General Relativity Physics Honours 2005 Dr Geraint F. Lewis Rm 557, A29

ecture Cosmology Cosmology studies a on-off phenomenon – the Universe. The fact that General Relativity can provide a dynamical description of the evolution of the Universe is seen as one of the great successes of the theory. While we live in a single instance of the Universe, the equations of relativity describe a whole range of potential universes whose evolution depend upon their energy content. In this course, we will look at the GR formulation of cosmology and the potential forms of evolution.

ecture Cosmology Western society viewed the Universe as infinite in size, but finite in age, naturally explaining Oblers’ paradox: why is the sky at night dark? If the Universe was infinitely old and unchanging, then the Earth should receive radiation from an infinite number of stars, and the night sky should be as bright as the surface of the Sun. With a finite age, we only receive light from a finite number of stars. An alternative, however, is that the star light loses energy as it travels to the Earth, such as through a Doppler shift in an expanding universe. (Why can’t the energy just be absorbed by some intervening material?)

ecture Newtonian Cosmology Much of cosmology can be expressed in terms of Newtonian mechanics, although this was realised after GR cosmology. Consider the energy of a finite number of particles The final term is a cosmological repulsion force of the form F = 1/3  m i r i (something we will see in more detail later). Does this imply a preferred origin?

ecture Newtonian Cosmology If we assume the cosmological principle (more on this shortly) the view point from any point within the distribution, and motion must be purely radial (why?). Hence, at any time t we can write where S(t) is a scale factor which is the same for all the particles. With this we also have where H(t) is the Hubble parameter.

ecture Newtonian Cosmology The energy equation can therefore be written as where A, B & D are positive constants. If we rescale such that R(t)=  S(t), then this equation can be written as where (GR gives the same Friedmann equation)

ecture Newtonian Cosmology If E=0 then  is arbitrary (k=0). If E  0 then we can choose then k only takes on the value of § 1. Hence, we have a simple differential equation to describe the motions of the particles. If  =0, then there are three simple evolutionary states;

ecture The Cosmological Principle Relativistic cosmology is based on several simple principles. The Cosmological Principle states that “at each epoch, the universe present the same aspect from every point, except for local irregularities” So, there are no preferred locations, and no preferred directions to look, and so the Universe is homogeneous and isotropic. Note that this refers to spatial directions; the Perfect Cosmological Principle states that the Universe appears the same at all times and it is unchanging (this is the source of the well known “Steady State” model of the Universe).

ecture Weyl’s Hypothesis Simply put, Weyl’s hypothesis states that we can ignore all of the small scale irregularities in the Universe, and describe the energy distributions in the Universe as perfect fluids. The postulate requires particles in the fluid to follow non- intersecting time-like geodesics. We can also assign a population of observers that ride along with the fluid (i.e. they follow the same null geodesics). We will see that the proper time for these observers is the same as the cosmic coordinate time we use to describe the metric.

ecture Relativistic Cosmology Working with coordinates (t,x 1,x 2,x 3 ), we can slice the universe into a series of t=constant hypersurfaces. The Weyl hypothesis requires that the geodesics pass through each hyperslice the same spatial coordinates (x 1,x 2,x 3 ) i.e. the geodesics are orthogonal to the hypersurface. These define comoving coordinates. The line element is then where t is the world or cosmic time. Note that hypersurfaces of constant t are surfaces of simultaneity (but the view from a point is related to its past light cone!).

ecture Relativistic Cosmology The cosmological principle forces us to consider universes with constant spatial curvature; if curvature varied from place-to-place then different points would distinguishable. Furthermore, it also forces h  to be of the form otherwise different directions would be distinguishable. Given these facts, a general form for the metric for the universe can be derived.

ecture Constant Curvature The curvature tensor for a 3-space of constant curvature is where K is the curvature. The Ricci tensor is then The most general spherically symmetric line element in 3-d is

ecture Constant Curvature The non-vanishing elements of the Ricci tensor are For constant curvature, this gives with the solution

ecture Robertson-Walker Metric Combining to get the full relativistic line element, we see where k=-1,0,+1 (this requires a coordinate change of the form r’=| K | 1/2 r to eliminate K). This is the Robertson-Walker line element (and associated metric) which is the foundation of modern cosmology.

ecture Robertson-Walker Metric The Robertson-Walker metric describes any homogeneous and isotropic universe, irrespective of its content (in terms of energy distributions). The specifics of the energy densities, however, govern the choice of k and the evolution of R(t). We will look at this in more detail with the Friedmann equations.

ecture 3-Space Geometry (22.8) The value of k determines the spatial geometry. We will consider the k=+1 case in detail (but read the others in the textbook). k=+1: The geometry is a 3-sphere (analogous to the surface of a 2- sphere in 3-d). Introducing  such that r=sin  then We can embed this 3-d surface into 4-d Euclidean space with the coordinates (w,x,y,z)

ecture 3-Space Geometry This yields Hence we can think of R(t) as the radius of the Universe. Remember that this radius is only apparent in a 4-d view. Note that the total volume of the Universe is then V=2  2 R 3 (t o ). Hence the Universe has finite volume, but it contains no edges. In terms of topology, it is closed and unbounded. When considering the time axis, the overall space-time is cylindrical.

ecture 3-Space Geometry k=0: Here the 3-Space is flat. In terms of topology the space is open and unbounded i.e. you can travel for ever in a straight line, and never come back to the same point. Hence, the universe is infinite in extent (and always has been!) k=-1: Following a similar argument for the spherical space, we can describe this case as a 3-hyperboloid. Here, however, the geometry is open and unbounded. These completely describe the potential geometries of the universe (given the Robertson-Walker metric).

ecture Propagation of Light Our view of the Universe depends upon the propagation of light through the curve space-time metric. To understand this, we need to consider the paths of null geodesics. Suppose an observer sits at r=0 and we consider radial light rays. Given that ds=0, then Suppose a light ray is emitted at a time t 1 at a distance r 1 and is received today (r=0) at t 0

ecture Propagation of Light Remembering that for a comoving source at distance r, the coordinate is fixed, then Hence, in an expanding universe there will be a redshift

ecture Hubble’s Law If we consider a nearby source, then we can write then Hence, we can see we can derive Hubble’s law from the Robertson-Walker metric. Hubble’s constant (which is not a constant) is now known to be 72 km/s/Mpc.

ecture Cosmological Distance Generally, the idea of “distance” in cosmology is rather complicated. The instantaneous proper distance, d=R(t)r, is of little practical use. One of the most commonly used is the luminosity distance, which is akin to the r 2 law for radiation. Unlike Euclidean cosmology, there is an additional term of (1+z) for the redshifting of photons, plus an additional (1+z) as the emission rate is time dilated. The resulting definition is

ecture Cosmological Distance Note that the textbook is wrong! The luminosity distance is related to other distances through Here d t is the transverse proper distance and equals the proper distance in a flat (k=0) universe. In a non-flat universe, d t depends upon the curvature. While we will not look at the derivation, another common cosmological distance is the angular diameter distance which tells us the angle subtended by a object with proper length L.

ecture Friedmann Equations We shall not consider the detailed derivation of Friedmann’s equations, but we will consider universes described by the Robertson-Walker metric and assuming Weyl’s postulate then; where  is the proper density in the special coordinate system in which u a =(1,0,0,0). For normal matter, p=0, meaning that the energy in motion is negligible to that in the form of rest mass.

ecture The Cosmological Constant Einstein found that field equations could be extended such that Einstein added this term on the left, introducing a curvature that drives objects apart (i.e. anti-gravity!). This allowed him to make the cosmos static. The modern interpretation, however, places this term on the right i.e. a fluid in a universe where p=-  will provide the same term in the field equation. In modern cosmology, it appear that  is the dominant energy density in the universe today.

ecture Friedmann Equations Taking the field equations, Friedmann derived the differential equations which govern the evolution of the scale factor. Assuming a universe with a general fluid and a classical cosmological constant , then (exercise) Assuming the fluid is dust-like (p=0) then we can combine these to give the generalized Friedmann equation.

ecture Friedmann Equation Assuming the fluid is dust-like (p=0) then where C=8  R 3  /3. Amazingly, this is the same form as the Newtonian version we saw earlier. Hence, the evolution of the Universe can be determined by specifying the parameters in this equation and integrating to determine R(t).

ecture Deceleration The rate of change of H(t) can be encapsulated in the Deceleration Parameter such that If q>0 then the cosmos is slowing down over time. Hubble’s constant H o and q can be determined by observations of distant sources as they modify Hubble’s relation.

ecture Quintessence In deriving the Friedmann equation, we have assumed the Universe is composed of two fluids, dust (p=0) and the cosmological constant (p=-  ). In general, we could put any mix of fluids into the Universe. Each of these are parameterized by an Equation of State Where  =0 for dust and  =-1 for the cosmological constant. It is found that  <-1/3 causes the Universe to accelerate. Determining  (and its potential evolution) is a major goal of cosmology.

ecture Cosmological Models There are several analytical solutions to the Friedmann equations (but in general they have to be determined numerically), although a qualitative understanding of the evolution of R(t) can be found from recasting the Friedmann equation as an energy equation; Hence a sketch of the form of the potential energy term will reveal if the universe expands for ever or contracts etc. Various cosmological models are summarized in Fig But which one do we live in?

ecture Our Universe The Friedmann equation can be written as where  c = 3H 2 /8  G is the critical density. With this, the spatial curvature is related to the energy densities in the Universe. So we can work out the various energy densities in the Universe and compare it to this critical value to determine k. Note that a spatially flat universe (k=0) implies  /  c = 1.

ecture Cosmological Supernovae Different combinations of the cosmological parameters predict different relations between apparent brightness and redshift. The cosmological supernova project has found that the Universe appears to be flat (k=0) with 28% of the energy in matter and 72% in a dark energy component (possibly a cosmological constant). Baryons are ~5% of the energy.

ecture Why Flat? The Big Bang model, as derived from the Friedmann equations, predicts that the Universe was denser and hotter in the past. However, it does not predict the values the energy densities in the Universe. Worryingly, randomly selecting energy densities result in a non- flat universe which, in general, expands or collapses too quickly. A solution can be found in the idea that the Universe underwent a period of rapid inflation in its very early history.

ecture Why Flat? The idea is related to GUTs and the symmetry breaking that gave us the forces we see today. At this time, the energy density in the Universe is dominated by a vacuum energy density V o scalar field. Hence the Friedmann equations can be written as Integrating this we find R = R o exp(H t) and we see the Universe expands exponentially. This occurred ~ secs and inflated the Universe by ~ Hence if the Universe is curved, it is so large that this curvature is impossible to measure. Inflation also explains why the Universe appears so homogeneous.

ecture Into the Future Given the current make-up of the Universe, it will expand for ever with accelerated expansion. We can, therefore, chart the future years, all stars extinguished years, protons decay, leaving only black holes years, the black holes evaporate After this, the Universe is thin sea of electrons, positrons, neutrinos and photons that hardly ever interact. It will become a sad, cold, lonely place, a little bit like Canberra.

ecture Anthropic Principle Weak AP: The observed values of all physical and cosmological quantities are not equally probable but they take on values restricted by the requirement that there exist sites where carbon-based life can evolve and by the requirements that the Universe be old enough for it to have already done so. Strong AP: The Universe must have those properties which allow life to develop within it at some stage in its history. Final AP: Intelligent information-processing must come into existence in the Universe, and, once it comes into existence, it will never die out.