Geometry of SpaceTime Einstein Theory of Gravity II Max Camenzind CB Oct-2010-D7
Textbooks on General Relativity
Geometry of SpaceTime II Connection and curvature on manifolds. Sectional Curvature. Geodetic Deviation Fromula Tidal field of Gravity. Ricci, Einstein and Weyl tensors. Cartan formalism. Einstein s Field Equations. Metric Theories of Gravity. Einstein-Hilbert action & Brans-Dicke. Solar System tests of General Relativity.
Riemann Curvature and Torsion
Curvature Identities
Curvature Local Expression 1
Spin Connection and Curvature Cartan s Equations Exterior derivative: d² = 0 β = f(x) α dβ = df α + f dα Cartan s equations: Ta = dθa + ωab θb = 0 : 1st struct equ Ωab = dωab + ωac ωcb : 2nd struct equ Cartan s equations provide a very powerful method to calculate the connection and curvature of spacetime.
Curvature Local Expression 2 Connection 1-form
Spin Connection of a 2-Sphere
Spin Connection of a 3-Sphere
Sectional Curvature
Sectional Curvature of Surface Normal Planes of Principal Curvature Tangent Plane Surface in E²
Congruence of Geodesics Connecting Vector =
Geodesic Deviation in SpaceTime Suppose we take a congruence of time-like geodesics labeled by their proper-time and a selector parameter ν i.e. xa=xa(τ,ν). We can define a tangent vector and connecting vector ξ such that We now need to make use of a Riemann tensor identity Remember:
Equation of Geodetic Deviation Way to measure Curvature Setting Xa = Za = Va and Ya = ξa, then the second term vanishes as Va is tangent to the geodesic and is parallel transported (DvVa = 0). the third term vanishes as the derivative with respect to τ and ν commute. The result is Tidal acceleration Tidal force
Equation of Geodetic Deviation How to measure Curvature Contracting with the spatial basis vectors e results in an equation which is only dependent upon the spatial components of the orthogonal connecting vector η This form relates geometry to the physical separation of objects and will give the same results as for the Newtonian tidal force equation. This is the basis e.g. for measuring gravitational waves.
Story so Far Equivalence principle implies Special Relativity is regained locally in a free-falling frame. Cannot distinguish locally a gravitational field from acceleration and hence we should treat gravity as an inertial force. Following SR we assume free particles follow timelike geodesics, with forces appearing though metric connections. The metric plays the role of a set of potentials. We can use these to determine a set of (tensorial) second order PDEs.
Story so Far Genuine gravitational effects can be observed (nonlocally) where there is a variation in the field. This causes particles to move of converging/diverging geodesics described by the Riemann tensor via the geodesic deviation equation. The Riemann tensor involves second derivatives of the metric, and hence it may appear in the field equations (but, it has 20 components). There is one meaningful contraction of this tensor (Ricci tensor) which is related to the Einstein tensor and only has 10 components Ricci tensor.
Bianchi Identities & Contractions ardebc+ crdeab+ brdeca = 0 hom. ED An important contraction is the Ricci tensor Rab = Rcacb Further contraction gives the Ricci scalar: R = gabrab = Raa These definitions lead to the Einstein tensor: Gab = Rab - ½Rgab Obeys the contracted Bianchi identity: bgab = 0.
The Ricci Objects From the Riemann tensor, we can define the Ricci tensor. It is defined through contraction of the tensor with Itself. This may seem strange, but if we have a mixed tensor, as we do here, this is a perfectly welldefined operation: R a bcd R Contraction over a and c a b ad = Rbd We may further contract the Ricci tensor, to the Ricci scalar. However, since the 2 indices are covariant, before we can contract, we have to raise one index. The metric helps us here to give: g Rbd = R ab a d R Contraction Over a and d a a = R
The Weyl Tensor in Dim n The Weyl tensor is in any Riemannian manifold of Dim n: Components of the raised index Riemann tensor, Rabcd = g ae R e bcd The Metric Cabcd 1 = Rabcd ( g ac Rdb g ad Rcb + gbd Rca gbc Rda ) + n 2 1 + ( g ac g db g ad gcb ) R ( n 1) ( n 2 ) The Ricci Tensor Components of the Weyl Tensor The Ricci scalar
Einstein s Complete Theory 1915 Einstein s Field Equations couple matter to curvature Trajectories of freely falling bodies (mass and massless) Geodesic Equation Free-Fall of TestBodies Levi-Civita connection Also Photons follow Null geodesics.
Summary of the 4 Key Principles of General Relativity
Limits of General Relativity Quantum Theory of Gravity GR is undeniably very successful in explaining current observations within the known Universe. However, it is also undeniable that GR has inherent limitations. When applied to the interior of black holes and the earliest moments in Big Bang cosmology, GR predicts an infinitely strong gravitational field known as a singularity. The whole concept of space-time breaks down at the singularity, which justifies the need for a Quantum Gravity theory to give sensible descriptions of physics within this extreme region of space-time. General Relativity is a classical theory of gravity that presumes a deterministic concept of motion within spacetime, and inherently disagrees with currently accepted concepts of Quantum Mechanics in the absence of gravity.
Limits of GR Planck Units
Planck Units 2 Under these conditions, classical GR is no longer valid has to be merged with Quantum Gravity (Loop Quantum Gravity or String Theory).
Planck Length and Planck Time Compared to Subatomic Length Scales far away! By combining Planck s constant h, the universal gravitational constant G, and the speed of light c, it is possible to compute the smallest length and time scales imaginable, known as the Planck length and Planck time. Res LHC Note: This statement holds provided that h, G, and c are truly constant over the entire age of the known Universe!
Quantum Cosmos Area & Volume quantized??? Pre-Big-Bang Collapse Big-Bang Time Our Universe Expansion
Quantum Cosmos near Planck-Time Space is quantized on the level of the Planck-Length Planck-Cells no Continuum
The Many Roads to Quantum Gravity There exist numerous approaches towards Quantum Gravity: String Theory (ST) Loop Quantum Gravity (LQG) Path Integral Method Twistor Theory Non-Commutative Geometry Causal Set Theory Causal Dynamical Triangulations Regge Calculus etc... All the approaches listed are very theoretically motivated and generally do not agree with each other. They are often difficult to interpret.
Best-Selling Popular Books Critical of String Theory
The Person of the 21st Century? CONNECTIONS: Quarks to the Cosmos Beyond Einstein and the Big Bang
The Hilbert Action
Action for Scalar Fields
On the Lambda Term
On the Lambda Term 2
Sign Conventions
General Relativity is Metric Theory P1: Space of all events is a 4-dimensional manifold endowed with a global symmetric metric field g (2-tensor) of signature (+---) or (-+++). P2: Gravity is related to the Levi-Civita connection on this manifold no torsion. P3: Trajectories of freely falling bodies (local inertial frames) are geodesics of that metric. P4: Any physical interaction (other than gravity) behaves in a local inertial frame as gravitation were absent (covariance).
Spacetime geometry is described by the metric gµν. The curvature scalar R[gµν] is the most basic scalar quantity characterizing the curvature of spacetime at each point. The simplest action possible is thus Varying with respect to gµν gives Einstein's equation: Gµν is the Einstein tensor, characterizing curvature, and Tµν is the energy-momentum tensor of matter. Einstein s Gravity is Metric
Scalar-Tensor Gravity is Metric Introduce a scalar field φ (x) that determines the I strength of gravity. Einstein's equation n t is replaced by variable Newton's constant extra energy-momentum from The new field φ (x) is an extra degree of freedom; an independently-propagating scalar particle (Brans and Dicke 1961; Quintessence models). φ
Brans-Dicke Theory is Metric
The new scalar φ is sourced by planets and the Sun, distorting the metric away from Schwarzschild. It can be tested many ways, e.g. from the time delay of signals from the Cassini mission. Experiments constrain the Brans-Dicke parameter ω to be ω > 40,000, where ω = inf is GR.
f(r) Gravity
MOdified Newtonian Gravity Potential Wells are much deeper than can be explained with visible matter This has been measured for many years on galactic scales Kepler: v=[gm/r]1/2
MOdified Newtonian Dynamics - MOND Milgrom (1984) noticed a remarkable fact: dark matter is only needed in galaxies once the acceleration due to gravity dips below a0 = 10-8 cm/s2 ~ ch0. He proposed a phenomenological force law, MOND, in which gravity falls off more slowly when it s weaker: 1/r2, a > a0, F 1/r, a < a0.
Bekenstein (2004) introduced TeVeS, a relativistic version of MOND featuring the metric, a fixed-norm vector Uµ, scalar field φ, and Lagrange multipliers η and λ: where Too complicated cannot be true!
Mike Turner 2007
Summary SpaceTime is the set of all events, it has the structure of a pseudo-riemannian manifold with a metric tensor field g. Einstein s gravity assumes the connection to be metric, i.e. the Levi-Civita connection. Freely falling objects follow geodesics on this manifold, also self-gravitating ones (SEP). The Einstein tensor is coupled to the energymomentum tensor of all type of matter in the spacetime (including fields and vacuum).