1. General Relativity Physics Honours 2010
Florian Girelli
Rm 364, A28
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2. Geometry Covariant derivative
Geodesic equation and geodetic precession
Curvature
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3. 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
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4. 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
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5. Applications of covariant derivative:
Geodesics,
Geodetic precession,
Lense Thirring effect.
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6. 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.
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7. Parallel transport on the sphere on geodesics (big circles: meridians and equator)
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8. 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.
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9. 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
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10. Geodetic precession
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11. 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)
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12. 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;
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13. 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.
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14. 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
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15. 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.
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16. 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.
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17. 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.
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18. Slowly Rotating Spacetime
We won’t have time to consider the derivation of the effect in general, but you can read it in Section 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;
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19. 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;
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20. 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.
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21. Back to covariant derivatives
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22. 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?
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23. 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)
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24. 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.
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25. Lecture Notes 9

26. 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!)
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27. Riemann Tensor Notation: Bianchi Identity (proof written in inertial
frame to neglect
Christoffel terms)
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28. 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
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29. Einstein Equations
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30. 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.
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31. 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
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32. 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
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33. 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:
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34. 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.
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35. 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
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36. 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
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37. 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.
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38. 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.
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39. 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.
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40. 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/ ):
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…
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41. 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)
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42. Einstein Equation
We (finally) get the Einstein equation (for zero cosmological constant): matter tells how spacetime to curve.
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43. 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/
Tiny modifications of Schwarzschild metric
Chapter 10.2
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44. GR is not alone…
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