Geometry ofSpaceTime -

Einstein Theoryof Gravity II

Max CamenzindCB Oct-2010-D7

Textbooks on General Relativity

• 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.

Geometry of SpaceTime II

Riemann Curvature and Torsion

Curvature Identities

Curvature Local Expression 1

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 = dωa

b + ωac ωc

b : 2nd struct equ

Spin Connection and Curvature Cartan´s Equations

Cartan´s equations provide a very powerful methodto calculate the connection and curvature of spacetime.

Connection 1-form

Curvature Local Expression 2

Spin Connection of a 2-Sphere

Spin Connection of a 3-Sphere

Sectional Curvature

Planes ofPrincipal Curvature

Normal

TangentPlane

Surface in E²

Sectional Curvature of Surface

=

Congruence of GeodesicsConnecting Vector

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:

Geodesic Deviation in SpaceTime

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

Equation of Geodetic DeviationEquation of Geodetic Deviation Way to measure Curvature

„Tidal acceleration“ „Tidal force“

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.

Equation of Geodetic Deviation How to measure Curvature

• 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 time-like 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.

Story so Far …

∀ ∇aRdebc+∇cRdeab+∇bRdeca = 0 hom. ED• An important contraction is the Ricci

tensor Rab = Rcacb

• Further contraction gives the Ricci scalar: R = gabRab = Ra

a

• These definitions lead to the Einstein tensor: Gab = Rab - ½Rgab

• Obeys the contracted Bianchi identity: ∇bGa

b = 0.

Bianchi Identities & Contractions

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 well-defined operation:

Contraction over a and c

abcd

ab d bdaR R R → =

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:

Contraction Overa and d

ab a abd d ag R R R R= → =

The Ricci Objects

The Weyl tensor is in any Riemannian manifold of Dim n:

( )

( ) ( ) ( )

12

11 2

abcd dac adb cb bd bc

ac d

ca d

b ad c

b aa cd

b

C R R Rg g g gn

g g g gn

R

R R

n

= − − + − +−

+ −− −

Components of the raised index Riemann tensor, e

abcd ae bcdR g R=

The Ricci scalar

Components of theWeyl Tensor

The Metric

The Ricci Tensor

The Weyl Tensor in Dim n

Einstein’s Field Equations couple matter to curvature

Geodesic Equation Free-Fall of Test-Bodies Levi-Civita connection

Also Photons follow Null geodesics.

Trajectories of freely falling bodies (mass and massless)

Einstein´s Complete Theory 1915

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 validhas 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 scalesimaginable, known as the Planck length and Planck time.

• Note: This statement holds provided that h, G, and c are truly constant over the entire age of the known Universe!

Res LHC

Quantum CosmosArea & Volume quantized

???

Pre-Big-Bang Collapse

Time

Pre-Big-Bang Our Universe Expansion

Big-Bang

Quantum Cosmos near Planck-Time

Space isquantizedon the levelof the Planck-LengthPlanck-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 MethodTwistor TheoryNon-Commutative GeometryCausal Set TheoryCausal Dynamical TriangulationsRegge 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

Beyond Einstein and the Big Bang

CONNECTIONS: Quarks to the Cosmos

The Person of the 21st Century ?

The Hilbert Action

Action for Scalar Fields

On the Lambda Term

On the Lambda Term 2

Sign Conventions

• 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).

General Relativity is Metric Theory

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

Introduce a scalar field φ (x) that determines thestrength of gravity. Einstein's equation

is replaced by

Scalar-Tensor Gravity is Metric

Int

The new field φ (x) is an extra degree of freedom;an independently-propagating scalar particle(Brans and Dicke 1961; Quintessence models).

variable “Newton's constant” extra energy-momentum from φ

Brans-Dicke Theory is Metric

The new scalar φ issourced by planets andthe Sun, distorting themetric away from Schwarzschild. It canbe tested many ways,e.g. from the time delayof signals from theCassini mission.

Experiments constrain the “Brans-Dicke parameter” ω to be

ω > 40,000 ,where ω = inf is GR.

f(R) 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 Gravity

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.

where

Too complicated cannot be true !

Bekenstein (2004) introduced TeVeS, a relativistic versionof MOND featuring the metric, a fixed-norm vector Uµ ,scalar field φ , and Lagrange multipliers η and λ:

Mike Turner 2007

• 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 energy-momentum tensor of all type of matter in the spacetime (including fields and vacuum).

Summary