Lecture 13

Cosmology Cosmology: The Hot Big Bang Model We are now in the position to begin a sophisticated, although elementary, quantitative study of modern cosmology. Unfortunately, we will not be able to undergo a full analysis involving Einstein’s equations. However, we will obtain many results via classical arguments. Notions of Cosmology in 1905 The idea that the various spiral, fuzzy nebulae observed by Messier and others actually lie outside our own Milky Way galaxy only came about in the 1700s with the proposition of Immanuel Kant. Even the fact that the Milky Way, which shimmers in the night sky, is comprised of stars was discovered in the 1500s by Galileo Galilei with his new telescope1. The debate about the status of the fuzzy nebulae raged up until the late 1920s with well-respected astronomers on both sides of the issue. Around the turn of the 20th Century ever more powerful telescopes were being constructed and the fuzzy nebulae could be seen to contain stars. Their distance from the Earth was speculation. Whether one thought of the “island-universe” concept, in which the nebulae are all within the Milky Way (and thus the Milky Way comprised the entire universe), or that our Milky Way was a typical galaxy like those observed; most agreed that the universe was infinite in extent, static, and uniform in its distribution of matter. This was the predominant view up through 1916 when Einstein pondered relevance of his theory to the state of the universe. That the universe was infinite and static stemmed from the basic analysis of the universe using Newtonian mechanics. Assume that the universe is finite, then there would exist a center of mass towards which all objects would be attracted. Such an unstable universe was thought to be untenable, for if this were so, the universe would not be ever lasting. The most palatable resolution was to have the universe be infinite in extent with a uniform distribution of mass on the largest scale: there is no center of mass and a static universe is possible. This idea of an infinite universe is the most common view of lay people. For if the universe were not infinite, how does it end? A barrier? Then what lies on the other side? You may already notice that this line of thinking is rooted in a prejudicial Euclidean viewpoint (more on this latter). Natural Philosophers knew that there were problems with an infinite universe consisting of a uniform distribution of stars. For instance, Olber’s Paradox states that if the universe were like this, then in any direction you point, the line from your finger will terminate on a star. Thus the night sky ought to be ablaze with starlight! Some thought that interstellar dust would block much of the light. However it was known that if the dust absorbed this light, it would heat up and radiate thermally. Thus, this dust would become infinitely hot over time. Hence the notion of a static universe stood on shaky ground but was still the overwhelming viewpoint up to 1929. This is the viewpoint that Einstein clung to when he first considered the cosmological implications of general relativity. 1

Recent historical studies have turned up poems, predating Galileo, which allude to this fact.

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The Cosmological Constant The problem that Einstein observed was that a static universe was not a solution of his equation (just as it is not in Newton’s theory). Einstein was so entrenched in his cosmological beliefs that he modified his equation so as to allow a solution. He found that if he added a constant (multiplied by the metric tensor for mathematical consistency) a static universe was a possible solution. This constant, Λ, was called the cosmological constant and the Einstein equation took the form, G µν =

8πG µν T − Λg µν c4

Notice that this constant pervades the entire universe (it does not depend on position nor time) and thus can be seen as an additional energy density that exists everywhere. The sign indicates that if Λ is positive, then this energy density is negative (like having a negative mass) and provides repulsion amongst all objects. This repulsive energy density could balance the attraction due to all of the matter and energy in the universe. It was soon discovered, however, that such a solution is unstable, any slight perturbation away from equilibrium would drive the state away from equilibrium. (Think of balancing a pencil on its tip; this is an equilibrium situation since all forces point in the vertical direction. However, the slightest nudge will make the pencil fall). There was soon another reason that this model would needed to be tossed. In 1929, Edwin Hubble published a paper giving evidence that the universe was expanding. This was a blow to Einstein’s static universe conception and obviated the need for the cosmological constant. The inclusion of the constant was, quote, “my greatest blunder” 2. Others explored how such a constant influences cosmological kinematics. As you may have heard, there is recent evidence supporting its reintroduction. We will return to a more detailed discussion of this constant later. Edwin Hubble and the Expanding Universe Since Hubble’s discovery is of such importance we will spend some time discussing the events that led up to it. Afterwards we will return to the theoretical aspects of cosmology and modify the picture somewhat. There are two main inputs into Hubble’s conclusion of 1929; distances to the spiral nebulae and their relative velocity. Some of the data was found earlier. Relative Velocities In 1912 Vesto Slipher measured the spectra of several spiral nebulae and, from the Doppler shift formula, determined their relative velocity. Most of these were found to be moving away from us and a few approaching us (nowadays we know that he measured only nearby galaxies whose relative velocity is dependent upon, not only on universal expansion but the nearby, random, attraction of these objects). Of the 41 spiral nebulae measured, 36 were found to be receding (M31 and M32 are moving towards us). Several of the relative speeds were found to exceed any relative velocity measured in the Milky Way. These results were not interpreted as any evidence of extra-galactic objects at the time. 2 It may be that Einstein never said this phrase and was only attributed to him by Pais (in his book on 20th Century physics) but it does encapsulate his sentiment at the time. This quote is so pervasive nowadays that it is taken as fact. This is not a course in the history of science, thus we will not bother with such points further.

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The first suggestion of a link between recessional velocities and an expanding universe was given by Willem de Sitter in 1917in his theoretical study of general relativity. While Einstein devised a universe that was static by introducing the cosmological constant, de Sitter’s model had a cosmological constant yet no matter. Although assumed static (not true), when matter was introduced they accelerated away from each other. In this way there may have been thought a relation to Slipher’s observed redshifts. In 1922 Carl Wirtz proposed making just such a relative velocity – distance relation. To measure the distance he used galactic diameters as a gauge and from this he found that the recessional velocity grew with distance. However, the distance gauge is a very crude measure of distance as galaxies have varying diameters. From 1922 on through 1929 a wide range of cosmological models of expanding and contracting universes were devised. In fact, a year before Hubble’s paper, Howard Robertson had made the same conclusion based on Slipher’s work and Hubble’s previous distance determinations. However the debate between the placement (within or outside of the Milky Way) of the spiral nebulae raged on. For example, in 1920 the National Academy of Science hosted a debate on the spiral nebulae. Two prominent astronomers of the day took opposite views: Heber Curtis argued for extra-galactic spiral nebulae (not the first argument, which goes back to Kant), and Harlow Shapley argued that they must reside within the Milky Way. Shapley’s convincing argument was that there had been observed novae within these nebulae, just like some observed within the Milky Way: if these were beyond the Milky Way, these stellar explosions would have to be enormously powerful – thousands of times more powerful than the known novae. Hence it is absurd to think that these spiral nebulae could lie outside of the Milky Way. So how does the shift in spectra determine the relative velocity? As was mentioned earlier, elements given off telltale frequencies of light as they change their quantum state. This often occurs when the gases are heated, such as in stars. Thus the composition of stars can be determined no matter where they are to be found. In addition, as a continuous spectrum of light frequencies (such as from a heated blackbody – like a star) pass through an intermediate gas cloud, certain frequencies are absorbed as the gas is excited into a higher state. When an object emits EM radiation and is in motion with respect to the Earth, the observed velocity of light is the same, as mandated by special relativity. However, just as with sound waves, the frequency of the light will differ depending upon the relative velocity – the Fizeau - Doppler effect. *** Distance determinations The beginnings of the determinations of the distance to the spiral nebulae have origins decades before Hubble’s paper. In the first decade of the 20th Century Henrietta Leavitt, an astronomer at Harvard College Observatory, was assigned the task to investigate variable stars observed in the Magellanic clouds. Variable stars are ones whose brightness varies with time. Her research focused on a particular type of variable star called Cepheid variables, these are generally yellow giants with periods of days to months. The distance to the Magellanic clouds was determined previously through other methods, and all stars within them are assumed to be at the same distance. In 1908 she made the important discovery that

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the brighter (larger) Cepheids had longer periods and determined a relation between the intrinsic brightness of the star and its period. Thus, by measuring the period of a Cepheid variable you can determine its intrinsic brightness and by employing the inverse square law you can determine how far this star is away from us. This opened up a whole new method to determine large-scale distances and beginning in the 1920s Edwin Hubble began looking for Cepheid variables in the spiral nebulae. Using the 100 inch telescope on Mount Palomar he began tabulating distances to spiral nebulae. Note that these results settled the age old debate whether the spiral nebulae lie inside or outside of the Milky Way. For example, he soon found the distance to the Andromeda galaxy (M31) as being 10 times as far from us as the Megallanic Clouds. In 1929 he had enough data … ***

Theoretical introduction Einstein’s theory of general relativity opened the door to a once speculative area of research known as cosmology. Cosmology is the study of the nature, development, and future of the universe as a whole. Up until the general theory, cosmology was an area of research left to philosophers and religious thinkers. At the time of general relativity’s unveiling, the universe was thought to be static, unchanging, and infinite in extent. This view was so prevalent at the time that it dictated the nature of Einstein’s equation. Einstein was aware that his new theory could go far beyond all others in describing the nature of the universe. The state of the universe could be checked against solutions of his equation for validity. It was soon determined that a static universe of homogeneous density could not be a solution of Einstein’s equation. (The case of an infinite dust field of uniform density is often the first example shown in examining solutions of Einstein’s equation). Einstein, aware that his theory did not support his view of cosmology, modified his equation to support such a solution. This was done by adding the only possible form, a constant, Λ, multiplying the metric, to his equation, R µν − 12 g µν R + Λg µν = T µν . This new constant, the Cosmological Constant, provides for a static uniform universe with proper adjustment. Soon afterwards (1929), Edwin Hubble discovered that all galaxies are receding away from us, a fact interpreted as proof of an expanding universe. This result was a clear blow to the idea of a static universe and Einstein was forced to retreat from the addition of the Cosmological Constant. As will be discussed shortly, the cosmological constant is a point of constant debate and current trends are reviving the idea of its inclusion. A solution of Einstein’s equation had been found by Freidmann in 1922 (and later Lemaitre in 1927) and others which supported an expanding universe as a solution. This was a triumph for the theory to be able to describe the evolution of the universe. The formulation of a solution for the evolution of the universe was developed by Robertson and Walker in the 20s. This formulation was based on several plausible criterion for the universe. First, the matter in the universe was taken to be a uniform dust. That is, the scale was taken to be great enough that galaxies can be considered as specks of dust. Second, we are not in a preferred place in the universe (a point made several times during this course). Hence our view is not a special one. In all directions galaxies are observed to be uniformly distributed, leading to a homogeneous view. From this we can conclude that no matter where you are in the universe, matter (i.e. galaxies) are homogeneously distributed. Again, any perceived inhomogeneity could be used to pick out a special place in the universe. The third 4

fact is that no matter in which direction we look, galaxies are viewed to move away from us equally in all directions. Hence, in all directions the view of the universe is the same. The universe is isotropic. With these three conditions, the solution for the metric can be limited to a specific form. This form is called the Robertson-Walker metric,  dσ 2  ds 2 = c 2dt 2 − R(t )  + σ 2d Ω 2  . 2 1 − kσ  where σ = r/R(t). This metric describes possible forms of the universe on the large scale. The factor k is simply a sign representing the form of the curvature. It takes values of +1, -1, or 0, for positive, negative, or flat curvatures respectively. This metric, with these values of k, is the only metric for which space is isotropic and homogeneous. We will look at each of these forms in turn. [See appendix 3 for details on how the form of this metric is determined]. k = 0. For this form, at any time t the metric for the spatial part is 2 dl = R(t )  dσ 2 + σ 2 d Ω 2  , this is the space of flat Euclidean space. It extends out to infinity

but its scale, given by R(t), expands continually. k = +1. This is the case of a closed three-dimensional space of constant curvature. This space is closed, meaning that there is no boundary of this space, in addition the three dimensional space is not embedded into a fourth spatial dimension. Put another way, fourdimensional spacetime is not embedded in a five dimensional space. There is no extrinsic curvature, only the intrinsic curvature measured locally as discussed before. k = -1. This case corresponds to a hyperbolic, open space, of constant negative curvature. This type of space is not realizable as embedded into a space of one additional dimension. It is too difficult for us to envision this case. This space has no boundary as well but it is open, going off in particular direction you will never return to the same region. The scale factor R(t) basically tells us how the space involves in time. As the positively curved surface is the easiest to visualize, we will concentrate on this case. With this in mind we can limit the number of spatial dimensions down to two and draw an analogy with the surface of the balloon. In this case, the universe started from a point and evolved outwards. The entire universe is limited to the surface of this sphere. The two dimensional closed surface. Though we picture this sphere embedded in a third dimension, in fact it is not, we do this solely for our own benefit. It is important to understand that there is no other way ‘to look’ but within the surface of this sphere.

With this in mind note how the galaxies appear within this surface, they all expand away from each other. In addition, there is no center point of expansion, no point from which all is 5

‘exploding’ from. Every point is the center of the universe. Put another way, you are the center of the universe (but so is everyone else). The question is, which of these cases is our universe actually in? This is an open research question. The experimental data point to one particular case but the results are not conclusive yet. The three cases point to different futures for the universe.

For the positively curved case, the ‘gravitational attraction’ of all of the matter in the universe will cause the expansion to slow and reverse itself. For k=0, the universe expands to an asymptotic limit of a static universe. For k=-1, the universe continues to expand. There is not enough time to discuss cosmological dynamics but now we introduce the concept of the factor called Ω. For some critical density of the universe, the universe will reverse itself and collapse back into a ‘big crunch’. This critical density signifies the zero curvature case. To determine the state of the universe, the observable density of all matter is compared to this critical density. The ratio is given the name Ω = ρ/ρc. If Ω is less than 1 the universe is open, if it is greater than 1 than it is closed. To relate the observed expansion to the scale factor R(t) the Hubble constant is defined as the rate of change with time of R(t) to the value, H(t) = 1/R*∆R/∆t. This is not a constant but can change with time obviously. It is possible to relate the redshift factor, z, to this constant. (Basically, given a redshift the distance can be obtained). The value is not known precisely and its value is an active area of current research. Recently, data has been obtained by examining supernovae in far off galaxies that seem to suggest that the universe is actually accelerating in its expansion. This is not represented in the diagram above and requires other models to help explain the results, if there is one. These results may involve either bringing back the cosmological constant to help explain it, or another round of inflationary expansion (another interesting topic in which there is not enough time to discuss).

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Future Directions To this date, general relativity has passed all experimental tests with flying colors. Further more accurate tests will be conducted in the coming years, (Gravity Probe B, Gravity Wave detectors). The areas of active research, where there are still some unanswered questions, concerning general relativity are cosmology and black holes. (This is not to say all other questions have been answered). The structure and geometry of the universe is still an open question. It is becoming clear, however, that to answer these questions requires probing further and further back in time. This means probing closer to the initial singularity that was the big bang. In this regime the foundations upon which general relativity rest begin to exhibit quantum properties. The other area where this comes into play is at the singularities inside of black holes. The interplay of general relativity and quantum mechanics is a very tricky one to understand. A full theory incorporating both of these theories is yet to be developed, and may not be in our lifetime. To a certain extent certain aspects of quantum theory can be incorporated into a general relativistic description of cosmology at early times. However, this can only be pushed back so far, to about 10-42 seconds after the big bang. This may not seem to be a problem since it is such a short time, however what happens before this time dictates what happens after and a complete picture can not be developed until a theory of quantum gravity is developed. Why do we need to bring these two theories together anyways? This is a good question which could alleviate the whole problem. There are a small number of researchers who propose this view. There are technical problems with this viewpoint and, thus, does not really make the problem go away. As was mentioned above, in order to explore regimes near singularities both of these theories come into play. We can get an idea of the scale of the regime by making some basic arguments. One result of quantum theory is that energy conservation can be violated for very short times. The limit of this violation imposed by quantum mechanics is related to the famous Heisenberg Uncertainty Principle, here between time and energy. The statement is that a quantum fluctuation of energy can occur for an interval of time such that the product of the change in energy and the time interval is less than the value of Planck’s constant, i.e. ∆E ∆t ≤ = = 1.054 × 10−34 Js . Thus the larger the energy created (particles ‘pop out of the vacuum’) the shorter the time it has before they must annihilate back into the vacuum. The range over which this fluctuation can effect nearby objects is limited by the speed of light, R < c∆t. From the uncertainty = = = relation above we see that the range is R ≤ c∆t = c = c . The last term is = 2 E M pc M pc known as the Compton wavelength for a particle of mass M. This is a measure of the ‘fuzziness’ of a particle. Consider now a quantum fluctuation which occurs whose ‘radius’ Rp is less than the Schwarzschild radius, 2GM/c2. This implies that the fluctuation is a black hole. To describe such an object requires both general relativity and quantum mechanics. The scale at which this occurs is generally viewed as the regime where quantum gravity must come into play. We can easily find the scale at which this occurs, the Planck scale, by equating the two distances (Compton wavelength and Schwarzschild radius),

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RSc = RCompton →

GM p c

2

=

= =c → Mp = = 2.18 × 10−8 kg M pc G

=G = 1.62 × 10−35 m 3 c To explore fluctuations greater than the Planck mass, or equivalently to measure distances shorter than the Planck length Rp, neither general relativity nor quantum field theory can be used alone. We can also get a sense of the energy of the fluctuation and the time limit by going back to the uncertainty relation. E p = M p c 2 = 1.22 ×1019 GeV Rp =

= = 5 × 10−44 s 2 M pc Again, to describe fundamental particles with an energy of the scale of the Planck energy or time intervals less than the Planck time requires a theory of quantum gravity. We can take this time limit as the approximate limit in exploring the initial conditions of the big bang. What this argument suggests is that we can push the separate theories of general relativity and quantum field theory (quantum mechanics united with special relativity) back to a time of about 10-42 s or so. Prior to this time physics is governed by the unknown theory of quantum gravity. ∆E p ∆t p < = → t p =

If a theory of quantum gravity is developed, the hope is that it will describe the initial conditions of the universe and answer all questions about its development. Hence, such a theory would be enormously powerful. Over the past 50 years it has become clear that such a theory is not going to be easily developed. What are the difficulties in uniting these two powerful theories? There are several different ways to point out the conflicts. First, it is clear that general relativity needs to have some modification on the extremely small, high energy scale. At the center of black holes and the beginning of the universe, the theory calls for a singularity. This singularity is a point of infinite spacetime curvature and energy density. Such singularities are mathematically unacceptable. However in low curvature, low energy, regions the general theory is an accurate theory. Quantum field theory (the unification of quantum mechanics and special relativity) on the other hand is an accurate theory on short distance, moderately high energy, scales. On the large scale, low energy scale, quantum mechanics transitions to classical mechanics - a transition that is not entirely well understood. There is much work today on this transition regime between quantum mechanics and classical physics. Another problem in bringing together these two theories is the question of what exactly is being quantized. To discuss quantization, first consider classical electromagnetic theory and its quantized form quantum electrodynamics. The process of quantizing the electromagnetic theory replaces the notion of an electromagnetic wave with particles (quanta) which mediate the electric and magnetic forces. The photon is the quanta of EM radiation. Classically it is electromagnetic waves (or electric and magnetic fields) which mediate the forces. This is what is being quantized. (The process is more complicated then simply replacing waves with particles but there is no space to discuss the details). For general relativity what is to be quantized? Recall that the Einstein equation gives the metric solution for a particular distribution of energy. The metric is the ‘field’ which mediates the gravitational force (we are drawing an analogy to electricity and magnetism, again, there is no gravitational force but the curvature of spacetime). So the quantity to quantize is the 8

metric itself. Or, put another way, spacetime itself must be quantized! This is surely a strange requirement. What does it mean to replace spacetime with quanta (gravitons) which mediate the gravitational force? What do these particles propagate through, since there is no longer a continuum of spacetime? Tying this together with quantum mechanics which employs time and space as parameters to describe the wavefunctions of the quanta – here being space and time itself. It is somewhat self-referential and causes difficulties in even beginning to construct a theory. Quantum theory relies on a spacetime background, however here we are doing away with such a concept and replacing it with discrete particles. There are several different programs actively pursuing the goal of quantum gravity. Some come from the side of quantum field theory, beginning by treating gravity as any other field they would quantize, modifying general relativity in the process (string theory falls under this category). Others come from the relativist side, preferring to alter the quantization program to fit in with general relativity. Still others think that such a theory is so radical that we should start from scratch and hopefully general relativity and quantum mechanics will fall out in the appropriate limits. Each tactic has benefits and sheds light on the problem as a whole but no one program has been successful yet. To go into the details of each would send us to far out of our path. On the web site are a few, somewhat non-technical, review articles by prominent researchers discussing the problem and the different approaches.

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Overall Summary We have come a long way from the classical view of relativity in the past two weeks. There were many complex concepts to get a hold of and some difficult problems to figure out. Hopefully, there were not any insurmountable barriers. In this last section a general overview of all we have covered is presented. The idea is to give you the ‘big picture’ of this whole program.

Special Relativity: -

Incompatibility between Maxwell’s theory and Galilean relativity, as demonstrated by the Michelson-Morley experiment, and gedankenexperiments, led Einstein to posit the two postulates of special relativity, 1) All physical laws are equivalent in all IRFs. Corollary: The speed of light in vacuum is the same, 3 x 108 m/s, in all IRFs.

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The Universe is not a 3 dimensional Euclidean space with a separate, one dimensional, time. For IRFs the universe is a 4 dimensional (differentiable) manifold endowed with a Lorentzian metric. In other words, the way to define ‘distance’ (what we call interval, s) between two events is via the metric equation, ∆s 2 = c 2 ∆t 2 − ∆x 2 − ∆y 2 − ∆z 2 . This establishes the geometry of spacetime. It was shown to incorporate Einstein’s two postulates.

The following consequences of this new geometry are, -

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Time Dilation. An observer (in IRF S) will observe time pass at a slower rate for objects traveling with respect to S (in frame S’). ∆t ' = γ∆t . Non-simultaneity of events. Two events which occur simultaneously in on e IRF (are spacelike separated), may not appear simultaneous in another IRF moving with respect to the first. This was also demonstrated via spacetime maps; two events which are spacelike separated do not preserve their temporal order in different IRFs. Timelike separated events maintain their order of occurrence in all IRFs. Length Contraction. An observer in S will observe lengths to contract along the direction of motion for objects moving with respect to S. L’ = L/γ. Absolute Speed Limit, c. All forces of nature travel at or less than c. No object, thing, or information, can travel faster than light in vacuum. This was demonstrated in several ways: cause and effect as well as energy conservation are violated if objects can travel faster than light. The addition of velocity formula also prohibits objects to move greater than c, v ' +V vx = x . v' V 1 + x2 c

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Principle of Maximal Aging. Of all worldlines connecting two timelike separated events, the one which is an IRF will have the greatest proper time.

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Velocity through spacetime (4-velocity). Defined as tangent to a worldline and equal to the interval between two nearby events divided by the proper time between the two events. The components are, ∆t ∆x ∆y ∆z ∆s 2 v µ : (c , , , ) → | v µ |2 = 2 = c 2 . ∆τ ∆τ ∆τ ∆τ ∆τ The magnitude of the 4-velocity is always c.

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Momentum (4-momentum). Defined as mass times 4-velocity, analogously to classical physics. The new component (ct) is related to the energy, as its low velocity limit gives KE/c. pµ = mvµ.

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Relativistic Energy. (A scalar). Defined as the first component of the 4momentum times c.

E = γmc 2 = (mc 2 ) 2 + ( px c ) + ( p y c ) + ( pz c ) . 2

2

2

The rest energy gives us the famous equation E = mc2. -

Energy-Momentum 4-vector. Also called ‘momenergy’ in the text. Defined simply as, qµ = cpµ = mcvµ. As the magnitude of vµ is always c we see the magnitude of the energy-momentum 4-vector is always mc2.

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Momentum Conservation. (3-momentum conservation). In one IRF, the 3 momentum is conserved in all interactions if no outside forces act.

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Energy Conservation. Relativistic energy is conserved in all interactions (unlike in the classical case).

- 4-Momentum conservation. In addition to the energy and three momentum being conserved, the 4-momentum is conserved in all collisions. Spacetime plots of energy vs. momentum allowed for an easy method to find momenta and energy in different IRFs. General Relativity: -

Newtonian Gravity Incompatible With SR. Since, under Newton, it is mediated instantaneously causality will break down.

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Equivalence Principle. Physical Laws are identical in a uniform gravitational field and in a local uniformly accelerating reference frame. [Gravity ÅÆ Acceleration].

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[Curvature ÅÆ Acceleration]. By the strict definition of IRFs, a free falling frame in a gravitational field is an IRF. In the presence of gravity worldlines of IRFs are curved.

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Principle of Maximal Aging. The worldlines of IRFs have the greatest proper time between two events.

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Geodesics. The natural trajectories for objects when no outside forces are acting upon them are ones of maximal proper time. (Gravity no longer seen as a force). Lines with maximal proper time are unique and are called geodesics. (Analogs of lines of shortest distance in 3-space).

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There is no longer a gravitational force only the natural movement of objects through curved spacetime (following geodesics).

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[Energy ÅÆ Curvature]. The curvature of spacetime arises from the energy density (mass), as dictated by Einstein’s Equation, 8πG G µν = 4 T µν , c where Gµν is the Einstein tensor describing the curvature of space and Tµν is the stress-energy tensor describing the distribution of energy and momentum.

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The solution to the equation for a particular distribution of energy is the metric gµν. This metric yields the geodesics, which are the solutions.

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Recall what the metric components were for a general metric equation, ∆s 2 = g tt c 2 ∆t 2 − g xx ∆x 2 − g yy ∆y 2 − g zz ∆z 2 − g tx c∆t ∆x − g xy ∆x∆y − ....... These components contain all of the information about the geometry of the space under consideration.

Consequences of the above were shown from the equivalence principle, -

Gravitational Time Dilation. An observer far away from any source (masses) will observe time to run slower at points closer to the source.

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Gravitational Red Shift. This can be understood from the previous effect. If an object emits light at frequency f, (cycles per second), the far away observer will receive them at a greater frequency, since the time has slowed. This frequency shift is called ‘red shift’.

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Gravitational Lensing. This comes directly from the fact that now light bends around massive objects.

- Gravitational Waves. There was no time to derive the form, but considering small variations in the metric, Einstein’s equation takes the form of a wave

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equation, much like Maxwell’s last two equations. This implies that objects with changing energy density should emit gravitational waves.

- Schwarzschild Solution. The solution of Einstein’s Equation outside of a spherically symmetric mass M is given by the Schwarzschild metric. This metric is the form used to find general relativistic effects near Earth. dr 2  2GM  2 2 ds 2 =  1 − c dt − − r 2d Ω2 . 2  rc   2GM   1 −  rc 2   The consequences first gotten from the equivalence principle can now be quantitatively obtained from this form. -

Black Holes. It was noted that this metric breaks down when the term in parentheses vanishes. This is merely a mathematical singularity. An object falling in will not experience any significant in crossing the Schwarzschild radius. By a classical argument, this radius is seen as the point where the escape velocity equals the speed of light. Hence nothing can escape once inside of the horizon.

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Inside the Horizon. By examining the solution inside of the horizon it was seen that once inside of the horizon, all timelike worldlines terminate on the singularity at r =0. However, the Schwarzschild coordinates are in terms of coordinates for an observer at infinity. This makes descriptions of the inside suspect, since no information can escape. To make since, new coordinates which are more natural for an in falling observer were used (Eddington-Finkelstein coordinates). With these coordinates it was again observed that all worldlines terminate at the singularity.

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Cosmology. A quick discussion of how GR relates to the history of the universe was presented. Using some guiding principles (isotropy and homogeneity) the Robertson-Walker metric was motivated as a metric for the universe as a whole.

Punchline: The one main point that you should come away with is that special and general relativity are nothing but geometry.

What was Left Off Of course we could not cover all of relativity in the short time we had, so here is a list of some topics which are traditional presented in a more formal course. Some were left off due to their complexity, others because of time limitations. Special Relativity:

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Relativistic Acceleration. This can be easily derived from the formula for addition of velocities. E and B fields. How do electric and magnetic fields behave under Lorentz transformations. Fluid dynamics.

General Relativity: - Other solutions/metrics. (Kerr metric, Kerr-Newman metric). - Energy and momentum in general relativity (especially with the Schwarzschild metric). - Further explorations of black holes. (Read Taylor and Wheeler for this). - Further explorations of cosmology.

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Appendix 1 Spaces of Constant Curvature. 2-Sphere. [Note: This appendix is adapted from chapter 2 of “The Early Universe” by Edward Kolb and Michael Turner. Calculus is used in this appendix.]

In order to demonstrate how the form of the Robertson-Walker metric arises for isotropic, homogeneous spaces we will examine the simple case of 2 dimensional spaces. For 2 dimensional spaces, isotropic and homogeneous clearly leads to spaces of constant curvature, (if the curvature were not constant, then different points would not be the same). Hence we want to examine spaces of constant curvature. There are three cases to consider, positive, negative, and zero curvature. A space zero curvature is simply the flat plane (R2) and so we do not discuss this case. A two dimensional space of constant negative curvature (H2) can not be pictured in 3 dimensions, it requires 3 additional fictitious dimensions to display such a surface. Due to this difficulty, we will not discuss this case here. A two dimensional space of constant positive curvature is simply the two sphere (S2). Where ever you go the curvature is the same. To visualize this surface requires one additional fictitious dimension, hence this is the case we will examine. To visualize a two sphere, we generally embed it into a three dimensional Euclidean space. Let’s carry this out explicitly. Once placed within a fictitious three-dimensional space, the equation for a two sphere is simply, x12 + x22 + x32 = R 2 To consider this in the three Euclidean dimensions we note the metric for that space is, dl 2 = dx12 + dx22 + dx32 . To embed this surface and find the metric on the sphere, (coordinates x1 and x2), we use the equation for the surface and substitute it into the Euclidean metric. Thus, ( x dx + x dx ) ( x dx + x dx ) 2 x3 = R 2 − x12 − x12 → dx3 = 1 1 2 2 or dx32 = 1 2 1 2 2 22 . Inserting into the R − x1 − x2 R 2 − x12 − x22 metric we get, ( x1dx1 + x2 dx2 ) 2 dl = dx + dx + . R 2 − x12 − x22 Changing coordinates to polar coordinates in the x3 plane, x1 = r’cosθ, x2 = r’sinθ, and x32 = R 2 − r '2 we get for the metric, 2

2 1

2 2

R 2 dr '2 + r '2 dθ 2 . 2 2 R −r' The last step is to define a new coordinate, r = r’/R we get,  dr 2  dl 2 = R 2  + r 2 dθ 2  . 2 1 − r  Now compare this to the form of the Robertson-Walker metric for the case of positive curvature (k=+1). To extend this to a three-dimensional space of constant curvature is dl 2 =

straightforward.

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Interpreting this space as that of an expanding universe we see that as R changes, the space remains isotropic and homogeneous. The coordinates r and θ are unchanged as the scale factor R increases. These coordinates are called ‘comoving’ coordinates. The space of constant positive curvature is bounded, you can orbit the space and return to the same point. Spaces of constant negative and zero curvature are open, or unbounded. For the case of zero curvature, the scale factor is merely that, it represents the scale between nearby points. The space itself is infinite, quite a different picture from the k = +1 case. Now if the intrinsic geometry of the universe is determined to be the flat, k=0, case it is still possible for the global, or extrinsic, geometry to be different. For example, the topology of the surface may be different, say a cylinder or torus. In these cases the local geometry is flat but one can orbit the universe and return to the starting point. This possibility leaves open the case of extra dimensions called for by string theory. In these, as yet unproven, theories, extra dimensions exist to make the theories consistent. To explain the lack of perception of these dimensions it is explained that they curl up into tiny closed spaces of dimensions too small to explore with modern accelerators. Hence, the universe may not be simply a four dimensional manifold endowed with the characteristics described before. There could be several more dimensions in the topology of a torus, or cylinder, of very small radius. These dimensions do not affect the large-scale structure of the universe.

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