General relativity

General relativity: intro

• Why do all masses fall at same rate? All normal forces (e.g. electrical, friction, elastic...) don't produce same accn in all bodies.

  F = ma = mg so a = g
  

• Are we really sure the m's are the same? This concerned Newton

• The first m (inertial mass mI) measures how hard things are to accelerate (2nd. law), the second (gravitational mass mG) measures gravitational force

  F = mIa F = mGg

• Pseudo-forces (e.g. centrifugal force) behave the same way. $$ \color{red}{ F = \frac{{mv^2 }}{R} = ma} $$ so all bodies undergo same centrifugal accn.

mI = mG

F = mIa = mIg 

so a = g only if the "inertial mass" the gravitational mass.

• Can demonstrate this is true to 1 part in 10¹² (Eötvos experiment).
• Special relativity said you cannot do an experiment to decide if you are moving.
• General says that you cannot do an experiment to distinguish between a gravitational field and an acceleration (!!!!!!!!!)

• You cannot distinguish them
• 

• 
• Suppose you are in an accelerating elevator, and a beam of light is shot horizontally, it will appear to fall..
• Suppose you are in a stationary elevator, and a beam of light is shot horizontally, it will fall..
• You cannot distinguish the two. Light gets affected by gravity?

General relativity:

• What is a straight line?
• Which is the straight line?

Dynamics:

A Body continues at rest or in a state of uniform motion unless acted on by a force.

• Uniform motion means in a straight line.

• A geodesic in Euclidean space ≡ straight line ≡ shortest path.

• Geodesic: shortest time path between 2 points --> straight lines in a flat space. It is "extremal path".
• Can either say:
• There is a force called gravity which acts on all energies (and hence attracts light)

• There is no such thing as gravity, it's just that masses distort space-time in their neighbourhood

• Massive bodies follow timelike geodesics so planets are actually moving in "straight" lines in a curved space...
• Either way, don't jump off tall buildings: you can be just as dead in a curved space!

• "Lenses extend unwish through curving wherewhon till unwish returns on its unself" e.e.cummings

Gravitational Red-shift

A ball thrown up near the earth's surface will lose energy.

• light emitted from floor of elevator with frequency ν_E,
• hits ceiling after a time t = h/c.
• during this time, lab (elevator) has accelerated to a speed u = gt,
• so the light will have a red-shift $$ \color{red}{ z = \frac{u}{c} = \frac{{gh}}{{c^2 }} \Rightarrow \frac{{GM}}{{c^2 r}}:\lambda _o = \lambda _E \left( {1 - \frac{{GM}}{{c^2 r}}} \right)} $$

This is another consequence of the equivalence principle: confirmed in numerous experiments over the last 40 years. Implies that clocks run slow in gravitational fields $$ \color{red}{ t' = \frac{t}{{\left( {1 - \frac{{2GM}}{{c^2 r}}} \right)^{1/2} }}} $$

(Confirmed by Rebka-Pound using Mossbauer techniques in 1960. An atom is a good clock:

A consequence: time stops at the edge of a black-hole for an external observer.

Gravitational force

• Fact that orbits are closed is "coincidence" not true for any potentials except r², 1/r and 1/r²

• Hence get "rosette" orbits.

• Much less dramatic in practice: perihelion (closest approach to sun) of Mercury advances by 43" arc/century

And light gets does bent:

this is a very large cluster of galaxies, which acts as a very large (and rather bad!) lens. It produces several images of a much more distant galaxy

Geometry of Curved spaces

Note we have carefully avoided saying what we mean by a curved space

• If you take the example of the 2-D curved surface of the Earth, this is embedded in a 3-D space. Hence If a massive body curves space, it almost implies extra dimensions.
• In fact we can carry out tests to decide if we live in a "normal" 3-D space (Euclidean) e.g.

• parallel lines
  angles of a triangle add up to 1800 α + β + γ = π These are experiments that we can almost do. (Gauss tried the 2nd!).

• circles don't satisfy

  C = 2πa
  

• In fact,

  C = 2πx = 2πRsin(a/R) = 2πa(1-a2/6R2)
  

• i.e. by measuring the circ. of large circles, we can determine if we live in a curved space. (a physicist would say we are measuring distant-dependence corrections to π!)
• Can also have negative curvature K<0. Triangle formula gets corrected in similar fashion: Max value in positively curved space?

So we conclude that Relativity (Special and General) works because

• It contains Newtonian Gravity
• It agrees exactly at very low speeds
• It predicts perihelion advance of Mercury
• It predicts the bending of light by gravity (which Einstein predicted)
• It predicts gravitational waves (that even Einstein hadn't foreseen)
• Back to Newton!