Geometry of the Universe: Cosmological Principle God is an infinite sphere whose centre is everywhere and its circumference nowhere Empedocles, 5 th cent BC Homogeneous Cosmological Principle: Describes the symmetries in global appearance of the Universe: The Universe is the same everywhere: - physical quantities (density, T,p, ) Isotropic Universality Uniformly Expanding The Universe looks the same in every direction Physical Laws same everywhere The Universe grows with same rate in - every direction - at every location ``all places in the Universe are alike Einstein, 1931
Friedmann-Robertson-Walker-Lemaitre Universe for an expanding Universe with matter density ρ(t) pressure p(t) cosmological constant Λ [or elusive dark energy ρ v (t)] dynamics is ultimately set by the geometrical term k:
Geometry of the Universe The three possible geometries of the Universe
The Universe is a Riemann manifold, i.e. a mathematical space in which every point has a neighborhood which resemble Euclidean space, but in which the global structure is more complicated, and on which distances can be defined through the introduction of a metric (which allows generalization of Pythagoras theorem) Einstein s equations: gravity modifies the curvature (i.e. distorts space-time)
Building the metric in the simplest 2-D isotropic curved space: the 2-sphere (S 2 ) - setting: and introducing the dimensionless coordinate r r /R:
Summarizing: k=1 k=0 (R ) k=-1 now repeat the same game with one more dimension
fictitious get rid of x 4 so that: and switch to radial coordinates: with the additional redefinition: r=r /R, so that:
Robertson-Walker metric k=1 k=0 k=-1 R(t)= scale factor
The equivalence principle at every space-time point in an arbitrary gravitational field it is possible to choose a locally inertial coordinate system such that, within a sufficiently small region of the point in question, the laws of nature take the same form as in unaccelerated Cartesian coordinate systems in the absence of gravitation. the above statement implies that, for any particle under the influence of gravity, a transformation exists to a coordinate system ζ a where the particle is (locally) inertial (i.e. a free fall system), i.e.: (proper time)
x a =inertial coordinates ζ a = free fall coordinates =affine connection (geodesic equation) Affine connection can be expressed as a function of the metric: Christoffel symbol i.e. g μν is the gravitational potential, since its derivatives determine the force Γ
Equivalence principle general covariance 1. recover special relativity when g μν η μν and Γ 0 2. form invariance of equations under general coordinate transformation x x tensor transformations under x x (repeated indices are summed): ex: etc define covariant derivative ( μ does not transform the right way): ex: for instance: Γ is not a tensor, but the combination below is: =0 in free fall system, so this is true everywhere
An aside: not all relevant quantities are tensors. Example: the metric determinant: transforms as a scalar density of weight -2, i.e. as a scalar, apart from a Jacobian factor to the power of -2: since: the product: is an invariant
Curvature tensor: only tensor that can be constructed from the metric tensor and its first and second derivatives (Riemann-Christoffel tensor) symmetric by λ ν or µ k antisymmetric by λ µ or ν k cyclicity: R λµνk +R λkμν +R λνkμ =0 Ricci tensor Ricci scalar Bianchi identity:
N.B.: a non-constant metric tensor g μν (x) does not automatically signal the presence of a force, i.e. of a gravitational field. For example, the Minkowski metric can be expressed in polar coordinates: however, obviously a proper transformation exists for which: i.e. g μν (x) is equivalent to η μν. globally, not just locally Necessary and sufficient conditions for a metric g μν (x) to be equivalent to Minkowski metric η μν : 1. R λ μνk=0 everywhere 2. @ some point g μν has 3 positive and 1 negative eigenvalues gravitation curves space
The non relativistic limit writing: comparing with the newtonian limit: one gets: (φ=gravitational potential)
time dilation gravitational effect special relativity 1. a moving clock ticks slower due to special relativity 2. and ticks even slower deep in a gravitational potential a negligible effect unless in extreme conditions anyway right?
The GPS system: general relativity is among us! # of visible satellites from 45 ⁰ latitude 24 satellites, at least 4 of them visible at the same time from any point on Earth position obtained from timing info by solving the equation system: Cesium clock tick on satellite +45 μsec/day -7 μsec/day =+38 μsec/day (nsec precision required) CUMULATIVE EFFECT: neglecting gravitational effect would add an error in position at a rate of ~ 10 km/day!
so most taxi drivers would not agree on the fact that gravitational time dilation on Earth is a negligible effect because they would take you to the wrong place! N.B. : not properly a test on General Relativity, but only of the equivalence principle (shared by all metric theories)
Einsten s equations G μν : gravitational constant energy-momentum tensor tensor contains only g g or 2 g (more on this later ) G μν ;ν =0 (from energy momentum conservation T μν ;ν =0 ) weak limit: with so that Poisson s equation is recovered: Einstein s eqns+robertson-walker metric=friedmann equations
The energy momentum tensor density and current for the energy-momentum four-vector p α. Consider a system of particles with energy-momenta p α n. The density of p α is defined as: while its current is given by: so that combining the two: i=1,2,3 (with ) using: while writing: one gets: (symmetric) ( ) (tensor)
conservation law for T αβ : ( force density ) if particles are free: for local collisions: : sum over particles interacting with m does not depend on time if momentum is conserved in collisions again: T αβ / x β =0