4.2 METRIC. gij = metric tensor = 4x4 symmetric matrix

Transcription

1 4.2 METRIC A metric defines how a distance can be measured between two nearby events in spacetime in terms of the coordinate system. The metric is a formula which describes how displacements through a curved manifold can be translated into distances it relates coordinate separations to lengths. Whether we use the term or not, we are all familiar with metrics. For example, using the Pythagorean theorem, we can write the distance between two points in ordinary 3D space as This is the so called Euclidean metric. In the special theory of relativity this becomes generalized to the distance between two spacetime points: This is the Minkowski spacetime. Its infinitesimal version can be written dx0 cdt gij = metric tensor = 4x4 symmetric matrix

2 <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> where the only non-zero component of gij are the diagonal terms (-1, 1, 1, 1). Note that the line element is invariant under Lorentz transformations =1/ p 1 v 2 /c 2 v as well as under coordinate transformations

3 Observers in relative motion disagree on spatial separation Observers in relative motion disagree on time separation Observers in relative motion agree on spacetime separation Metric turns observer-dependent coordinates into invariants! If two points x i and x i +dx i can be connected by a light ray, then photons propagate along null trajectories!

4 Example: In a homogeneous and isotropic Universe there are no off-diagonal terms in gij. I shall give the argument that proves that there are no terms of the form, say, dxdt. Let s set dy=dz=0 so that ds 2 =dx 2 c 2 dt 2. If we now set ds=0, we have dx/dt=±c, which is the equation of a light ray propagating along the positive or negative x-axis ( null trajectory''). Let us suppose now we have a metric of the form ds 2 =dx 2 + dxcdt c 2 dt 2. If we set ds 2 =0 for this metric, there will be two very different solutions for dx/dt: dx/dt=(c/2) ( 1 ± 5). In this new case, because of the dxcdt term, the symmetry of the flat space case is destroyed.

5 Another way of thinking about the metric: when handed a vector (say connecting two grid points) we think of a line with an arrow (direction) attached, the length of the line corresponding to the distance between 1 and 2. This notion is rooted too firmly in flat Euclidean space! In actuality, the length of the vector depends on the metric. In the figure, the number of lines crossed by a vector is a measure of the vertical distance traveled by a hiker. Vector of the same apparent 2D length corresponding to identical coordinate separations corresponds to different physical distances. Contour map of Mauna Kea. Closely spaced contours near the center corresponds to rapid elevation gains. The two thin lines correspond to hikes of very different difficulty even though they appear to be of the same length.

6 k=r 2 > 0 is the Gaussian curvature The great advantage of the metric is that it can incorporate gravity! Instead of thinking of gravity as an external force and talking about test particles moving in a gravitational field, we can include gravity in the metric and talk of test particles moving freely in a distorted or curved spacetime. metric on the surface of a 2D sphere: R R R R dl The polar angle θ is measured from a fixed zenith direction, and the azimuth angle (in a reference plane that passes through the origin and is orthogonal to the zenith) is measured from a fixed reference direction on that plane. new radial coordinate: r=rsinθ flat space

7 R R R R

8 negative Gaussian curvature

9 <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> FRIEDMANN-ROBERTSON-WALKER (FRW) METRIC The metric for an expanding spacetime that has homogeneous and isotropic spatial sections takes the Robertson-Walker form ds 2 = c 2 dt 2 + a 2 (t) dr 2 1 kr 2 + r2 d 2 + r 2 sin 2 d 2 where (r, θ, φ) are time-independent (comoving) spherical coordinates and t is the (cosmic) time since the Big Bang measured by comoving observers who are at rest with respect to the matter around them. The curvature constant k (with dimension length 2 ) determines the geometry of the metric: it is positive if the universe is closed, zero if it is flat, and negative if it is open. The metric is non static because of the time dependence of the scale (or expansion ) factor a(t). k>0 k<0 k=0

10 <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> The non static character of the metric can be made more explicit by calculating the physical (or proper) distance at time t from an observer at the origin to a point at comoving radial coordinate r, t,, = const dr ds = ap 1 kr 2 Z d p ds Since r is time-independent, the proper distance increases with a(t). The rate of change of the proper distance between two comoving observer is then Hubble law NB This statement is independent of GR. We ve only used symmetry so far!

11 The FRW metric describes all of the possible geometries of a homogenous isotropic Universe, but not all possible topologies. Geometry has local (visible Universe) structure, while topology only has global structure. Definition. A surface s geometry consists of those properties which do change when the surface is deformed. For example, curvature, area, distances, and angles are all geometric properties. k>0 Λ=0 k=0 k<0

12 The concordance cosmological model assumes that the Universe possesses a simply-connected topology It is a common misconception to describe a flat or hyperbolic Universe as necessarily open (i.e. infinite). topology

13 Take a flat Euclidean surface. That means, if we draw a triangle on the surface, its angles will sum to 180 deg. Now roll that surface up into a cylinder. The surface has now acquired extrinsic curvature because of the particular way it is embedded in a higher dimension. However its intrinsic curvature (that belonging to the surface alone) has not changed; it is still intrinsically flat. To see this consider any figure that you might have drawn on the surface. Within the surface, nothing about the figure is disturbed. If the figure conformed to Euclidean geometry before being rolled up, it will conform to Euclidean geometry after being rolled up. k=0 k=0 Gauss's Theorema Egregium: When surfaces are bent (but not stretched!), because measurements of lengths and angles on them remain unchanged, their Gaussian curvatures will not change either. In technical terms, the Gaussian curvature is invariant under isometries.

14 4.3 COSMOLOGICAL REDSHIFT The FRW line element vanishes for two events connected by a light signal; for photons moving along a radial trajectory (dθ=dφ=0), ds 2 =0 implies Without loss of generality, we can place the observer at the origin of the coordinate system. A light pulse leaving a source at comoving coordinate r e at time t e will arrive at the origin r=0 at a later time t 0 given by Photons emitted at a later time t e +δt e will arrive at time t 0 +δt 0 after traveling the same comoving distance, so the integral will not change:

15 For small δt e and δt 0, the previous equation implies time dilation! This time dilation also applies to wavelengths (think of light pulses separated by one period), so not due to Doppler! The expansion of the universe therefore stretches photon wavelengths by a(t 0 )/a(t e ), a factor generally denoted with (1+z): where we have used again the usual convention a(t 0 ) = 1.

16 In terms of the emitted and observed frequencies, the relation is NB By measuring z we obtain no information on when the light was actually emitted! Notice how the redshift we observe for a distant object depends only on the relative scale factors at the time of emission and observations, not on the rate of change of the scale factor at those times! re Cosmological time dilation effects can be directly observed in the light curves of Type 1a SNe.

17 Time dilation of supernova light curves. The left panel shows light curve points from high-redshift (blue) and nearby (black) supernovae. The right panel shows the same after removing the time dilation expected from redshift. From Goldhaber et al. 2001, ApJ, 558, 359.

18 4.5 GRAVITY WHERE THE ACTION IS The laws of Newtonian mechanics can be formulated in terms of a variational principle called the principle of least action. Consider the simple case of a particle of mass m moving in 1D in a potential Φ(x). The equations of motion are summarized by the Lagrangian Newton s law can be expressed as the Lagrange s equation

19 Consider the possible paths between an event (xa,ta) and an event (xb,tb.) For each path construct a real number called its action: Φ(x)=0 The action is an example of a functional a map from functions [in this case x(t) s] to real numbers. Among all the curves connecting A and B, those that minimize the action δs=0 satisfy Lagrange s equation A particle obeying Newton s law follow a path of extremal or least action. The motion of a free test particles in curved space-time can be summarized by a similar variational principle the principle of extremal proper time: A geodesic is an extremum of the action on the set of curves.

20 The proper time along a timeline world line between events A and B is World lines that minimize the action between A and B are those that satisfy Lagrange s equations σ=parameter x i =x i (σ) geodesic equation is the relativistic analog of

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22 All freely falling particles follow geodesic paths in curved space-time. The distribution of matter (or more generally, stress-energy) determines spacetime curvature. Space-time tells matter how to move. (Along geodesic paths) Matter tells spacetime how to curve. (Field equations) Compare to equivalent description of Newtonian gravity: Gravitational force tells matter how to accelerate. (F = mia) Matter tells gravity how to exert force. (F = GMmg/r 2 ) mi=mg

23 4.6 COSMIC DYNAMICS FRW To solve for a(t) and k we must substitute this metric into Einstein s field equations, differential equations relating the metric functions gij to the density and pressure of matter the GR analogs of the Newtonian Poisson equation Einstein tensor stress-energy tensor Here the Riemann tensor Rij and the Ricci (curvature) scalar R = g ik Rik are functions of the metric tensor gij and its first two derivatives, and give the curvature of space. Tij is the stress energy tensor it measures the relevant properties of all matter in the Universe. As the stress energy tensor is assumed to be symmetric:

24 there are potentially 10 Einstein equations (the number of independent components of a 4 x 4 symmetric matrix). energy density energy flux shear stress pressure momentum density momentum flux If the metric has additional symmetries, the number of independent Einstein eqs. may be much less. A homogeneous and isotropic perfect fluid is characterized by a density ρ and pressure p. Such a fluid has a stress energy tensor given by

25 If the metric is of FRW type, there are only 2 distinct equations: Relates the curvature of the Universe to its matterenergy content and expansion rate. Gravity pushes instead of pulls if pressure is negative and p< ρ/3. Friedmann equation acceleration equation

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30 The cosmological debate (SS vs. BB) acquired religious and political aspects. Pope Pious XII announced in 1952 that big-bang cosmology affirmed the notion of a transcendental creator and was in harmony with Christian dogma. Steady-state theory, denying any beginning or end to time, was in some minds loosely associated with atheism. Gamow suggested steadystate theory be part of the USSR Communist Party line, although in fact Soviet astronomers rejected both steady-state and big-bang cosmologies as "idealistic" and unsound. Hoyle himself associated steady state theory with personal freedom and anti-communism!

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36 The universe's ostensible fine tuning is the result of selection bias: i.e., only in a universe capable of eventually supporting life will there be living beings capable of observing any such fine tuning, while a universe less compatible with life will go unbeheld. WAP states that the Universe must have the age and the fundamental physical constants necessary to accommodate conscious life. As a result, it is unremarkable that the universe's fundamental constants happen to fall within the narrow range thought to be compatible with life...

37 Universe s Fine Tuning ΩRAD ΩM Ω Λ Selection Effect: AP? e

38 A=5 5 Li3 A=8 8 Be4 UNSTABLE! STABLE ISOTOPES 7 Li, 9 Be

39 2γ The net energy release of the process is MeV. Because the triple-alpha process is unlikely, it requires a long period of time to produce carbon. One consequence of this is that no Carbon was produced in the Big Bang because within minutes after the Big Bang, the temperature fell below that necessary for nuclear fusion. Ordinarily, the probability of the triple alpha process would be extremely small. Hoyle s resonance