ASTRONOMY 9: HISTORY OF COSMOLOGY

Handout #20

J. E. Baker
UC Berkeley, Spring 2000

General Relativity (1915)

A Brief History of Gravity
- Aristotle: natural tendency of motion towards/away from center of Earth
- Kepler: a kind of magnetism
- Descartes: cosmic vortices
- Newton: mysterious instantaneous action at a distance, F=GMm/d²
Einstein fits gravity into the spacetime framework of relativity
General theory of relativity
The Principle of Equivalence (1907)
- Imagine two situations:
  
  1.
  You are standing in a stationary elevator on Earth. All falling objects accelerate downward at $1g\approx 10$ m/s².
  2.
  The same elevator is floating through space with no gravity, but the elevator is being accelerated upward at 1g.
- Einstein's equivalence principle: there is no experiment you can do inside the elevator to distinguish these two situations! (Note: we are ignoring tidal forces, more on this later.)
- Special relativity works in freely falling reference frames--inertial frames are accelerating
- By transforming to freely falling frame, can get rid of gravity!
- This expresses the equivalence of inertial mass m_i and gravitaional mass m_g:
  - Newton's law of gravity: F = GMm_g/d² = m_g g
  - Newton's second law: F=m_i a
  - Contrast with electric force $F \propto q_1 q_2/d^2$ : m_g is a gravitational ``charge'' analogous to q
  - Electric charge has nothing to do with inertial mass; why then is m_i=m_g? No experiment has ever measured $m_i\neq m_g$ .
  - In Newtonian physics, this was an unexplained mystery.
  - In general relativity, it implies that gravitation is simply bent spacetime!
Curvature of Spacetime
- Imagine two elevators floating through empty space
  1.
  Moving with constant velocity
  - Inertial frame, thrown ball follows straight-line path
  2.
  Accelerating upward
  - Non-inertial frame
  - Ball thrown horizontally at speed v follows curved (parabolic) trajectory
  - Imagine v=c: light beam also follows curved path!
- By principle of equivalence, (2) is exactly the same as gravity $\Rightarrow$ gravity bends light!
- Or: gravity curves space and light follows the shortest path (``geodesic'') in the curved space
- Gravity also curves time!
  - Imagine clocks at center and edge of a disk rotating at high speed
  - By special relativity, clock at edge ticks slower
  - But this is an accelerated frame, again equivalent to gravity
  - So clocks in a gravitational field run slower
- Trampoline analogy for curved space
  - Imagine a 2-d plane universe (would have to think in 4-d to visualize curved 3-d space!)
  - Soccer ball follows straight-line motion on flat trampoline
  - If you're standing on the trampoline, the surface is curved
  - Soccer ball naturally follows curved path, could orbit you (if no friction)
  - Einstein's interpretation: Earth orbits Sun not because of force of gravity, but because it is following shortest path in curved space!
- Tidal forces
  - A uniform gravitational field would be just like tilting the trampoline with no one standing on it: space is still flat, but everything accelerates in one direction
  - Gravity is directed towards the center of massive objects, giving rise to tidal forces
  - Objects on sides of elevator accelerate towards each other a little, since they're really accelerating towards Earth's center (not exactly ``down'')
  - Usually small, but can be large (eg, falling into black hole)
  - Curvature of space (departure from special relativity) really represents the tidal forces
- The Equations of G.R.
  - Equation of the universe: $G^{\mu\nu} = 8\pi T^{\mu\nu}$
  - Einstein: ``I never realized that so many Americans were interested in tensor analysis.''
  - $G^{\mu\nu}$ : geometry, curvature of 4-d spacetime
  - $T^{\mu\nu}$ : description of matter/energy densities, pressures, stresses, ...
  - Matter tells space how to curve, and space tells matter how to move
- Mach's Principle
  - Are you rotating, or is the universe rotating around you?
  - Who decides what the inertial frames are?
  - Ernst Mach's answer: the overall distribution of matter in the universe
  - Mach's principle is mostly, but not fully, embodied within G.R.:
    - Spacetime curvature can exist independent of any mass (eg, gravitational waves)
    - Solutions to Einstein's equations for the geometry are not always unique for a given mass distribution
Tests of the theory
- Not easy, since differences between Newton and Einstein only become large for very strong gravity (like near a black hole)
1.
Arthur Eddington (1919) famously observed bending of starlight around the Sun during a solar eclipse
- Deflection of only 1.75 arcseconds!
- Einstein becomes popular hero
2.
Shift in perihelion (closest point to Sun) of Mercury's orbit
- Newtonian gravity predicts 5514 arcseconds per century
- Observed 5557 arcsec/century, 43 too much!
- Agrees well with Einstein's GR prediction
3.
Gravitational waves
- Propagate at speed c, no more Newtonian action at a distance!
- Like ripples on a pond, fluctuations in curvature of spacetime
- Still building large experiments which may detect them directly (LIGO)
- Joseph Taylor (1993 Nobel Prize work): already detected indirectly because they carry energy away from the orbits of binary pulsars (mass of Sun but radius of S.F.!)
4.
Gravitational redshift
- Clocks in stronger fields tick more slowly
- So distant observer measures lower frequency for radiation coming out from inside a gravitational field, so longer wavelength (redshift)
- Easily observed using spectra of white dwarf stars (mass of Sun but radius of Earth)
Black Holes
- Gravitational fields get stronger as objects become more massive and smaller
- We know gravity deflects light; can imagine making an object so dense that not even light can escape! This is a black hole
- Schwarzschild showed that this happens if an object is smaller than R_S = 2GM/c²
- For the Earth this is only 1 cm; for the Sun, about 3 km!
- Nothing can travel faster than light, so nothing inside the event horizon (E.H.) can have any causal influence on the outside
- For non-rotating black hole, event horizon is a sphere at radius R_S
- Light cones tilt sideways as you approach the black hole: space and time are interchanged!
- Observer on outside sees you fall slower and slower as you go in, infinite time to cross E.H.
- Black holes have almost certainly been discovered
  - Large ones (10⁶- $10^9\times$ mass of Sun) sit at the centers of many galaxies
  - Smaller ones (1-10 $\times$ mass of Sun) result from deaths of massive stars in supernova explosions
  - Can't be seen directly, but matter falling in gets heated to very high temperature and emits high-energy radiation before crossing E.H.
- Wormholes might form, could they be used to zip across the universe?
  - Probably not, they seem to be unstable and trajectories through them are spacelike (requires faster than light speed)
  - Might be able to hold one open and go through, but only if negative mass/energy exists (pretty weird!)
Global geometry
- G.R. can describe not just local spacetime curvature (eg, due to Sun or galaxy), but global curvature of the whole universe!
- Fifth postulate of Euclidean (ancient Greek) geometry:
  - Given a line and a point not on the line, one and only one line can be drawn through the point so that it is parallel to the line
- Some theorems: angles of triangle add to 180 $^\circ$ , circumference of circle is $2\pi r$
- If we modify the 5th postulate (but keep the other 4), we get interesting new curved geometries
- Studied in late 1800s by Riemann and others
- Three basic possibilities for a homogeneous universe:
  1.
  Accept Euclid's 5th postulate
  - Space is ``flat'', ordinary geometry works
  2.
  Replace ``one and only one line'' with ``no lines''!
  - Example: surface of the Earth (again, have to think of 2-d universes in order to visualize curved space)
  - ``Straight lines'' (shortest distance between two points) become great circles (lines of longitude, but not latitude)
  - Different great circles always intersect twice, no parallel lines!
  - Sum of triangle angles is $>180^\circ$ , $C<2\pi r$
  - Universe has positive curvature
  3.
  Replace ``one and only one line'' with ``infinitely many lines''!
  - Example: saddle (not very good analogy, only works near saddle point)
  - Straight lines diverge, so can have infinite number of parallels
  - Sum of angles is $<180^\circ$ , $C>2\pi r$
  - Universe has negative curvature
- Note: we visualize these geometries by embedding 2-d space in a 3-d space, but in general the embedding is not necessary

About this document ...

This document was generated using the LaTeX2HTML translator Version 98.1p1 release (March 2nd, 1998)

The command line arguments were:
latex2html -up_url ../../notes.html -up_title Astronomy 9 Lecture Notes -split 0 -t Astronomy 9 (Spring 2000): Handout 20 -dir /coma8/jbaker/public_html/courses/ay9/week10/handout20 handout20.tex.

The translation was initiated by jonathan baker on 2000-04-05

Up: Astronomy 9 Lecture Notes

jonathan baker
2000-04-05