General relativity, 7

Transcription

1 General relativity, 7 The expaning universe The fact that the vast majority of galaxies have a spectral reshift can be interprete as implying that the universe is expaning. This interpretation stems from the Doppler effect in which the relative motion of an emitter an a etector prouces a frequency shift of the etecte light with respect to the emitte light. The frequency shift formula can be written as a wavelength shift formula: λ = 1+ β 1 β can be written in terms of relative spee as z = 1+ β 1 β 1/ 2. Now, the galactic spectral shift factor z = λ 1/ 2 1. For β << 1, the square root on the right han sie can be evaluate using the binomial expansion (what else?, with the result being z β. In 1931, Hubble publishe observe reshift ata for 20 or so galaxies containing visible Cephei variables, which coul be use to etermine the galactic istances ( D from Earth. The z -values he measure range up to about 0.05 an the corresponing istances range to about 100 Mly. Hubble foun a remarkably linear relation: z KD, where K was a constant. If the galactic spectral shift can be interprete as ue to motion, then, except for a very small number of close-by galaxies that seem to be moving ranomly relative to the Milky Way, most galaxies seem to be moving away from us the farther, the faster. Assuming that there is nothing physically unique about the Milky Way, this galactic recession shoul be observable from any galaxy. Thus, Hubble s z-d relation appears to imply the universe is expaning more-or-less equally in all irections. Since Hubble s original work, many more galactic reshifts have been measure using, appropriately enough, the Hubble Space Telescope. In aition, a new kin of stanar canle namely, light from supernovae type Ia has greatly expane our istance measuring capability. At present, the Hubble z-d rule is well establishe for z up to about 0.1. It is conventional to write z KD as v KcD = H 0 D, where H 0 is known as the Hubble constant. The best-measure value of H 0, base on reshifts of about 200 galaxies, is 72±7 km/s/mpc. One Mpc (mega-parsec = 3.262x10 6 ly;; thus, z = v c = 0.1 correspons to D 4x10 8 ly. More recently, ata from the WMAP satellite (more about WMAP later using a ifferent, inirect measurement puts the value of H 0 at 70.0±2.2 km/s/mpc. Notice that the imensions of H 0 are LT 1 /L = T 1, so 1 H 0 is a time. Example: 1 ly = ( c (1 y = km-y/s. 1 Mpc = km-y/s = kmy/s. Thus, 1 H 0 = 1 Mpc/70 km/s = km-y/s/70 km/s = y. As we will see this is an extremely important value. Friemann-Lemaître-Robertson-Walker (FLRW universe We woul like to have a theoretical framework for unerstaning Hubble s observations as well as the other central facets of moern cosmography, in particular, the cosmic microwave backgroun an the observe cosmic hyrogen-helium-euterium-lithium abunances. Because Einstein s general relativity is so successful at joining relativity an gravity for a single massive object, it is natural to ask whether GR can also provie a usefully preictive escription GR 7 1

2 of gravity at the cosmological level. The starting point for such a moel is the assumption that there are, on average, no special places in the universe (on average. Thus, the moel assumes that the universe is spherically symmetric an uniform everywhere. (That approximation is more believable on large length scales where clumpiness becomes less obvious. That might be 10s or 100s of millions of light years. The goal is to fin a solution to Einstein s equations that is vali insie the uniform mass-energy istribution of such a universe. Alexaner Friemann prouce the first solution. His 1922 paper on the subject annoye Einstein, because the size of Friemann s universe ha to be ynamically changing something Einstein knew was wrong. Remember, this was before Hubble iscovere other galaxies an their associate re shifts;; at that time, common wisom ictate that the universe must be static, always as it now appeare, an infinite in extent. (Such a universe poses a ilemma: every irection in the sky shoul intersect a star, so the night sky shoul be bright in every irection which it clearly is not. This is known historically as Olbers Paraox. As we will see, Hubble s result fixes this problem. In fact, Einstein alreay knew that his original 1915 version of general relativity ha to escribe a ynamically evolving universe, an to stop it, he propose a moification that inclue a cosmological constant term a kin of fuge factor that mae his equations time-inepenent. Einstein wrote a letter to the journal in which Friemann s paper appeare saying that Friemann ha mae a mathematical error. Friemann was sure that he han t an for two years trie to get Einstein to retract his criticism. Eventually, Einstein recante, but still maintaine that Friemann s calculation was physically irrelevant. Friemann ie shortly after, without seeing the vinication of his work that woul result from Hubble s. In 1927, Georges Lemaître, traine as both a physicist an a Catholic priest, reinvente Friemann s solution, without having rea Friemann s paper. He sent a manuscript to Einstein etailing his expaning universe, which elicite, in essence, the response: Yes, I know this. Friemann has alreay showe it to me. His physics was wrong an so is yours. In a subsequent paper, Lemaître went even further. He propose that if the universe is now expaning, earlier it must have been enser an hotter, an that all of the matter in the universe might have originate from the ecay of one giant raioactive atom. Of course, once Hubble s finings became known, Einstein ha to reconsier his objections. Famously, he calle introucing his cosmological constant the biggest mistake of his life (as we will see, getting ri of it was even bigger! an warmly an publicly embrace Lemaître s primeval atom iea ( this is the most beautiful an satisfactory explanation of creation to which I have ever listene. (Lemaître was inucte into the Pontifical Acaemy of Science in 1936 by Pope Pius XI for proposing a plausible scientific rationale for the creation story in Genesis, though Lemaître never claime such a connection himself. Finally, in 1935 Howar Robertson an Arthur Walker prove that the solution of Einstein s gravitational equations by Friemann an Lemaître is the only one possible insie a spherically symmetric, uniform mass istribution. Curiously, (in America the solution is now often referre to as the Robertson-Walker metric, but more properly shoul be attribute to the whole cast of characters. In the literature, this attribution is usually abbreviate FLRW. Though not the usual convention, it is more poetic an hopeful to invert the R an W, as we o subsequently, giving FLWR the flower universe. In the FLWR universe, proper time is given by ( τ 2 = ( 2 a(t ( 2 2 r 1 k(r r 0 + r 2 θ 2 ( 2 + r 2 sin 2 θ ( φ 2. (1 GR 7 2

3 Note that in (1 time is the same everywhere, so all clocks are always synchronize. The parameter r 0 is a proper length scale (e.g., the raius of the visible universe at some reference time, such as the present. The space part of (1, without the a -multiplier, marks off coorinate istances, physical istances at the reference time. In this s-t, free-falling objects are assume to be at fixe coorinates (r,θ,φ an physical istances between them change in time accoring to the scale factor a(t, which is imensionless. The parameter k can be 0, ±1. If k = +1, the spatial part of the corresponing s-t is close (i.e., is finite in extent an positively curve, much like the surface of a sphere (although here the surface has three imensions. If k = 1, the space part is infinite in extent an negatively curve. If k = 0, the space part is infinite in extent an flat. The FLWR s-t has a constant spatial curvature equal to k r 0 2. In the FLWR s-t, galaxy clusters are like the proverbial raisins in a loaf of raisin brea that is rising: as the ough expans it causes the raisins to grow increasingly farther apart. In the FLWR s-t, the expaning ough is space itself;; the raisins, the galaxies, are going along for the rie. FLWR kinematics: galactic reshift an energy ensities Note that (1 is a kinematic statement: it harbors certain qualitative consequences for how light an matter move about in the universe. Equation (1 oes not tell us, however, anything irectly about the values of a or k, or where they come from. The situation is a little like the statement, near the surface of Earth all objects fall along parabolic trajectories. From this information one can erive certain consequences, for example, that the acceleration each boy experiences is inepenent of position an time. You coul also preict how things like times-of-flight an maximum ranges woul epen on initial conitions, an from these you coul etermine g E if you ha enough trajectory ata. You coul not, however, use that information to etermine Newton s law of gravity that is, g E s origin. In the same sense, to etermine where a an k come from we nee ynamics, namely, Einstein s equations in which correct values of cosmic masses an energies are inserte. By itself, (1 is useful for escribing cosmological kinematics, primarily associate with how light propagates through a universe with a varying scale factor. Matter in the FLWR s-t is continuously istribute, like a flui or ust. We can imagine that every point in s-t has a bit of that flui attache to it. The FLWR s-t eals with enormous istances an times;; we can therefore interpret these bits of flui as galaxies or better, clusters of galaxies. In the FLWR metric all places in the universe are ientical, so all can equally serve as the spatial origin. We might as well, therefore, take r = 0 to be at the Milky Way. Suppose a crest of a light wave is emitte from a galaxy at coorinate position r = R relative to the Milky Way at time T e. The next crest is emitte at T e + ΔT e. The first crest is etecte by us at r = 0 at time T ( now ;; the secon reaches us a little later at time T + ΔT. During the time it takes light to travel to us, the scale factor of the universe changes, so the time between crests at etection is not the same as at emission: ΔT ΔT e. As a result, there s a spectral shift ue to the change in a. To etermine the effect of this, recall that light travels through s-t with zero elapse proper time. Set τ an the angular terms in (1 equal to zero. T Then a(t = 0 r T + ΔT T = R 1 k(r r 0 2 e, because the coorinate (not physical istance T e + ΔT e a(t GR 7 3

4 travele (the mile integral is the same for both crests. The time between crests of light is much, much shorter than the time for light to go from one galaxy to another, so the last integral can be approximate by T + ΔT T e + ΔT e a(t T T e a(t + ΔT a(t ΔT e. In other wors, ΔT ΔT e = a(t, the ratio of the perio of etecte light to that of the emitte light is the same as the ratio of the scale factor at etection to the scale factor at emission. Since wavelength of light is irectly proportional to perio we can write λ = a(t. More importantly, we can express the galactic spectral shift as z = a(t 1. (2 Equation (2 is a irect result of the kinematics of the FLWR s-t. To evaluate z for any particular signal requires knowing the two a s. On the other han, we know from Hubble s an others work that z for all sufficiently istant galaxies is > 0, so it must be that the ratio of the a s is > 1. The scale of the universe is apparently greater (at least at this moment in the universe s history as time goes on. In other wors, the FLWR moel s-t tells us that positive z implies cosmic length scale expansion. In this moel, the galaxies are all receing, not because they were all once together at the same point an a mighty explosion hurle them out into space, but because, like the raisins in the rising ough, the istance scale of space is increasing. Equation (2 leas irectly to Hubble s law z H 0 D c, for small values of z. Small z means that can be approximate by the first two terms of a Taylor expansion: a(t a z = 1 a a ΔT, where ΔT = T T e. Inserting this into (2 prouces ΔT. The quantity ΔT is approximately the istance, D, light travels between emission an etection (at least for nearby galaxies an the coefficient 1 a a = H 0 c Hubble constant (ivie by c. It is customary to set a(t = now = 1. When that s one is the Equation (2 can be expresse as z = 1 1 or as a(t 1 e = (1+ z. (3 The larger is z, the smaller is a, an the oler must be the light we observe for that source now. Measure values of D, augmenting Cephei istances with istances inferre from supernovae type Ia, show that Hubble s z H 0 D c hols pretty well for z up to about 0.1, but that z increases faster than D for larger values. A irect interpretation of this phenomenon is that a change from its current behavior to a ifferent one sometime in the past. We will see in a bit what the contemporary unerstaning of this shift in a was ue to. Before turning to how a is linke to gravitational ynamics, it is important to note another purely kinematic consequence of the FLWR s-t. The universe is fille (in the uniform GR 7 4

5 cosmic sense with matter/energy in various forms, all of which contribute to the cosmic gravity. Some of the matter/energy is highly relativistic e.g., photons, for sure, an possibly also neutrinos. Such entities travel at or near the spee of light an their contribution to the total gravity is associate with their kinetic energies. Some forms of matter/energy are nonrelativistic, like atoms an ark matter, for example, an their contribution is funamentally from rest energy. As the scale factor of the universe changes the relativistic an non-relativistic energy ensities change also, but ifferently. To a very goo approximation, the relativistic particles carry so little energy now that they on t interact with non-relativistic matter anymore. In that event, the number of photons, neutrinos, electrons, an nucleons are all fixe. The energy ensity in non-relativistic matter is approximately just the rest energy of matter per unit volume. Volume is proportional to the scale factor, a, raise to the three-power. Thus, we can 1 write the energy ensity of non-relativistic matter as ρ m (T = ρ m0 a(t, where the subscript 0 means, value measure at the present time. The rest energy (if any of a highly relativistic particle is negligible an its total energy can be expresse in terms of its e Broglie wavelength as hc λ. As a changes, the wavelength changes also. This leas to a ifferent energy ensity 1 epenence, namely, ρ r (T = ρ r0 a(t 4. Though toay the ensity of non-relativistic matter/energy is estimate to be about 5500 times larger than that of relativistic matter/energy, at some point in the past the two must have been equal. That woul have occurre at an a - value of a R = ρ r0 ρ m0 = 1/ 5500, provie the number of photons then is the same as toay. In fact, the number of photons change before this a -value an there s goo evience that a R is actually more like 1/3200. For times earlier than when a R = 1/ 3200 the universe must have been ominate by relativistic particles photons, neutrinos, an other particles whose kinetic energies were much greater than their rest energies. That early epoch of the universe is calle raiation ominate. 3 GR 7 5