General Relativity and Space-time

(See the section on black holes for an introduction to relativity.)

• Einstein's General Theory of Relativity is needed to build cosmological models.
• The Universe is regarded as a 4 dimensional space-time: 3 space dimensions plus time.
• The distribution of mass and energy throughout the Universe determines the shape of space-time.
• The shape of space-time determines how mass and energy will move.

Mass tells space-time how to curve, and space-time tells mass how to move.

John Wheeler

Isotropic and homogenous

• A homogenous Universe is one that has the same properties everywhere, e.g. the number of galaxies in a given volume of space.
• The Universe appears isotropic if it appears the same everywhere.
• If we look at the largest scales, e.g. by counting the numbers of high redshift galaxies in different parts of the sky, we find that the Universe is isotropic.
• Cosmologists are only interested if the Universe is currently isotropic and homogeneous on the very largest distance scales.

The Cosmological Principle

• Solving Einstein's equations from General Relativity requires assumptions.
• The cosmological principle states that the Universe is isotropic and homogeneous.
• If we suppose we are not located in a special place in the Universe, then isotropy makes homogeneity seem reasonable.
• Cosmologists charactertise the Universe in terms of a density and pressure.
• Other quantities, such as temperature, can be derived from knowledge of density and pressure.
• The pressure of the Universe is taken as zero. The distance between stars and galaxies are such that there are no frequent collisions that you'd find in a gas like air.
• Remember that density here refers to both mass and energy.

Pythagoras in flat 2D

Source: Andrew Conway CC-BY SA 3.0

s²=x²+y²

• The square of the distance between two points is equal to the sum of squares of the x and y co-ordinates.
• This is a simple example of a metric - a formula for calculating distance.
• In this case we are calculating distance on a 2 dimensional flat surface.
• The shortest distance between two points in flat space is a straight line.

Distance on a sphere

Source: Andrew Conway, adapted from this by Cffk

CC-BY SA 3.0

Great circle distance

• Consider two cities A and B separated by λ in longitude with latitudes L_A and L_B.
• Pythagoras's theorem cannot be applied - squaring the differences in longitude and latitude an adding them will not give the distance between A and B.
• The shortest distance between A and B is not a straight line but the arc of a great circle - a circle with the same diameter as the sphere.
• Notice that we are still considering a 2D surface, but it is curved.
• If A and B are really close together then Pythagoras becomes a useful approximation - space is locally flat.

The Metric for a flat 3D space

The distance squared ds² between two points is given by

ds² = dx² + dy² + dz²

• dx is the difference in x - likewise for dy and dz.
• The d means that the differences are very small and have to be added up using calculus (using integrals).
• The metric is Pythagoras's theorem applied locally.

Flat space-time metric

ds² = dx² + dy² + dz² - c²dt²

• Notice that a time difference dt is now included and is multiplied by the the speed of light squared and has a minus sign.
• This metric tells you the "distance" between two events.
• An event is defined by a location in space (x,y,z) and time t.
• Events associated with a signal travelling at the speed of light will have ds=0.
• Two events in which one event caused the other must have ds² <0 - this is equivalent to saying nothing can travel faster than light.
• This is the metric used in Special Relativity.

Cosmology space-time metric

ds² = R²(dx² + dy² + dz²) - c²dt²

• This is the metric used in cosmology.
• R is the scale factor and can vary across time and space.
• For example, if R increases with time t, then the Universe is expanding.
• If R is a function of x, y or z then space-time will be curved.

Scale factor and redshift

Recall from before, that if λ₀ is the stationary wavelength and λ is the observed (shifted) wavelength, then the redshift is given by

z = (λ - λ₀) / λ₀

The correct explanation for this is not the Doppler effect due to the source's motion, but because the scale factor has changed, which tells us that:

z = (R - R₀) / R₀

where R is the scale factor at observation and R is the scale factor as the light was emitted.

Solutions to Einstein's equations

• Einstein's equations can be solved to relate the density of the Universe ρ, the scale factor R and Hubble's constant H.
• Hubble's constant turns out to be equal to the rate of change of R divided by R.
• And a critical density can be defined in terms of H and the constant G, from Newton's law of gravitations, as follows:

ρ_crit = 3 H² / 8 π G

The critical density

ρ < ρ_crit

• The Universe expands forever, though the expansion slows.
• Space-time is curved, and the Universe is open and infinite.

ρ > ρ_crit

• The Universe's expansion slows and until it eventually starts to contract.
• Space-time is curved, and the Universe is closed and finite.

ρ = ρ_crit

• The Universe expands forever, though the expansion slows, getting closer to halting.
• Space-time is flat, and the Universe is open and infinite.

The Cosmological Constant

• In Einstein's original description of the Universe using general relativity, he included a constant Λ (capital lambda) - the cosmological constant.
• It allowed him to describe the Universe as being a steady state - this was before Hubble's work and evidence of an expanding Universe.
• The cosmological constant can be intepreted as causing a repulsion in the Universe.
• Einstein later referred to it as his greatest blunder.
• So far we have assumed Λ=0.

A cosmic surprise.

• For many years, observational efforts in cosmology were directed at determining whether the density was above or below the critical density.
• Best estimates suggested we live in a flat Universe, near critical density.
• Efforts were also made to measure the deceleration of the expansion.
• But then, in 1998, observations of distant type Ia supernova revealed that the expansion of the Universe was accelerating.
• This can be described in the models if Λ>0, but there is no clear physical interpretation of what might cause this repulsion; for this, and other reasons, we are forced to consider that Universe is mainly composed of dark energy, distinct from dark matter.