Theory of General Relativity

• Expansion on the concept of Special relativity

• Special: Inertial perspectives are Equivalent

(unaccelerated)

• General: All perspectives are equivalent

Let’s go back to Newton

• F = G M1 M2 / R2

• F = ma

• When we solved this we “cancelled” the masses – why?

Why is the mass for gravity the same as the mass for

acceleration?

Equivalence

• Inertial mass = gravitational mass

• New thought: when in free fall – (an accelerating frame)

you feel weightless, but this is an inertial frame – or is it?

• Gravity = acceleration

General Relativity

• An accelerated environment is equivalent to a

gravitational environment

• Examples: An elevator going up –

A spaceship speeding up in

space

So what?

• This all seems obvious – but

work out the implications –

new Gedanken experiment

• Shine a light across the

elevator

But acceleration = gravity

• So a stationary elevator in a

grav field must produce the

same result –

• GRAVITY BENDS LIGHT ?!!

We can do this another way

• Light “falling” down a gravity

field gets blue shifted (gains

energy) while light going

“against” gravity gets

redshifted (loses energy)

Gravity Affects Light

• It also slows down clocks!

• Gravity “bends” light

• Light gains and loses energy as it moves through

gravitational fields

• Clocks in strong gravity move slowly

Light curves?

• How can light curve from gravity? – it has no mass

• How can it not travel in a straight line?

• New way of thinking – gravity is not a “force” as Newton

would describe it – it is a change in geometry

Geometry and Mass

• This is a revolution – the geometry of space-time is

changed by the presence of mass – light moves along a

geodesic in this altered geometry – the shortest distance

between two points

• Mass and space-time are linked

• This is a field theory – not a force theory

Einstein’s Theory

• G = 8G/c4 T

• English translation: Local Geometry (curvature) of space-

time is defined by the distribution of mass and energy (T)

Easy – right?

• This equation is so hard to solve, that a closed form

solution has only been done for a handful of extremely

simple cases

• Point mass, spherical mass, black hole – uniformly

distributed material in a homogeneous and isotropic

condition

Is it true?

• GR makes several testable predictions

bending of starlight

orbit of mercury

gravitational redshift

binary pulsar decay – gravitational radiation (gravity

does not act “at a distance”)

1919

verification of

GR showing

bending of

starlight

Cosmological Principle

• The universe is homogeneous and isotropic on large

scales –

• True?

Cosmology, finally!

• Can solve G=8G/c4 T under the assumption of the

cosmological principle: theoretical cosmology

• Observations to test the predictions of the above:

observational cosmology

BRAIN BREAK

Cassini Mission

Titan

True colorVisible light

Titan

Radar Map

Temperature of Titan

Pressure on Titan

Phase Diagram for Methane (CH4)

Titan

View from the ground

Modeling the Universe

The real way

• Apply the cosmological principle to the universe: matter

and energy are uniformly distributed

• Solve Einstein’s Equations of General Relativity under

these assumptions

• We will not be doing this

Our Way

• Apply the cosmological principle –

• Solves Newton’s equations for such a circumstance

• Show the real relativistic answer (which is a little bit

different) and just accept it

Concepts needed

• Scale Factor: R(t)

• Co-moving coordinates (expand along with the universe)

distances between objects remain constant (D)

• Distance between two objects (freely expanding)

– Then: R(then) • D

– Now: R(now) • D

Concepts (continued)

• Scale factor changes with time (universe is

expanding/contracting)

• Rate of change of R =

• This can be thought of as speed

(there, you just did calculus!)

R

More concepts

• The rate of change of is

just the

acceleration:

R

R

Newtonian Escape Velocity

• Consider a small particle (mass m) at the surface of a

sphere with mass M (M >> m)

• If the particle has kinetic of (1/2 m v2) equal to it

gravitational potential energy (GMm/R) then it could just

escape from the gravitational field (and have zero kinetic

energy when it got to R =

• Solving for v, we find:

v2 = ( 2GM/R) = escape velocity

(Newtonian)

Now consider the universe, expanding at a

rate

R

• Kinetic energy per unit mass

(energy density = (½ mv2 m = ½

v2) define this quantity as

• So (twice) the energy density of the

universe =

2 = 2

= 2GM/R + 2

[ - GM/R = ]

• What does this mean?

• Expansion of the universe is related

to its gravitational potential energy

at it’s “ kinetic energy at infinite

size”

R

Consider the possibilities2

= 2GM/R + 2 • As R gets larger, the 2GM/R

term gets smaller and smaller,

if < 0, then at some point,

= 0, and the expansion stops

(it never gets to R = )

• If = 0, then

gets smaller and smaller, and

the universe expands for ever,

but just barely

If > 0 it expands forever

R

R

R

Now lets get rid of M

• The mass of the universe

is just:

M = 4/3 R3 , where is

the mass density of the

universe

2= 8/3 G R2 + 2

R

Now – add Einstein

• If we had done the full relativistic solution:

2= 8/3 G R2 – kc2

k = curvature +1, -1, 0

since R just a scale factor, when can choose our coordinates to make k = one of these values

Friedmann Equation

R

Other changes in relativistic solution

• is the mass-energy density (not just mass)

• If = just mass this is a “standard model”

Standard Model Basics

• k = +1, positive curvature, closed universe (re-collapse)

• k = 0, flat curvature, open universe (expand forever)

• k = -1, negative curvature, open universe

The Hubble constant

• Remember: v = H0 • d (H0 = v/d)

• v = change in distance divided by change in time

= R(then)•D – R(now)•D / (t(then) – t(now))

v = ΔR•D/ Δt

continuing

• v = ΔR•D/ Δt

• H0 = v/d (d = R•D)

• H0 = [ΔR•D/ Δt] / R•D = [ΔR/ Δt] / R =

/R = H0

If we use R(today) and (today)

R

R

The Hubble “constant”

• The Hubble constant varies with time!

• It is the constant of proportionality between

speed and distance

• The ratio of /R = H at any time,

= H0 at the present time

R

Why I want to know H• Rearrange the equation:

2= 8/3 GR2 – kc2

H2 = 8/3 G - kc2/R2

If k=0, then =crit, and

crit = 3H2/(8G)

~ 10 H atoms/cubic meter

= / crit

R

Friedmann equation

2 = 8/3 GR2 – kc2

The rate of expansion (contraction) of the universe

is a function of its gravity and its curvature

k= curvature term, = 0, +1, or -1

(4/3R2 = M/R) where M is mass of the universe

If is the mass density

R

• = /crit where crit is the density necessary to just close

the universe

• When = 1 (by definition) k = 0

• = 1 + kc2 / (H2R2)

and curvature

• = 1 + kc2 / (H2R2)

Clearly:

if k = 0, = 1 for all time

if k = +1, > 1 for all time, but changes

if k = -1, ,< 1 for all time, but changes

New information

= -4/3GR + R/3

acceleration = -GM/R2 + mysterious new term

Is the cosmological constant, appears from Einstein’s

equations

Note that looks like in its application- a mass/energy

density

R

Friedmann equation +

2 = 8/3 GR2 – kc2 + R2/3

R

Two basic equations2

= 8/3 G R2 – kc2+ R2/3

Kinetic energy = gravitational energy – curvature of space + energy of vacuum

Or

Kinetic energy – gravitational energy – vacuum energy = - curvature

R

= -4/3GR + R/3

RAcceleration = gravitational force (attractive) + vacuum force

Two basic equations

ሶ𝑅

𝑅

2

= 8/3 G – kc2/R2+ /3

Kinetic energy = gravitational energy – curvature of space + energy of vacuum

Or

Kinetic energy – gravitational energy – vacuum energy = - curvature

ሷ𝑅

𝑅= -4/3G + /3

Acceleration = gravitational force (attractive) + vacuum force

Rules for Standard Cosmology

• Standard cosmology means mass dominates over energy, and the cosmological constant = 0

• Three cases:

k = -1, neg curve, < 1, expand forever, infinite

k = 0, flat curve, = 1, expand forever, infinite

k =+1, pos curve, > 1, collapses, finite

Observational cosmology is the attempt to measure these quantities,

now, and over cosmic time.

How fast is the

universe

expanding?

Is the rate of expansion

changing, and if so, by

how much?

How much mass is

there in the

universe?

What is the

curvature of

space –

positive,

negative or

flat?

Inferred by solving the

equations, I have no idea

what this actually is.

The Einstein Static universe

• 2 = 8/3 GR2 – kc2 + R2/3

• This is what Einstein got, and wasn’t very happy about it –

• He wanted = 0, and = 0 (static universe)

• This meant = 4G and k = 4GR2/c2

R

R

R

The Einstein Static universe

• No expansion, positive curvature, finite size – infinite in

time requires a specific value of the cosmological

constant

• But it is unstable – look what happens if you change R

What is ?

• Looks like an energy density – but it is constant – doesn’t go down as does

• A positive value for implies a force that wants to push the universe outward – an negative value wants to collapse the universe

• The R/3 term increases as R, gets more important as the universe expands – less important in a small universe

Any physics to all of this?

• can be interpreted as an energy density associated with

the vacuum: As space grows, there is more vacuum,

therefore more total energy associated with vacuum

• This is pretty hard to believe – certainly seems to violate

conservation of energy

New possibilities for a universe

• Unlike the standard model, the addition of the

cosmological constant alters the relationship between k,

, and the fate of the universe:

• De Sitter universe: no mass, positive cosmological

constant: exponential, scale free-expansion, no big bang

Steady State model -

• De Sitter Universe + matter

• Requires the constant creation of mass to keep the

product of R a constant

• No deceleration: H is unchanging throughout time – no

beginning or end –

• Perfect cosmological principle

• Disproved if the universe is accelerating or decelerating

A negative

• Causes all models to collapse, regardless of other factors:

as R increases towards infinity, always wins

LeMaitre Model

• Choose close to, but just off from crit = 4G

• This causes the universe to “hover” at a near constant

size for a long time, before expanding again

• Popular when the value of H0 was inconsistent with the

age of the earth

So what happens in the far future?

• Second Law of thermodynamics: Entropy always

increases

• Entropy = disorder

• Defines positive time

• Heat Death

• Big Crunch