Ideas of Modern Physics

1
Ideas of Modern Physics
The final steps to the theory of general
relativity were taken by Einstein and David
Hilbert at almost the same time.
2
Review

• Origins of the general theory of relativity
• The principle of equivalence
• Consequences for geometry

3
Today

• The general theory of relativity
• Mechanics in relativity
• Cosmology and holes in space-time

http//antwrp.gsfc.nasa.gov/apod/ap001031.html
Galaxies falling into a hole.
4
Idea behind geometric theory

• Matter bends space and time.
• Bending on a 2-d surface is characterized by
  curvature (C1/R_1R_2)) at each point which
  says how the surface accelerates away from a
  plane. Bending in higher dimensions is
  characterized by curvature in each hyper plane at
  each point. How can we relate curvature to matter?

5
Einsteins solution

• Einstein guessed that the curvature functions
  (meter-2) are proportional to the local energy
  and momentum densities (kg/m3).
• The proportionality constant that emerged from
  comparison with Newtonian theory in the limit of
  weak gravitational fields is 8pi 8piG/c2 where
  G is Newton's constant.

Differential tensor field equations not shown!
6
Curvature of space

• We can estimate the magnitude of the curvature of
  space at a distance r from a single static mass M
  by dimensional analysis. Assume
• C MG ca/rb
• with integers a, and b. This quantity has
  dimensions of inverse area if a-2, and b3.

7
Near the Earth

• The ratio of the curvature of space on the
  surface of the Earth to the curvature of the
  surface of the Earth is
• C/(1/r2) MG/(c2r) 7e-10
• The curvature of space near Earth is so small as
  to be usually unnoticeable.

8
Gravitational time dilation

• At 10,000 km, a clock should run 4.5 parts in
  1010 faster than one on the Earth. Comparing
  timing pulses from atomic oscillator clocks
  confirms the gravitational time dilation in 1976
  to within 0.01. Corrections are now standard in
  the synchronizing satellites.

http//hyperphysics.phy-astr.gsu.edu/hbase/relativ
/gratim.htmlc1
9
Mechanics in general relativity

• Einsteins theory reproduces Newtonian mechanics
  when curvature is small.
• Relativity is manifest as an approximate 1/r4
  correction to Newtons 1/r2 force law, largest
  at small r so near a mass.
• Relativistic effect are observed in satellite
  motions.

10
Precession of Mercury

• Deviations from closed elliptical orbits result
  from deviations from Newtons 1/r2 law. The axis
  of the orbit of Mercury (closest to the Sun so
  most sensitive) precesses due to the other
  planets. Relativity explains an excess of 40
  arcsec per century as calculated by Einstein.

11
Gravomagnetism

• Relativity theory requires a velocity dependent
  component to gravitational interaction rather
  like magnetism in electromagnetic theory.
• A spinning mass behaves as a gravomagnet and two
  spinning masses interact through gravomagnetism.

12
Gravitational radiation

• Einsteins theory addresses gravitational action
  at a distance as follows.
• If a mass is jiggled, ripples of space-time
  curvature carry the signal.
• Gravitational radiation carries energy and
  momentum and wiggles stuff in its path.

13
Evidence for gravity waves

• In 1974, Joseph Taylor and his student Russell
  Hulse discovered a binary neutron star system
  losing energy as expected from gravitational
  radiation.

14
Direct detection

• LIGO is a collection of large laser
  interferometers searching for wiggles generated
  by exploding stars.

http//www.ligo.caltech.edu/
15
Black holes

• Light falls in a gravitational field. A
  sufficiently strong field should bind light into
  orbits. When escape velocity v2GM/r1/2c,
  nothing can emerge.
• Inside the Schwarzschild radius
  r_s2GM/c2(1/2), light itself cannot escape.
  An object condensed within its Schwarzschild
  radius is called a black hole.

16
Theory of compact objects

• Matter in the earth is held up by electric
  forces.
• In the Sun, matter is held up by pressure
  associated with high temperature by nuclear
  fusion reactions.
• At higher density when fusion burns out, gravity
  can squeeze electrons into protons forming a
  neutron star.

17
Black hole formation

• At sufficient density, no known force prevents
  the complete collapse of matter to a point- a
  hole.
• Matter falling into black holes provides the only
  explanation for energetic objects seen throughout
  the universe.

http//antwrp.gsfc.nasa.gov/apod/ap010921.html
Black hole roaming the Milky Way.
18
Artists rendition

• Spinning black hole with accretion disk.

http//antwrp.gsfc.nasa.gov/apod/ap011029.html
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A hot black hole candidate
http//antwrp.gsfc.nasa.gov/apod/ap021005.html
20
Example black hole candidates

• Two galaxies colliding, central huge black holes
  merging.

http//antwrp.gsfc.nasa.gov/apod/ap021128.html
21
Geometry of the universe

• The surface of a cylinder is finite in one
  dimension. The surface of a sphere is completely
  finite. Einstein suggested our universe might be
  finite like the surface of a sphere. It might
  have some kind of donut or more complex topology
  at large or microscopic scales.

22
Astrophysics and cosmology

• In Einsteins theory, geometry and matter go hand
  in hand. To discover the geometry of the
  universe, we have to look at the matter and
  understand what we can see.

http//antwrp.gsfc.nasa.gov/apod/ap021012.html