction of second of arc from its original path
  • First measured by British expedition in May 1919; newspapers announced end of Newtonian era
  • Measured in almost every total eclipse since 1919
  • Also validated by deflection of radio waves from spacecraft and natural radio sources
• Gravitational redshift
  • Lorentz contraction and time dilation also occur in intense gravitational fields
  • Atom can be thought of as small clock
  • If clocks run slower in intense gravitational field, wavelength of electromagnetic radiation emitted by atoms shifted toward longer wavelengths or red end of spectrum - gravitational redshift
  • Has been measured for Sun, small massive stars, and in binary star systems
  • Agrees with Einstein prediction
• Gravitational radiation
  • Rubber sheet model: running finger (a mass) along sheet produces local moving curvature
  • Moving curvature is source of waves in spacetime continuum that propagate through space at speed of light - gravity waves
  • Attempts to measure on Earth not conclusive that effect has been observed
  • Binary star system: gravity waves carry energy out of system causing orbital period to decrease
  • Decreasing orbital periods for neutron-star binaries (pulsars) measured; reasonable agreement with predictions of general relativity

19.8. Explaining Other Forces With Geometry

• General relativity explains action of gravity by changing geometry (curving) of spacetime
• Geometrical explanation exist for other forces like electromagnetic force?
  • Einstein spent last 40 years trying to find ways in which general relativity theory could be extended to explain other forces
  • No success for Einstein
• Kaluza-Klein theory
  • Theodor Kaluza (1885-1954), German mathematician
  • 1922, represented electromagnetism as curvature of a 4th spatial dimension
  • Found comfortable co-existence of Einstein's general relativity equations and Maxwell's electromagnetic equations in 5 dimensional spacetime domain
  • Why don't we have evidence for 4th spatial dimension if it exists?
  • Oskar Klein (1894-1977), German mathematician, clarified Kaluza theory
  • 1926, hypothesis, don't see 4th spatial dimension since it is rolled up into small cylinder
  • Analogy, roll sheet of paper up into tight cylinder like a soda straw; from a distance cylinder looks like straight line
  • Klein hypothesis is that thickness of in 4th spatial dimension too small for us to perceive
  • Why limit to 5 dimensions? Why not many "rolled-up" dimensions?
  • Such has been proposed, but Kaluza-Klein theory not seen as promising direction

19.9. Black Holes

• Black hole (Wheeler 1969) - a star whose density is so great that spacetime is warped such that there are no exit paths only entrance paths, i.e., spacetime folds in over itself and isolates region from rest of spacetime
• Event horizon is boundary between what can escape warped spacetime and what can not; radius known as Schwarzschild radius; if mass is in solar masses, then Schwarzschild radius in kilometers is given by R(s) = 3M
• For non-rotating star collapsing to become black hole (Schwarzschild black hole), matter
  • Grows fainter and redder
  • Stretched in radial direction
  • Looses all identity except for mass, angular momentum and electric charge; lost are such identifying attributes as temperature, pressure, density, chemical composition, etc.
• For clock falling into black hole, clock runs slower as it approaches event horizon; stops on event horizon
• Does collapse continue inside event horizon until mass concentrated in zero volume, known as a singularity; do not know
• Rotating black hole (Kerr black hole)
  • Two event horizons joined at poles of rotation axis
  • Spacetime dragged around spinning black hole; observer outside of outer event horizon rotates relative to distant stars
  • Energy can be extracted from objects falling into rotating black holes; spin of black hole slows down
• Black holes found as member of binary star system
  • Visible companion usually blue supergiant
  • Visible companion transfers matter to unseen black hole companion
  • Matter forms large accretion disk around black hole; accretion disk several orders of magnitude larger than black hole
  • Accretion disk emits X-rays as matter slows down to enter event horizon
  • Best candidate for black hole is known as Cygnus X-1 (first X-ray source found in constellation of Cygnus); visible blue supergiant known as HDE 226868