1. General Relativity Physics Honours 2005
Dr Geraint F. Lewis
Rm 557, A29

2. The Story Thus Far…
Several inter-related concepts from non-Euclidean geometry
Characterizes the manifold
Describe straight/extremal motion over the manifold
The covariant derivative
How do we relate these to the physics of gravity?

3. Lifts & Rockets (Ch 10)
Remembering the equivalence principle, Einstein stated that there is an equivalence between being in an accelerated rocket and being in a gravitational field (i.e. the space-time of special relativity is recovered for a stationary rocket and free-falling observer in a gravitational field).
There are subtle differences in the two situations due to non-uniform gravitational fields (Fig ). Non-uniformities cause the test masses to approach each other in a free-falling lift.
In relativity this variation is described by the Riemann tensor and the equation of geodesic deviation.

4. Newtonian Deviation
We can use Newtonian physics to study the motion of a pair of neighbouring particles in three space. In cartesian coordinates we have
where the greek indicies run from 1!3. Parameterized in terms of Newtonian time we can describe the particle motions as
where (t) is a small connecting vector.

5. Newtonian Deviation We can write the equations of motion as
where  is the gravitational potential. The expression for the connecting vector is Taylor expanded about the path of the first particle (and so the derivatives are on the x path).

6. Newtonian Deviation Subtracting these two expressions
It is straight forward to see that Laplace’s equation is
Can we determine a “geometric” form of these equations?

7. Principles of Relativity
Space-time can be represented by a non-Euclidean manifold with appropriate metric and Riemann tensors.
The geometry of the manifold is determined by the distribution of mass/energy.
Particles travel along geodesic paths that are extremal and are the straightest possible (parallel transport of tangent vector over the manifold).
Massive particles follow time-like geodesics (U.U=1), a light rays follow null geodesics (U.U=0).

8. Geodesic Deviation
Suppose we take a congruence of time-like geodesics labeled by their proper-time and a selector parameter  i.e. xa=xa(,). We can define a tangent vector and connecting vector , such that
We now need to make use of a tensor identity (6.11)
Remember:

9. Geodesic Deviation Setting Xa=Za=va and Ya=a then
the second term vanishes as va is tangent to the geodesic and is parallely transported (rv va = 0).
the third term vanishes as the derivative with respect to  and  commute (eqn 10.15).
The result is

10. Geodesic Deviation
Remember, this is a 4-d equation, but to compare to Newtonian we need to consider the 3-d space components. We can project out the 3-d part by firstly defining a projection operator
[Appendix D in Spacetime & Geometry by Carroll discusses projection operators (or more precisely tensors) in detail].
The orthogonal connecting vector a then is given by

11. Geodesic Deviation
This reduces the geodesic deviation equation to (pg 138)
This is still a 4-d equation, and to compare with Newtonian we need to extract the 3-d information. To do this we need to define an orthogonal frame of unit vectors
and eoa = va (remember these are 4-vectors).

12. Geodesic Deviation
The local basis can be parallely propagated (i.e. by attaching gyro-scopes to ensure Deia/D=0). These vectors satisfy the orthogonality relations
The spatial components can then be extracted by taking the dot product of the vector with each spatial basis

13. Geodesic Deviation
Contracting with the spatial basis vectors results in a equation which is only dependent upon the spatial components of the orthogonal connecting vector
This form must relates geometry to the physical separation of objects and must give the same results as the Newtonian deviation equation.

14. Vacuum Equations
Remembering that the Newtonian vacuum field equations were K=0 we can seek the equivalent in relativity. We can choose a special coordinate system such that eia = ia and
This is a scalar quantity and must be zero in all coordinate systems, and hence, in general

15. Vacuum Equations Einstein proposed, therefore, that
are the vacuum field equations of general relativity.
Remember that this seemingly simple equation represents a set of second order PDEs, allowing the metric to be determined.

16. Story So Far…
Equivalence 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.

17. 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. There is one meaningful contraction of this tensor (Ricci tensor) which is related to the Einstein tensor.

18. Full Field Equations
Matter (& energy) are a source of gravitation and the field equations must be modified in its presence. As we will see next, this is described by the symmetric rank 2 energy-momentum tensor Tab. In special relativity the continuity equation is
where the RHS corresponds to the extension of the SR result to a curved space-time with minimal gravitational coupling.

19. Full Field Equations
Remembering that the Einstein tensor is subject to the contracted Bianchi identities,
which suggests that
Self study Sec 12.10,  = 8 G/c4

20. Orthonormal Basis
It should be clear that an observer does not “see” coordinates associated with the metric (i.e. coordinate time vs proper time).
In the previous lecture we used orthonormal coordinates. These are very important as
We calculate in coordinate coordinates
We interpret in orthonormal coordinates
Hence, measurements by an observer are made in orthonormal coordinates. But how are such coordinate systems established?

21. Orthonormal Basis In our coordinate basis (as defined by the metric)
Noting that a general vector on the manifold is given by
This means that

22. Orthonormal Basis
Remember that any small patch on the manifold should appear flat and hence can establish a (local) Minkowski coordinate system.
Hence any vector on this patch can be written as
Of course, there is substantial freedom in the definition of the orientation of the orthonormal basis, but the situation is simplified if it has the same orientation as the coordinate basis.

23. Orthonormal Basis
If the metric is symmetric (i.e. the coordinate basis is orthogonal) then in the coordinate basis we can define four 4-vectors which represent the unit vectors of the orthonormal basis.
which gives
and hence it is straight-
forward to convert between the vector components in each basis.

24. Energy-Momentum Tensor (12)
The field equations for General Relativity are given by
where Tab is the Energy-Momentum Tensor. This tensor contains the information on energy density, momentum flux density and stresses in matter. These are the “sources” of the gravitational field. In this course we will look at
incoherent matter
perfect fluid
electromagnetic field

25. Dust
The simplest matter is noninteracting and incoherent (dust). This is characterised by a proper density 0 (the density as measured by a comoving observer) and a 4-velocity ua=dxa/d. The simplest 2nd rank tensor we can construct from these is
In special relativity, an observer at rest with respect to the dust sees the energy density as simply 0.

26. Dust In Minkowski coordinates, ua=(1,v) with =(1-v2)-1/2 & d=dt/.
This quantity, denoted by , is the energy density of moving matter as seen by an observer at rest in the Minkowski frame. Note that the energy density is proportional to 2 as the relativistic mass increases and the volume element decreases by the same factor due to length contraction.

27. Dust Writing out the tensor in full we see
Remember, these are 3-velocities! The various terms in this tensor can be interpreted as
0j terms: energy flux in the j-direction
jk terms: flux of j-momentum in k-direction

28. Dust The conservation of energy and momentum can be written as
Setting a=0 this corresponds to the equation of continuity
With a=1,2,3 we can recover the pressureless Navier-Stokes eqn

29. Energy Momentum Tensor
We can transform these continuity equations from the flat space-time of special relativity to the curved space-time of general relativity by replacing the normal partial derivative with the covariant derivative (the principle of minimal gravitational coupling).

30. Perfect Fluid
Considering a perfect fluid introduces one additional quantity, the scalar pressure p(x). Clearly, as p!0, the perfect fluid becomes incoherent matter. The simplest tensor that represents this is
where Sab is some symmetric 2nd rank tensor built from similar tensors associated with the fluid. We can assume

31. Perfect Fluid
For this energy momentum tensor to reduce to the p0 form of the Navier-Stokes equation (=r p instead of 0) and the continuity equation in the nonrelativistic limit then
Generally, p and  are related via some equation of state.

32. Maxwell’s Equations Maxwell’s equations are
We can introduce the Maxwell tensor
and the current density 4-vector

33. Maxwell’s Equations
The 1st two of Maxwell’s equations can be written as
and the 2nd pair as
With this a ja = 0 is the charge continuity equation

34. Energy Momentum (again)
While we will not derive the E-M tensor, in source free regions
given the correct field equations (pg 162 of textbook).
Specific components in special relativity are

35. The Newtonian Limit
In the Newtonian limit (low gravity & low velocity) the equations of relativity must reproduce Newtonian physics. In this limit, the metric can be written as
In a time t a body moves a distance x << c t = x0 and

36. The Newtonian Limit Considering the motion of a massive particle so
In the Newtonian limit, d» dt and dx/cdt» O()
and

37. The Newtonian Limit With this, the geodesic equation reduces to
(greek indicies denote spatial components). This becomes
Where the right hand term is the Newtonian equation. Hence

38. The Newtonian Limit The full metric in the Newtonian limit is given by
This can be derived from the Schwarzschild solution in the limit when
Note that in the limit, the potential !0 and so the metric becomes flat.

39. The Field Equations (Ch 13)
The field equations are given by
Both tensors in the field equations are symmetric, implying that they embody 10 2nd order PDEs, connecting 20 quantities.
Wheeler summarized these equations as
Matter tells space how to curve
Space tells matter how to move

40. The Field Equations
Consider the vacuum field equations, Gab=0, for gab. We have 10 equations for 10 unknowns. But not all the terms are independent, and are connected via the contracted Bianchi identity.
Hence, we really have only 6 equations for 10 unknowns and it appears the equations are under-determined!
However, this is to be expected as we cannot expect the equations to determine a preferred coordinate system (ie we have freedom to transform the coordinates). With this (& when Tab0) a coordinate system can be used to give 6 equations for 6 unknowns.

41. The Field Equations
The field equations are non-linear and hence you cannot superpose solutions. This is easily seen with the Schwarzschild solution
This robs physicists of the usual tricks you can employ in calculating physical properties. In fact, Einstein felt that there may be no solvable physical situations, but as we will see, Schwarzschild demonstrated that symmetries can be used to simplify situations and derive exact solutions.
The non-linearity of the field equations give us objects like massless eternal black holes.

42. The Schwarzschild Solution
The general spherically symmetric line element can be written
where  and  are functions of r and t and are determined
by the situation. The first few sections of Ch 14 justify the form
of this element, you should read it to get the idea.
Given the line element, the metric is

43. The Schwarzschild Solution
The nonvanishing components of the Einstein tensor are then
where dot is differentiation wrt t and prime wrt r.

44. The Schwarzschild Solution
The contracted Bianchi identity rb Gab = 0 imply that the final term vanishes if the other three do. Hence, the problem reduces to three independent equations in vacuo
Adding the first two gives ’ + ’ = 0 which means

45. The Schwarzschild Solution
Further,  is not a function of t and so the first equation is simply an ODE of the form & solution
The constant k by requiring the solution gives the weak field solution in the Newtonian limit. This yields

46. The Schwarzschild Solution
The final step involves a change of coordinates to remove the function h(t). Using a new time coordinate t’
Eliminates the function, leaving the Schwarzschild metric (after dropping the primes).

47. The Schwarzschild Solution
We have already examined many properties of the Schwarzschild solution, especially with regards to the physical implications.
The solution contains several singularities, how do you tell which are real and which are coordinate singularities?
This is revealed by the Riemann tensor invariant
This is finite everywhere (including r=2m) except r=0. This holds true in all coordinate systems (as it is a scalar) and so the singularity at r=2m is removable with a coordinate change.

48. Tidal Forces
Newton was able to show that tidal forces are due to changes in the gravitational force over space. Hence, the water of the Earth is drawn into an ellipsoid, explaining why there are two tides per day.
In GR we can consider the location of particles as they move along geodesics in curved space time. The geodesic deviation equation is
Where  is a small 3-vector connecting two nearby particles.

49. Tidal Forces
We can establish an orthonormal coordinate system in the Schwarzschild metric such that
Then the geodesic deviation equation becomes
The positive terms imply longitudinal stretching and transverse compression (i.e. distortion into an ellipse).

50. Tidal Forces
Notice that nothing weird happens at the event horizon (r=2m). Hence, you could cross the event horizon of a large black hole with no ill effects.
As r!0, then the terms in the tidal forces blow up. Hence the tidal forces increase as you fall towards the singularity of a black hole, and your body will be subject to increasing longitudinal stretching and transverse compression.
This process, which results in the atoms of your body being ripped apart is known as spaghettification.

51. Charged Black Holes (Ch 18)
While physically unlikely, charged black holes provide one of the few solutions to the field equations of relativity.
We seek a static, asymptotically flat, spherically symmetric solution of the Einstein-Maxwell field equations
Unlike the Schwarzschild solution, where Gab=0, in this case

52. Charged Black Holes
Hence, we are searching for a solution to the source free Maxwell’s equations (but containing an electric field).
Assuming spherical symmetry, the metric can be written as
and the Maxwell equation for a point charge is given by

53. Charge Black Holes
Insertion of this tensor into Maxwell’s equations give
where  is a constant. At large distances, where  and  ! 0, then E(r)» /r2 and we can identify  with the charge (in appropriate units).
The Maxwell tensor Fab can now be inserted into Tab and the field equations solved in a similar fashion to the Schwarzschild case (read through the approach in the textbook).

54. Charged Black Holes
The result of this procedure is the Reissner-Nordstrom solution
The solution is asymptotically flat and becomes the Schwarzschild solution in the limit that e!0.
Remember that >m then the event horizon vanishes and we are left with a naked, repulsive singularity.