General Relativity Physics Honours 2010 Florian Girelli Rm 364, A28 girelli@physics.usyd.edu.au Lecture Notes 5

Geometry Covariant derivative Geodesic equation and geodetic precession Curvature Lecture Notes 9

Covariant Derivative We need to look at the mathematical structure behind general relativity. This begins with the concept of the covariant derivative. Let’s start with (flat) Minkowski spacetime; Where the second expression gives us the derivative of the vector in the t direction. To compare the vectors at two points, we have had to parallel transport one vector back along the path to the other. In Cartesian coordinates, this is no problem as the components of the vector do not change. This is not true in general. Chapter 20.4 Lecture Notes 5

Covariant Derivative Remember, in general curvilinear coordinates the basis vector change over the plane. This change of basis vectors needs to be considered when calculating the derivative. Hence, the Christoffel symbol can be seen to represent a correction to the derivative due to the change in the basis vectors over the plane. For Cartesian, these are zero, but for polar coordinates, they are not. A vector field v is parallely transported along t if Lecture Notes 5

Applications of covariant derivative: Geodesics, Geodetic precession, Lense Thirring effect. Lecture Notes 9

Geodesics Imagine we have a straight line path though space time, parameterized by , and this path have a unit tangent vector u then Hence, geodesics are paths that parallel transport their own tangent vector along them (i.e. there is no change to the tangent vector along the path). Think about a straight line path through polar coordinates! Geodesics in curved spacetime are just a generalization of the above. Lecture Notes 5

Parallel transport on the sphere on geodesics (big circles: meridians and equator) Lecture Notes 9

Example: Free Falling Frames Orthonormal bases are parallel transported by definition This allows to construct freely falling frames. If we have someone falling from infinity radially inwards in the Schwarzschild metric, then Where the first component is the 4-velocity, but you should check for yourself that the other components are parallel transported. Lecture Notes 5

Geodetic precession Other vectors can undergo parallel transport. The spin of a gyroscope can be represented as a spacelike spin 4-vector. In our rest frame, The 4-spin and 4-velocity are orthogonal. This must hold in all frames. The 4-spin is parallel transported along the geodesic using Chapter 14 Lecture Notes 4

Geodetic precession Lecture Notes 4

Geodetic precession Let’s consider a gyroscope orbiting in the static Schwarzschild metric. We will see that General Relativity predicts that, relative to the distant stars, the orientation of the gyroscope will change with time. This is the geodetic precession. If the gyroscope is on a circular orbit then Orbital angular velocity φ R And so the 4-velocity is given by (Remember we are orbiting in a plane where =/2) Lecture Notes 4

Geodetic precession We can use the orthogonality to find And solving we find We can use the Christoffel symbols for the Schwarzschild metric and the gyroscope equation to find; Lecture Notes 4

Geodetic precession Combining these, we can derive the equation This is just the usual equation for an harmonic oscillator with frequency Clearly, there is something weird here. The components of the spin vector change with time, but the frequency is not the same as the orbital frequency. So, the spin angle and the orbital frequency are out of phase. Lecture Notes 4

Geodetic precession The solution to these equations are quite straight-forward and Where s.s=s2*. Let’s consider t=0 and align the spin along the radial direction and we need to look at the orthonormal frame of the observer; as the metric is diagonal, we can define the radial orthonormal vector is Lecture Notes 4

Geodetic precession The period of the orbit is P=2/ and therefore the component of spin in the radial direction after one orbit is So, after each orbit, the direction of the spin vector has been rotated by This is the shift for a stationary observer at this point in the orbit. We could also set up a comoving observer at this point (orbiting with the gyro) but the radial component of the spin vector would be the same. Lecture Notes 4

Geodetic precession For small values of M/R the angular change is For a gyroscope orbiting at the surface of the Earth, this corresponds to While small, this will be measured by Gravity Probe B. Lecture Notes 4

Rotating body In general stars and planets rotate and the Schwarschild metric does not describe the correct associated space-time. Instead one should consider the metric associated to a rotating black-hole: the Kerr metric. where These are Boyer-Lindquist coordinates. This is asymptotically flat and the Schwarzschild metric when a=0. Lecture Notes 4

Slowly Rotating Spacetime We won’t have time to consider the derivation of the effect in general, but you can read it in Section 14.5. For a slowly rotating spherical mass, the Kerr metric becomes Where J is the angular momentum of the massive object. Hence the form of the metric is deformed with respect to the one of Schwarschild, but what is the effect of this distortion? Firstly, lets consider this metric in cartesian coordinates; Lecture Notes 4

Slowly Rotating Spacetime Let’s drop a gyro down the rotational axis. We can approximate the Schwarzschild metric as being flat (as contributing terms would be c-5). For our motion down the z-axis (rotation axis) then The non-zero Christoffel symbols for our flat + slow rotation metric are simply When evaluated on the z-axis and retaining the highest order terms. From this, we can use the gyroscope equation to show; Lecture Notes 4

Slowly Rotating Spacetime Again, we have a pair of equations that show that the direction of the spin vector change with time with a period of This is Lense-Thirring precession. At the surface of the Earth this has a value of While small, this will be observable by Gravity Probe B. Lecture Notes 4

Back to covariant derivatives Lecture Notes 9

Covariant Derivatives for tensors We can generalize the covariant derivative for general tensors (This is straight forward to see if we remember that t = v w and remember the Leibniz’s rules). What about downstairs (covariant) component? Lecture Notes 5

Covariant Derivative for tensors One of the fundamental properties of the Christoffel symbols is In Special Relativity, the stress-energy tensor is conserved. This is naturally generalized to the curved case using the covariant derivative: This is not anymore properly a conservation of energy since if spacetime is dynamical, matter can exchange energy with it (« local conservation of energy » cf Example 22.7) Lecture Notes 5

Introducing curvature To measure curvature we analyze the parallell transport of the vector along a closed loop: curvature measures the failure of coming back to the same vector. Mathematically it can be obtained from is the Riemann tensor. Lecture Notes 5

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Riemann Tensor Writing this in a local inertial frame, it becomes Leading to some immediate symmetries (true in general); Instead of 256 independent components, this tensor really only has 20 (Phew!) Lecture Notes 5

Riemann Tensor Notation: Bianchi Identity (proof written in inertial frame to neglect Christoffel terms) Lecture Notes 5

Contractions Contracting the Riemann tensor gives firstly the Ricci tensor Contracting again gives the Ricci scalar We also have the Kretschmann scalar, which is the measure of the underlying curvature If this is not zero, the spacetime is not flat! For the Schwarzschild metric, we have Lecture Notes 5

Einstein Equations Lecture Notes 9

Einstein eq. and applications Up to now, you have seen: Space-time defined by a metric, (global) symmetries given by Killing vectors. Propagation of matter (particle) given by geodesic equation What is the analog of the Poisson equation? Geometry tells how matter should propagate. Matter tells how geometry should curve. Lecture Notes 5

Source of gravity = mass density =M/V ~ energy/volume. needs generalization to the 4d case. 3d volume: 4d volume: 3d volume embedded in 4d: Energy momentum To obtain a momentum from a 3d volume embedded in 4d, we need a tensor encoding the “momentum density”. Stress-energy-momentum tensor Lecture Notes 5

Stress-Energy-Momentum To understand what this means, consider flat spacetime at a constant time; this is a 3-d space with n=(1,0,0,0). energy density momentum density stress tensor: measures internal forces that part of the medium exert on the others (Cauchy 1822) force per unit of area ~ pressure Chapter 22 Lecture Notes 5

Examples Fluid: If we are at rest with respect to a perfect fluid then In flat spacetime, we can extend this to a moving fluid so It should be clear that in the rest frame, this becomes the above rest frame expression. Particle: it is not difficult to see that we have From a lagrangian: Lecture Notes 5

Curvature and gravity Einstein’s key idea is to understand that gravity is encoded in the geometric properties of spacetime: curvature. We consider two particles in a gravitational potential, first in the Newtonian formalism, then in the GR case and compare. Lecture Notes 5

Newton deviation In the Newonian picture, we have the Newton equation. Gravitational potential If we consider two nearby particles separated by a vector  Taking the leading terms, we get the Newtonian deviation between two trajectories Chapter 21 Lecture Notes 5

Geodesic Deviation The separation of two free falling objects gives a measure of the underlying curvature. We need to consider the paths of two nearby geodesics, with a 4-space separation as a function of the proper time along both curves. Where the Riemann Curvature tensor is Lecture Notes 5

Geodesic Deviation This is easier to understand in the free falling frame. We need project the deviation vector into the orthonormal frame Remembering what the 4-velocity is in the free falling frame, then Where the Riemann tensor has been projected onto the orthonormal basis. Lecture Notes 5

Geodesic Deviation In the weak field limit If we assume that our objects fall slowly (non-relativistic) along the geodesics, then the orthonormal and coordinate frame are approximately the same, so We can calculate the Christoffel symbols from the metric and then calculate the components of the Riemann tensor. Lecture Notes 5

Gravity is therefore related to the geometry of spacetime, Geodesic Deviation Keeping only the lowest order terms, we find In the weak field limit, non-relativistic limit, we recover the result from Newtonian physics. Remember, the Riemann tensor is something that describes geometry, but here it is related to something physical, the gravitational potential. Gravity is therefore related to the geometry of spacetime, ie its curvature. Lecture Notes 5

Looking for Einstein Equation We have now the tools to determine some equations of motion intertwining gravity and matter. The source should be expressed in terms of the stress energy tensor. Between 1910 and 1913, Nordstrom proposed to consider generalizations of the Poisson equation (arXiv:gr-qc/0611100): Wrong for many reasons.Eg gravitiational field carry energy and should self gravitate: non-linear effects Einstein and Fokker showed that this latter proposition can be put into a geometric shape This is the first geometrical candidate for gravity (scalar gravity). Not physical: no bending of light (metric is conformally flat), wrong precession… Lecture Notes 5

Looking for Einstein Equation Gravitational degrees of freedom are encoded by curvature (Riemann tensor, Ricci tensor, Ricci scalar), we look for the equations which give the Poisson equation in the Newtonian limit and are second order in derivatives in the metric. Einstein first tried But this fails to work since we have also the conservation of the stress energy tensor and Instead we have (thanks to the Bianchi identities) Lecture Notes 5

Einstein Equation We (finally) get the Einstein equation (for zero cosmological constant): matter tells how spacetime to curve. Lecture Notes 5

GR is not alone… Chapter 10.2 General Relativity is the « simplest » theory, but one can generate other theories compatible with experiments by adding scalar, vector fields, higher order contributions in curvature... Introduce Parametrized Post-Newtonian parameters: they determine how the gravitational theory candidate is different from Newtonian gravity (in the weak field limit). Experiments put constraints on the value of these coefficients. See for example arXiv:gr-qc/9903058. Tiny modifications of Schwarzschild metric Chapter 10.2 Lecture Notes 5

GR is not alone… Lecture Notes 5

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