3 Spacetime metrics. 3.1 Introduction. 3.2 Flat spacetime

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1 3 Spacetime metrics 3.1 Introduction The efforts to generalize physical laws under different coordinate transformations would probably not have been very successful without differential calculus. Riemann ( ) introduced multidimensional spaces and differential geometry, a necessary instrument in the development of the Theory of Relativity. In Euclidian space the distance between 2 points can be calculated from Pythagoras theorem. Under Special Relativity this is not an invariant, i.e. it changes its value when we measure it in different inertial systems. But it could be made invariant by adding time to space and work with spacetime instead of space. Hermann Minkowski defined the 4-dimensional spacetime in Flat spacetime In spacetime, the separation or interval s, defining the distance between two events like the Pythagoras theorem is defined by the metric (the definition of the separation and the sign convention varies. I will follow Rydens textbook): s 2 = c 2 τ 2 = c 2 t 2 + x 2 + y 2 + z 2 (19) t is the coordinate time, sometimes also called real time. xyz are the space coordinates. This space is non-euclidean but flat. It has a Lorentzian signature ( ) as opposed to Euclidean (++++). It is flat because a vector parallell transported along a closed curve always returns to its original position (cmp. e.g. to what happens if you move a vector on the surface of a sphere). τ is the proper time and may be regarded as your local, personal time. It is also called clock time and process time, because it measures the distance in time between two events, e.g. the swing of a pendulum from one position to another. τ is what your clock considers to be the time interval between two local events. t is the time interval you read from your clock between two events that take place elsewhere. If e.g. these processes happen in a coordinate system in motion relative to you or in a gravitational field stronger than your local field they will run slower. Thus, a clock riding along with a photon would stand still as viewed from an observer at rest (which is of course always the case since the speed of light is the same in all coordinate systems). Thus your clock would say that it would take an infinite amount of time to register the time between e.g. second marking ticks on the moving clock. 9

2 In the Minkowski spacetime we can define two regions related to our own position, one timelike and one spacelike. They are separated by the lightcone, defined by the incoming and outgoing light rays. Time runs along the symmetry axis of the lightcone and space is perpendicular to time. A separation between two events in spacetime is timelike if inside the cone and spacelike if outside. At the intersection the separation is lightlike, since this is the trajectory of a photon. Matter cannot travel along spacelike trajectories. Light and massless particles travel along the lightcone. The trajectory of a particle in spacetime is called a worldline. The trajectory of a particle in free fall in spacetime is called a geodesic. The position along the worldline is given by the proper time τ. Remember that the value of this parameter is independent of the observer, invariant under coordinate transformations. For two events occuring simultaneously τ 2 = ( x 2 + y 2 + z 2 )/c 2 = r 2 /c 2, where r is the proper distance between the two events. From this we can derive the Lorentz contraction formula. The Minskowski metric can be written: τ 2 = s 2 /c 2 = t 2 ( x 2 + y 2 + z 2 )/c 2 = η µν x µ x ν (2) µ,ν η is the metric tensor, describing the metric of the coordinate system. Tensors are convenient to use in differential calculus because the equations can be written in a more compact form. Here however, we will follow the textbook and refrain from using tensors in the following. By definition, the equivalence principle states that in a freely falling coordinate system, SRT applies and we can use the Minkowski metric to describe the spacetime. The metric can be used to derive the time dilation in SRT. Since v 2 t 2 = x 2 + y 2 + z 2 and using eq. 2 we obtain t = τ/ (1 v 2 /c 2 ) (21) A person who watches a clock onboard a spaceship in motion will notice that the time runs slower than on his own clock. In this way travelling salesmen will grow older more slowly than their family members at home. A business man using a few flight connections per week to visit his business partners will typically gain 1 5 s during his lifetime. The separation between 2 events A and B can be expressed as 1

3 B τ AB = dτ (22) A A free fall trajectory, a geodesic, requires that the proper time τ AB is minimized along a straight world line joining the two events. This statement is independent on which coordinate system is in use and is said to be generally covariant. To modify equations of physics such that they obeyed general covariance was one of Einsteins major goals. He stated: The general laws of nature are to be expressed by equations which hold good for all systems of coordinates. that is, are co-variant with respect to any substitutions whatever (generally covariant). 3.3 Curved spacetime If we carry out a coordinate transformation from an inertial system in free fall to an arbitrary coordinate system it can be shown that the time derivative of the velocity along a world line contains a term, the affine connection, that acts as an acceleration if the metric is not flat. Einsteins gravity can thus be regarded a property of curved spacetime. Cmp. Newton, who regarded the gravitational force as a result of action at a distance. The tensor that describes the curvature of spacetime is the Riemann-Christoffel (or simply the Riemann) tensor R α βγδ, corresponding to the Gaussian curvature, K, in three dimensions. In general relativity a relationship between the distribution and flow of massenergy in the universe and the geometrical properties of spacetime is established on the basis of the cosmological principle. This assumes that we can regard the universe as a perfect fluid. A perfect fluid is defined such that an observer in motion sees the fluid around him as isotropic. This will be the case if the mean free path between collisions is small compared with the scale of lengths used by the observer. For instance, the sound wave will propagate in air if its wavelength is large compared to the mean free path, but at very short wavelengths viscosity becomes important and the air stops acting like a perfect fluid. The connection between the density and flow of energy and momentum, given by the energy-momentum or stress-energy-momentum tensor, and the metric of spacetime, given by the Einstein tensor, are described by 1 independent non-linear partial differential equations called the Einstein field equations. 11

4 4 Cosmic kinematics 4.1 The geometrical framework From the cosmological principle it follows that spacetime has a constant curvature. This simplifies the modelling quite considerably of course. First we will have a look at the metric of a 2-dimensional surface embedded in 3 spatial dimensions. Most of the time I will use the same variable names as in the textbook (Ryden). In we defined the parameter x S k (r), which is the radial coordinate of a small circle on this surface projected on a plane and φ and θ are polar coordinates. Let us look at the metric of a 2-dim space with constant curvature as a function of the two spatial coordinates x and φ in the projected coordinate system instead of the r and φ grid on the surface itself as derived in sect. 2.2: ds 2 = f(x)dx 2 + x 2 dφ 2 (23) f(x) is an unknown function of x. In a flat, Euclidean 2-dim. space f(x) 1. In the case of a positively curved 2-dim space f(x) > 1 since for a circular area πs 2 > πx 2. Introduce the polar coordinate θ (sweeping along the meridian of a sphere if positively curved). Then the proper distance is d p = x Derivate with respect to x: f(x)dx = Rθ = Rsin 1 x R 1 1 f(x) = R 1 (x/r) 2 R f(x) = (24) (25) 1 1 (x/r) = 1 (26) 2 1 Kx 2 K is the curvature constant. Thus we may write the metric of a 2-dim. maximally symmetric surface as ds 2 = dx2 1 Kx + 2 x2 dφ 2 (27) The metric of a 3-dim space with constant curvature may be expressed: 12

5 ds 2 = dx 2 (1 Kx 2 ) + x2 dθ 2 + x 2 sin 2 θdφ 2 (28) Now introduce the scale-free radial distance σ, which in fact is the scale-free angular distance discussed below: σ = x/r (29) Let us also express the scale dependent curvature K in a form which contains the scale free curvature constant k as in eq. 18. k is independent of the expansion or the change of scale of the universe but contains the information about if the space is flat, negatively or positively curved: Now eq. 28 will read K(t) k R 2 (t) k = 1 k = k = 1 P os F lat Neg (3) ds 2 = R 2 dσ 2 ( (1 kσ 2 ) + σ2 dθ 2 + σ 2 sin 2 θdφ 2 ) (31) If we keep θ and φ constant we can express the proper distance, i.e. distance you measure with a rod (cmp. eq. 24), as d p = r dx σ = R dσ 1 Kx 2 1 kσ 2 the (32) Now let us switch to 4-dim spacetime and introduce the comoving coordinate system. Imagine this to be defined by objects in free fall, i.e. following their worldline, each of them carrying a clock. Then τ=t. One may think of a mesh, or a coordinate grid, following the objects in the dynamical evolution of the universe. In a 2-dim analogy of a positively curved expanding universe the positions of galaxies would be represented by points on a ballon that is inflated. But remember that in the general relativistic universe the expansion is not affecting the small scales (your body is not partaking in the expansion). The reason is that the Friedmann equations are based on the cosmological principle, assuming homogeneity. On smaller scales, of the size of a large cluster of galaxies, this is no longer correct. If we set the clocks according to the cosmological principle we may express the spacetime metric in this coordinate system: ds 2 = c 2 dτ 2 = c 2 dt 2 + R 2 dσ 2 (t)( (1 kσ 2 ) + σ2 dθ 2 + σ 2 sin 2 θdφ 2 ) (33) 13

6 This is the Robertson-Walker metric or simply the R-W metric. ds is also called the Robertson-Walker line element (1934). Remember that k has been chosen such as to become constant. The parameters used in this equation are frequently used in classic textbooks to express the metric. σ, Θ and φ are all comoving coordinates, i.e. they are like the coordinates of a grid that grows in size, following the expansion. We will now however switch to the notation used in Rydens book. Define R(t) = a(t)r (34) a(t) is called the cosmic scalefactor and is unitless. R is expressed in length units. In a positively curved universe R may be called the radius of the universe. But it is not physically existing in the 3-dimensional space we live in since we cannot simultaneously measure the distance R or point in the direction of R. In order to do this we would have to live in a 5-dimensional spacetime with 4 spatial coordinates. The R-W metric in Rydens notation reads ds 2 = c 2 dτ 2 = c 2 dt 2 + a 2 dx 2 (t)( (1 kx 2 /R) + 2 x2 dθ 2 + x 2 sin 2 θdφ 2 ) (35) or in a more compact notation where now and ds 2 = c 2 dτ 2 = c 2 dt 2 + a 2 dx 2 (t)( (1 kx 2 /R) + 2 x2 dω 2 ) (36) x = σ/r dω 2 = dθ 2 + sin 2 θdφ 2 The metric can also be expressed with the radial coordinate r instead of x ds 2 = c 2 dτ 2 = c 2 dt 2 + a 2 (t)(dr 2 + S k (r) 2 dω 2 ) (37) The coordinates used here (x, r etc.) are comoving coordinates. The x and r have dimensions of length while a is dimensionless and has the value a(t ) = 1 today. The time dependent proper distance is r d p = a(t) dr = a(t)r (38) 14

7 We obtain (see eq. 17) R sin 1 (x/r ) k = 1 d p = a(t)r(x) = a(t) r k = R sinh 1 (x/r ) k = 1 The proper velocity v p d p can be expressed as (39) d p = ȧ(t)r = ȧ(t) a(t) d p = H(t)d p (4) For small r, Hd p Hr. This relation is called Hubbles law (Hubble 1929). H is the Hubble parameter which at the present moment is called the Hubble constant H. A recent determination (Riess et al. 211, ApJ 73, 119) using the Hubble telescope, gives H =73.8±2.4 km s 1 Mpc 1. Instead of H, it is common to use the related dimensionless parameter h = H/ Cosmological redshift One of the fundamental observational supports of the Big Bang universe is the redshift of distant galaxies, manifested in the Hubble law. The most popular interpretation is that the redshift is caused by the expansion of the universe. This may be tested using the Tolman test. The prediction is that the surface brightness of an object should scale as (1+z) 4 (see above). The problem with performing the test is however that we have to use galaxies at large distances, which are affected by evolutionary effects for which we have limited information. Alternative interpretations of the redshift have been discussed (e.g. as a tired light effect or the new physics interpretation of the redshifts of certain QSOs by Halton Arp and colleagues). The trajectory of a light ray satisfies the equation c 2 dτ 2 = = c 2 dt 2 a(t) 2 dr 2 (41) Assume that the emission takes place at time t e and reaches us at t If two crests of the lightwave are emitted at t e and t e + t e and are received at t and t + t then t c t e dt r a(t) = dr (42) Since the relative change in R during the time span between t e and t e + t e is negligible, approximately the same condition holds for the next crest: 15

8 t + t c t e+ t e dt a(t) (43) c t a(t ) c t e a(t e ) t = a(t ) t e a(t e ) (44) Let the wavelength of the emitted radiation be λ e and that of the received λ. Then the cosmological redshift is obtained from 1 + z cosm = λ λ e = c t c t e = a(t ) a(t e ) = 1 a(t e ) (45) In addition to this redshift we also have to take into account the local redshift z local, due to the peculiar motion of the galaxy, typically of the order of a few hundred km s 1, and the gravitational redshift, z grav, e.g. when observing regions close to a black hole in the centre of active galactic nuclei: (1 + z obs ) = (1 + z grav )(1 + z local )(1 + z cosm ) (46) If we expand a(t) around a(t ) (see Ryden, chap. 7) we find a(t) = 1+(t e t )ȧ(t )+ 1 2 (t e t ) 2 ä(t )+... = 1+H (t e t ) 1 2 q H 2 (t e t ) q is the decelerationparameter: (47) q = (äa ȧ ) 2 t=t = ( ä ah ) 2 t=t (48) Inverting equation 47, we find a relation between the Hubble constant, the redshift, the deceleration parameter and the timespan which we will use later: and z = H (t t e ) + ( q )H 2 (t t e ) (49) t t e = 1 H [z ( q )z ] (5) 16

9 4.3 Distances We have already discussed the coordinate distance x, the dimensionless coordinate distance σ and the proper distance d p. Let us look at how different ways of determining distances link together. We are used to being able to determine distances using the apparent brightness of an object. The apparent brightness is simply proportional to the distance 2. In a curved spacetime this is not necessarily so. The luminosity distance, d L, can be defined as d L = ( L 4πf )1/2 (51) where L is the total luminosity (e.g. in Watt) and f is the apparent flux (e.g. in Watt/m 2 ), as we astronomers call it. In a R-W universe the light is spread out across a sphere of proper radius d p (t ) = r and a proper surface area, A p, of A p (t ) = 4πS k (r) 2 (52) The cosmological redshift will decrease the energy of each photon with a factor (1+z). In addition, the proper distance between the photons, and thus time between the reception of the photons, will also increase with a factor of (1+z). These two effects will cause a decrease of the measured luminosity with a factor of (1+z) 2. Thus the observed flux will be f = and luminosity distance to an object will be L 4πS k (r) 2 (1 + z) 2 (53) d L = ( L σ e 4πf )1/2 = S k (r)(1 + z) = R a(t e ) (54) Likewise we may define an angular diameter distance d A of an object of diameter l and apparent diameter δθ, which is equal to the coordinate distance S k (r) at the time of emission d A = l δθ = S k(r e ) = S k (r)/(1 + z) = R σ e a(t e ) (55) We can also define the parallax distance d P = R σ e / 1 kσ 2 e and the proper motion distance d M = R σ e Notice the simple relationships d A /d L = (1 + z) 2 and d M /d L = (1 + z) 1 These distance measures are the same at small distances d r. 17

10 The surface brightness s of an object with apparent diameter δθ can be expressed as s = 4f πδθ 2 = L 4πS k (r) 2 (1 + z) 2 πδθ 2 = This is the basis of the Tolman test. Ld 2 A π 2 S k (r) 2 (1 + z) 2 (1 + z) 4 (56) If we express σ e in t and t e and expand, using eq. 49 we obtain: l = LH2 4πz 2 [1 + (q )z +...] (57) m M = 25 5logH + 5logcz (1 q )z +... (58) This m-z-relation has been much used in cosmology to determine the parameters H and q under the assumption that M is idependent of z. H may then be determined at low redshifts but sufficiently large so it reflects the general cosmic expansion (z.1). q can be obtained from the shape of the relationship at high z. Thus small z H and larger z q. 4.4 Horizons The speed of light limits our vision of the universe. Below we will define two types of horizons related to this fact The particle horizon or object horizon Let us find an expression for the proper distance to the so called particle horizon, d P H, i.e. the distance to the most remote object in the universe we can observe today. That object emitted its light ɛ after the Big Bang at time t=. The trajectory of a light ray emitted from the object at this moment fulfills the condition dτ= cdt = a(t)dr (59) t d P H = c dt a(t) (6) 18

11 4.4.2 The event horizon Under certain circumstances the properties of the universe are such that there are objects that are so distant that we will never be able to see them. They are outside the event horizon. To find an expression which we can use to calculate this distance we simply take t min and t max as integration limits in eq. 6. If the universe expands forever, t max = and if the expansion stops and the universe starts contracting, t max is the moment of the Big Crunch or the gnab gib. 19