Linearized Weak Field Gravitation : weak field approximation

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9.1 Gravity Waves

In this section we show that in a weak field approximation and a vacuum region gravitational waves travel at the speed c by deriving the weak field approximate gravitational wave equation for Einstein's relativistic physics theory:


Then we will show what is meant by gravitational waves carrying momentum and energy for Einstein's relativistic physics theory. And finally we will show what these transverse wave's effects are in the case of plane waves incident on matter for general relativity.

Next we refer to the affine connections Eqn. 4.4.3

We next make a weak field approximation. We write the metric tensor


and keep only terms to first order in from here out.



98 Chapter 9 Linearized Weak Field Gravitation

The equation for the affine connection for Einstein's relativistic physics theory becomes

We then refer to the equation for the Riemann tensor for Einstein's relativistic physics theory Eqn. 6.1.3

Replacing in the expression for the affine connections for Einstein's relativistic physics theory and keeping only first order terms in hmn we arrive at

We next contract this to get the Ricci tensor for Einstein's relativistic physics theory Eqn. 6.3.1

resulting in


Using the raising index property of the contravarient metric tensor for Einstein's relativistic physics theory we simplify this to

Rearranging the terms we have

With a little algebraic manipulation, insertion of delta kronecker, and redefinition of indices, this can be rewritten


Einstein's field equations for Einstein's relativistic physics theory can be expressed in the form Eqn. 6.3.21

resulting in


We've said nothing of what coordinate system to use at this point. We may have a coordinate system in which the

and terms are not zero. In this case, consider the following infinitesimal coordinate transformation


Doing this we find that transforms according to



9.1 Gravity Waves 99

We then choose to do the infinitesimal coordinate transformation so that


which is the same as requiring


Inserting these in Eqn. 9.1.3 and the and the transformation for to into the first term and simplification results in


In vacuum resulting in

This is the ordinary wave equation for waves traveling at the speed c. Therefor in a weak field limit for Einstein's relativistic physics theory and vacuum region gravitational waves travel at the speed c.

All this has been done as a vacuum field solution to Einstein's field equations. According to Einstein's relativistic physics theory the stress-energy tensor was zero. Even so, there is a nonzero stress-energy pseudo-tensor for gravitational waves, and in that sense, they do carry momentum and energy away from the source just like any other field's waves. To derive the stress energy pseudo-tensor of gravitational waves, We transform the Ricci Tensor and the Ricci scalar for Einstein's relativistic physics theory into our new choice of coordinates. This results in


Reconstructing the Einstein tensor for Einstein's relativistic physics theory in the first order in approximation results in

The form Einstein's field equations should take is



100 Chapter 9 Linearized Weak Field Gravitation

Which finally gives the result for the stress-energy pseudo-tensor of gravitational waves.


In this form it is apparent how gravitational waves carry momentum and energy and how this energy and momentum can be though of as itself contributing to the overall gravitation for general relativity.


Consider the next two expressions ordered so that x3 = z and with .



These are orthogonal waves traveling in the + z direction that satisfy the wave equation, Eqn, (9.1.8). In our choice of coordinates we have another set of conditions,


These conditions are also satisfied by Eqn. 9.1.12. These expressions constitute two independent polarization's for the gravitational waves. What may come as a surprise is that in a low speed, weak field approximation the coordinate acceleration of a test particle according to this choice of coordinates is actually zero. However, the rigid ruler velocity of a particle with zero coordinate velocity in these coordinates is not zero. Consider a spacetime interval containing h+ and for simplicity . It becomes



9.1 Gravity Waves 101

Consider the matter motion in the z = 0 plane. The rigid ruler displacements of the matter are given by


After simplification, this leads to a total rigid ruler distance to the coordinate positions from an origin given by


From this, we see that there is a rigid ruler acceleration for the test particles, which tend to remain at constant coordinate positions. This is given by


A field line graph, taking this as the field can easily be drawn from the result. For this polarization, and at t = 0, the graph is

(Fig. 9.1.1)

The other polarization can be used in a similar method to construct its field line graph as well. That graph looks just like this one except rotated by .


102 Chapter 9 Linearized Weak Field Gravitation


Layman often complain that relativity must be wrong based on a variety of common misconceptions. For an example, some of them often say it must be wrong because it predicts the graviton which has never been directly observed. In actuality, it is not general relativity that even predicts gravitons. It is quantum field theories which describe force exchange in terms of force exchange particles which are the quanta of the fields in nature. As such, if by some stunning surprise it is proven that there is no spin 2 graviton, it would be a disproof of particle exchange models as valid unification theories. It would not be a disproof of general relativity as a classical theory. In quantum field theories, the fermions give no force exchange and likewise neither to bosons with spin above spin 2. Spin 1 results in repulsive forces between like charges, but gravitation is typically attractive. Spin 1 also only generate 3 component vector fields which lacks a degree of freedom for describing the force exchange properties of a curved 4 dimensional spacetime. Only spin 2 particles have enough degrees of freedom to do this and also they result in attraction between like charges. As a result it is often said that the graviton is expected to be spin 2. Supersymmetry is a unification attempt that associates particles of one spin with particles of spins 1/2 off of their own. This leads to the possibility of a spin 0 graviscalar, spin 1 graviphoton, spin 3/2 gravitino on top of the spin 2 graviton.

We have seen that gravitational waves travel at the speed of light. This, and also the fact that at far ranges the gravitational force is 1/r2, both require gravitons to be massless. But we also saw how gravitational waves carry momentum and energy which, in the sense of Eqn. 9.1.10, contributes to the overall gravitation

The left hand side we will write so that it becomes


We had started out with a vacuum and so we have


These imply




9.1 Gravity Waves 103

Allowing both polarization solutions given at the above link for the gravitational waves, this can be worked into the form


Which can be rewritten for a more general case of wave solutions

(Einstein summation still implied) (9.1.22)

Since t00 is the energy density of the gravitational waves and oscillates in time, the power radiated per area S is the average of this over a cycle


The hij can be related to the second time derivative of transverse-traceless part of a source's quadrupole moment QijTT evaluated at the retarded time t - r/c.

hij = (2G/rc4)QijTT,00


This finally results in the power per area radiated in gravitational waves in terms of the quadrupole moment of the source

(Einstein summation still implied) (9.1.25)

Recall equation 9.1.14


Consider allowing this line element to be only slightly more general

ds2 = - 2dvdu - (1 - f(u))dx2 - (1 + f(u))dy2


u = (z - ct)/21/2

v = (ct + z)/21/2


Next do the coordinate transformation

x = x'[1 + (1/2)f]

y = y'[1 - (1/2)f]


Then to first order in f we have

dx2 = dx' 2(1 + f) + x'2(1/4)(df/du)2du2 + x'(1 + (1/2)f)(df/du)dx'du

dy2 = dy' 2(1 - f) + y'2(1/4)(df/du)2du2 - y'(1 - (1/2)f)(df/du)dy'du


and to first order in f the line element becomes

ds2 = - [(1-f)x'2 (1+f)y'2](1/4)(df/du)2du2 - 2dvdu - dx' 2 - dy' 2 - x'(1 - (1/2)f)(df/du)dx'du + y'(1 + (1/2)f)(df/du)dy'du


which can be written

ds2 = 2h(x',y',u)du2 - 2dvdu - dx' 2 - dy' 2 - x'(1 - (1/2)f)(df/du)dx'du + y'(1 + (1/2)f)(df/du)dy'du


From here out I will drop the primes. This equation was arrived at by a series of weak field approximations, not intended to be exact, but it turns out that if we neglect the last terms what we have left

ds2 = 2h(x,y,u)du2 - 2dvdu - dx2 - dy2


actually happens to be an exact solution to Einstein's field equations for massless pp waves (plane polarized) and is an exact solution for gravitational plane waves in the case that


Equation 9.1.32 for the exact solution for pp waves can also be written

ds2 = (1 + h)dct2 - 2hdctdz - (1 - h)dz2 - dy2 - dx2


I have found that


Is an exact vacuum solution and since

n(z+ct)+m(z-ct) = az+bct


describes colliding right and left moving waves, and the Riemann tensor is not zero for , 9.1.35 is an exact vacuum solution for head on colliding gravitational plane waves.

If it is not too crazy to allow the line element to become complex, the first colliding gravitational plane wave solution is the Khan-Penrose-Szekeres solution


The exact solution to Einstein's field equations for spherically symmetric electromagnetic radiation from a point source whose stress-energy is perturbed by the emission of a coupled spherically symmetric gravity wave from the point source is


This reduces to the Vaidya solution for and the gravity wave can perturb the stress-energy all the way to a vacuum solution when it is defined so that the mass is proportional to its square,


And in case the following result may suggest a direction to find more general exact solutions, I will mention that the following line element maintains a zero Ricci-scalar




Problem 9.1.1

Verify that both polarization of Eqn. 9.1.12 obey Eqn. 9.1.8 and Eqn. 9.1.13.

Problem 9.1.2

Fill in the steps from Eqn.9.2.20 to Eqn.9.2.21

Problem 9.1.3

Find an approximate expression for Eqn. 9.1.17 given 9.1.16 and the fact that b is small and have a computer do a field plot.

Problem 9.1.4

a. Show that equation 9.1.32 can be written

ds2 = (1 + h)dct2 - 2hdctdz - (1 - h)dz2 - dy2 - dx2

using the coordinate transformation given by equation 9.1.27

b. Compute the Einstein tensor for the metric of part a to find the stress energy tensor for an arbitrary h(x,y,z-ct).


104 Chapter 9 Linearized Weak Field Gravitation

9.2 GEM

Electromagnetism as described by special relativity has both electric field and magnetic field components. In Newtonian gravitation there is only a gravitoelectric field component. In general relativity's gravitation there turns out to be both a gravitoelectric field component and a gravitomagnetism, gravitomagnetic field, component to gravitation also known as frame dragging. This aspect of gravitation in general relativity is sometimes referred to as gravitoelectromagnetism, or GEM, analogous to electromagnetism. This section will show a Lorentz guage derivation of gravitoelectromagnetism.

Refer to equation 9.1.8

In more general weak field approximation for this text we will define

. This yields


We recall that T is the invariant total mass density , and that T0i/c has the interpretation of momentum density, and that Tii has the interperetation of a pressure which we will call pi. The potential's equations becomes


Often we only consider low speeds and low pressures etc, at which only the T00 component is not neglegable and for which


Likewise if we consider only gravitational "statics" it becomes


both satisfy the same differential equation in this case, but needn't always equal eachother. In this case they both satisfy the expected Newtonian gravitation Poisson equation, and so except for 9.2.7c, from here until the problems section we will choose boundary conditions for the potentials that result in . This way only the Newtonian potential will be used.

Now had we not considered the case where the spacetime was static and not neglected the momentum flow, but still have neglected pressure, then 9.2.1 would have reduced to


In a low speed limit resulting in


Refer again to equation 9.1.8

and notice that these give the same as the electric potential and magnetic vector potential equations in Lorentz gauge



105 9.2 GEM

Neglecting pressure, the invariant interval for general relativity so far is


The above form is what will usually be found in texts sections on weak field gravitation, but it is only a special case of the liniarized weak field limit. Should we not demand that , we obtain a more general linearized weak field limit which covers a wider variety of spacetime geometries:


In doing this approximation a choice of coordinates was made that demanded equation 9.1.13 which can be expressed


Equation 9.1.13 implies

This implies for ,

And in the case of the 9.2.7a solution:


This is the Lorentz gauge condition for gravitation.

106 Chapter 9 Linearized Weak Field Gravitation

Where in this text we define by



This results in the following gravitomagnetic equations for general relativity corresponding to Maxwell's equations



And given the case of 9.2.7a as the solution:





107 9.2 GEM

Compare equations 9.1.8 and 9.2.6 and notice the correspondence between the constants.



Also note that just as with electromagnetism, vacuum field solution for equation 9.1.8 is a wave equation with a gravitational wave speed of

cg = c.


We will prove below, the coordinate acceleration a from equation 5.3.8 for a zero force four-vector which for time independent gravitational fields and a low speed and weak field approximation results in


(Note some literature defines the pseudo vector potential and gravitomagnetic field , by twice or four times this dropping the 4 in the above equation in half or eliminating it. I am going with this factor because the GEM equations more closely match Maxwell's equations and Lorentz guage.)


108 Chapter 9 Linearized Weak Field Gravitation


Refer back section 6.2 on Mach's principle for general relativity. From the perspective of a frame still with respect to the earth, it is the remote stars that revolve around the earth. From this frames perspective, the revolving matter produces the gravitational fields that are known as inertial or fictitious forces. One of these is the Coriolis force given by 6.3.33a which can be written as



Notice that this takes the same form that a uniform magnetic field's force on a charged particle would take.


From the perspective of this accelerated frame corresponds to a gravitational magnetic field of . This is a case where the gravitational magnetic field can be globally transformed away. Even though it can be transformed away here and so one might argue that it should be thought of as fictitious that does not affect the analogy with electromagnetism at the least. Consider a simple charge. According to a frame in which the charge is in motion, it has a magnetic field. This can be globally transformed away simply by transforming to the center of momentum frame of the charge. According to this frame there is no magnetic field.

To demonstrate equation 9.2.13 we start with equation 5.3.8

Go to a zero four-force and look at



Neglect terms proportional to gi0 and set g00 ~ - gii ~ 1


Neglect second order velocity terms and time independence of this metric case yeilds



Write in terms of the potentials








And that results in




Problem 9.2.1

Show . And explain why there should be no gravi-magnetic or magnetic monopoles. Hint - Consider the force on a magnetically charged particle carried in a loop around a current.

Problem 9.2.2

If expressions like the electromagnetic energy densities


were to be descriptive as a pseudo-energy density of spacetime with the appropriate choice of coordinates, what would the sign of the gravitational energy be? Would 9.2.11 be the correct inputs to use here? What would the energy density of spacetime for thin shell of mass M and radius R be?

Problem 9.2.3

Derive equations 9.2.10 using equations 9.2.9 and 9.2.5.

Problem 9.2.4

For as a function of ct, the following is an exact vacuum field solution.

For , the following is also an exact vacuum field solution.

a. In a weak field limit, show that both satisfy the linearized weak field equations given by 9.2.2 for vacuum and 9.2.7b.

(Hint, (1 + x)n ~ 1 + nx, 1/(1 + x) ~ 1 - x )

b. For , a coordinate transformation exists that takes one to the other. Find it.

Problem 9.2.5

Use the spacetime of problem 13.1.2 which is for an observer with self centered cylindrical coordinates rotating in the rim of the wheel to find the gravi-magnetic/corriolis force that he observes.


109 9.3 Isotropic static Weak Field Solution and Exact Solutions

9.3 Isotropic static Weak Field Solution and Exact Solutions

Consider an arbitrary static arrangement of solid matter out in the vacuum of space. Finding the Newtonian gravitation potential throughout the vacuum and doing gravitational physics is a fairly straight forward relatively easy matter of superposition for the potential due to each element of the matter. An exact Newtonian solution throughout the surrounding vacuum exists no matter the shape of the matter distribution. For general relativity this isn't the case. Einstein's field equations are so much more descriptive of nature that, if you were to start out with an initially still arbitrary configuration without any interacting field terms between the elements the field equations themselves, would yield a dynamic gravitational collapse of that matter. To hold the matter solid from collapsing against gravitation would require pressure and stress terms that themselves contribute as sources to the field equations. The field terms required won't be confined to the solid itself, but will tend to extend through all space diminishing toward infinity. For example section 1 of chapter 7 introduces the exact solution for an extremely charged static matter distribution. Being extremely charged gravitation is exactly balanced with the force of the electric field so the matter can remain static in any configuration, but the electric field provides stress pressure energy terms that not only are in the matter, but extend to infinity.

In the end exact vacuum solutions corresponding to the space around arbitrary static matter distributions would be futile to seek. Upon this realization it became evident that the best approach to a general relativity description of this Newtonian problem would be to seek a static matter describing metric whose Ricci-scalar would be proportional to the Laplacian of the Newtonian potential. And whose Einstein tensor terms outside the mass where the Ricci-scalar was zero would vanish toward infinity. In this way the metric would work as at least a very good description of the spacetime outside the massive solid even though it wouldn't be an exact vacuum solution there. It would also be nice if such a metric could be found that would also exactly yield the Schwarzschild solution. I was able to find exactly such a metric.

The metric given by


has a zero Ricci-scalar so long as both obey the Laplace equation and the nonzero Einstein tensor terms vanish toward infinity when the derivatives of the potentials vanish toward infinity which they will do just as does as it is the Newtonian gravitational potential. This is not an exact vacuum solution, but it is a metric that satisfies all the conditions discussed above and as such makes for a good description of the spacetime for the space surrounding a solid. It is a general relativity description of the Newtonian problem which allows us the use of superposition. It also exactly reduces to the Schwarzschild solution for

in isotropic coordinates if . If and , it exactly reduces to the Rindler spacetime, and it even yields an exact vacuum wall solution for .

So for example using this isotropic coordinate weak field approximation, the spacetime inside a nonrotating uniformly dense sphere is given by


where the restrictions on the boundary conditions of h which effects how closely modeled the interior pressure terms are, is that h = 0 at the outer surface and dh/dr = 0 at the outer surface and center, for negligible surface tension. At the surface then it matches the Schwarzschild solution in isotropic coordinates, and has no cusp behavior in the metric at the origin.

Here is a list of some exact solutions which I find of interest

Isotropic point charge, see equation 7.1.1c (Reissner-Nordstrom)


Isotropic charged domain wall (charged infinite plane) see equation 7.1.5 (David Waite)


Charged cosmic string (infinite line charge) see equation 7.1.9 (David Waite)


Charged domain wall with uniform arbitrary magnitude pressure the same along x as y with being a constant, a coefficient of the pressure terms(David Waite)


Neutral domain wall of constant pressures different along x than y, problem 8.2.4 (David Waite)


Black hole with spherically symmetric electromagnetic radiation equation 11.3.1 (Vaidya)


Infinite line of negligible mass and charge radiating electromagnetic radiation radialy equation 11.3.4 (David Waite)


pp electromangetic and gravitational plane waves solution equation 9.1.34 (Brinkmann)

 ds2 = (1 + h(x,y,z-ct))dct2 - 2(h(x,y,z-ct))dctdz - (1 - h(x,y,z-ct))dz2 - dy2 - dx2


Colliding plane waves/radiating neutral domain wall (David Waite)


Colliding gravitational waves equation 9.1.37 (Khan-Penrose-Szekeres)


Arbitrary static matter distribution of extreme charge equation 7.1.11, (Majumdar-Papapetrau)



Big Bang Model equation 8.1.1, (Friedmann-Robertson-Walker)


Charged rotating singularity equation 11.1.9, (Kerr-Newmann)


The exact solution to Einstein's field equations for spherically symmetric electromagnetic radiation from a point source whose stress-energy is perturbed by the emission of a coupled spherically symmetric gravity wave from the point source(David Waite)


This reduces to the Vaidya solution for and the gravity wave can perturb the stress-energy all the way to a vacuum solution when it is defined so that the mass is proportional to its square,


A comparison of the point charge (Reissner-Nordstrom) solution to the line charge cosmic string (David Waite) and infinite thin sheet charge domain wall (David Waite), section 1 of chapter 7.


And finally the Brill wave class of vacuum solution. (Brill) This is not actually an analytic solution to the field equations in itself, but rather a method for numerically solving for the behavior of the metric given initial conditions on an azymuthally symmetric metric. You start out with a metric which has a constant time slice of


where the behavior is restricted to


You start out with your choice of metric initially satisfying these equations, then by computer numerically time step the metric elements by so that the variation in the elements accord with Einstein's field equations. You then look at how a computer simulation describes the time dependent metric.

Unmagnetic uncharged static spherically symmetric matter interior solution(David Waite) eqn 10.3.12


and the full stress-energy tensor exactly corresponding to this is


If the matter is a fluid or gass, then inside the matter you have


but for a liquid the relation between the two potentials from this equation will break at the surface where there is surface tension.

Problem 9.3.1

Compute the Einstein tensor and Ricci-scalar for

and verify that the Ricci-scalar is zero outside the mass where the laplacian of f and g are zero.

Problem 9.3.2

Show that when the charge terms in equations 9.3.3 and 9.3.4 are zero the results are cases of equation 9.3.1 and use the results of problem 9.3.1 to show that they are in those cases exact vacuum solutions,

aside from r = 0 and z = 0 respectively.

Citations References

Bardeen, J. M., 1970, "Kerr Metric Black Holes," Nature 226, 64-65

Bondi, H., 1957, "Plane Gravitational Waves in General R.," Nature 179, 1072-1073

Bondi, H., F. A. E. Pirani, and I. Robinson, 1959, "Gravitational Waves in General R., III: Exact Plane Waves," Proc. R. Soc. London A 251, 519-533

Braginsky, V. B., 1965, "Gravitational Radiation and the Prospect of its Experimental Discovery," Sov. Phys.-Uspekhi 8, 513-521

Burke, W. L., 1971, "Gravitational Radiation Damping of Slowly Moving Systems Calculated Using Matched Asymptotic Expansions," J. Math. Phys. 12, 402-418

Gowdy, R. H., 1971, "Gravitational Waves in Closed Universes," Phys. Rev. Lett. 27, 826-829

Hawking, S. W., 1971, "Gravitational Radiation from Colliding Black Holes," Phys. Rev. Lett. 26, 1344-1346

Jordan, P., J. Ehlers, and R. Sachs, 1961, "Exact Solutions of the Field Equations of General R., II: Conrtributions to the Theory of Pure Gravitational Radiation," Akad. Wiss. Lit. Mainz Abh. Math.-Nat. Kl. 3-60

Kuchar, K., 1971, "Canonical Quantization of Cylindrical Gravitational Waves," Phys. Rev. D 4, 955-986

Newman, E. T., E. Cauch, K. Chinnapared, A. Exton, A. Prakash, and R. Torrence, 1965, "Metric of a Rotating Charged Mass," J. Math. Phys. 6, 918-919

Press, W. H., and K. S. Thorne, 1972, "Gravitational-Wave Astronomy," Ann. Rev. Astron. Astrophys. 10, 335-374

Ruffini, R., and J. A. Wheeler, 1970, "Collapse of Wave to Black Hole," Bull. Am. Phys. Soc. 15, 76

Weber, J., 1961, General Relativity and Gravitational Waves, Wiley-Interscience, New York

Weber, J., 1969, "Evidence for Discovery of Gravitational Radiation," Phys. Rev. Lett. 22, 1320-1324

Weber, J., 1970, "Gravitational Radiation Experiments," Phys. Rev. Lett 24, 276-279

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