Linearized Weak Field Gravitation

 Return to Modern Relativity

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 gravitational wave equation:

h^lsh'_mn,_ls = 0

(9.1.1)

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

Next we refer to the affine connections Eqn. 4.4.3

G^l_mn = (1/2)g^ls(g_ms,_n + g_ns,_m - g_mn,_s)

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

g_mn = h_mn + h_mn

(9.1.2)

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

 97

98 Chapter 9 Linearized Weak Field Gravitation

The equation for the affine connection becomes

G^l_mn = (1/2)h^ls(h_ms,_n + h_ns,_m - h_mn,_s)

We then refer to the equation for the Riemann tensor Eqn. 6.1.3

R^l_mrn = G^l_mn,_r - G^l_mr,_n + G^l_srG^s_mn - G^l_snG^s_mr

Replacing in the expression for the affine connections and keeping only first order terms in h_mn we arrive at

R^l_mrn = (1/2)[h^l_n,_rm - h^lsh_mn,_rs - h^l_r,_nm + h^lsh_mr,_ns]

We next contract this to get the Ricci tensor Eqn. 6.3.1

R_mn = R^l_mln

resulting in

R_mn = (1/2)[h^l_n,_lm - h^lsh_mn,_ls - h^l_l,_nm + h^lsh_ml,_ns]

Using the raising index property of the contravarient metric tensor we simplify this to

R_mn = (1/2)[h^l_n,_lm - h^lsh_mn,_ls - h^l_l,_nm + h_m^s,_ns]

Rearranging the terms we have

R_mn = - (1/2)[h^lsh_mn,_ls + h^l_l,_nm - h^l_n,_lm - h_m^s,_ns]

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

R_mn = - (1/2){h^lsh_mn,_ls - [h_m^s - (1/2)d_m^sh^l_l],_sn - [h^s_n - (1/2)d^s_nh^l_l],_sm }

Einstein's field equations can be expressed in the form Eqn. 6.3.21

R_mn = (8pG/c⁴)[T_mn - (1/2)g_mnT]

resulting in

- (1/2){h^lsh_mn,_ls - [h_m^s - (1/2)d_m^sh^l_l],_sn - [h^s_n - (1/2)d^s_nh^l_l],_sm } = (8pG/c⁴)[T_mn - (1/2)h_mnT]

(9.1.3)

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

[h_m^s - (1/2)d_m^sh^l_l],_sn and [h^s_n - (1/2)d^s_nh^l_l],_sm terms are not zero. In this case, consider the following infinitesimal coordinate transformation

x'^m = x^m + e^m

(9.1.4)

Doing this we find that h_mn transforms according to

h'_mn = h_mn - e_m,_n - e_n,_m

(9.1.5)

9.1 Gravitational Radiation 99

We then choose to do the infinitesimal coordinate transformation so that

h^lse_m,_ls = [h_m^s - (1/2)d_m^sh^l_l],_s

(9.1.6)

which is the same as requiring

h^lse_n,_ls = [h^s_n - (1/2)d^s_nh^l_l],_s

(9.1.7)

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

h^lsh'_mn,_ls = -(16pG/c⁴)[T '_mn - (1/2)h_mnT ' ]

(9.1.8)

In vacuum T_mn = T = 0 resulting in

h^lsh'_mn,_ls = 0

This is the ordinary wave equation for waves traveling at the speed c. Therefor in a weak field limit 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. 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 t^mn of gravitational waves, We transform the Ricci Tensor and the Ricci scalar into our new choice of coordinates. This results in

R'_mn = - (1/2)h^lsh'_mn,_ls

(9.1.9)

R' = - (1/2)h^lsh',_ls

Reconstructing the Einstein tensor in the first order in h_mn approximation results in

R'_mn - (1/2)h_mnR' = - (1/2)h^ls[h'_mn - (1/2)h_mnh'],_ls

The form Einstein's field equations should take is

R'_mn - (1/2)h_mnR' = (8pG/c⁴)t_mn

(9.1.10)

100 Chapter 9 Linearized Weak Field Gravitation

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

t^mn = - (c⁴/32pG)h^ls[2h'^mn - h^mnh'],_ls

(9.1.11)

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.

Consider the next two expressions ordered so that x³ = z and with w/k = c.

and

(9.1.12)

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,

[h' ^s_n - (1/2)d^s_nh' ^l_l],_s = 0

(9.1.13)

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 f₂ = 0. It becomes

ds² = dct² - (1 - bcos(kz - wt))dx² - (1 + bcos(kz - wt))dy² - dz²

(9.1.14)

9.1 Gravitational Radiation 101

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

dL² = (1 - bcos(wt))dx² + (1 + bcos(wt))dy²

(9.1.15)

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

L = [x² + y² + bcos(wt)(y² - x²)]^1/2

(9.1.16)

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 = ¶²L/¶t²

(9.1.17)

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 45° .

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

R'_mn - (1/2)h_mnR' = (8pG/c⁴)t_mn

The left hand side we will write G'_mn^[1] so that it becomes

G'_mn^[1] = (8pG/c⁴)t_mn

(9.1.18)

We had started out with a vacuum and so we have

G'_mn = G'_mn^[1] + G'_mn^[2] = 0

(9.1.19)

These imply

t⁰⁰ = -(c⁴/8pG)G'^00[2]

(9.1.20)

9.1 Gravitational Radiation 103

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

t⁰⁰ » (c²/16pG)[(h₊,₀)² + (h_x,₀)²]

(9.1.21)

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

t⁰⁰ = (c²/16pG)(h_ij,₀h_ij,₀)

(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

S = c<t⁰⁰>

(9.1.23)

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

h_ij = (2G/rc⁴)Q_ij^TT,₀₀

(9.1.24)

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

S = (G/8pr²c⁵)<Q_ij^TT,₀₀₀Q_ij^TT,₀₀₀>

(Einstein summation still implied) (9.1.25)

Exercises

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.

___________________________________________________________________________________________ 

104 Chapter 9 Linearized Weak Field Gravitation

Electromagnetism has both electric field and magnetic field components. In Newtonian gravitation there is only a gravi-electric field component. In general relativity's gravitation there turns out to be both a gravi-electric field component and a gravi-magnetic field component to gravitation. This aspect of gravitation is sometimes referred to as gravitomagnetism, analogous to electromagnetism. This section will show a Lorentz guage derivation of gravitomagnetism.

Refer to equation 9.1.8

h^lsh'_mn,_ls = (16pG/c⁴)[T '_mn - (1/2)h_mnT ' ]

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

f'₀₀ as (c²/2)h'₀₀, f'_ii as (-c²/2)h'_ii, and f_i as - c²h'_0i. This yields

Ñ ²f '₀₀ - ¶ ²f '₀₀/¶ct² = (8pG/c²)[T₀₀ - (1/2)T]

Ñ ²f '_ii - ¶ ²f '_ii/¶ct² = - (8pG/c²)[T_ii + (1/2)T]

Ñ ²f _i - ¶ ²f _i/¶ct² = - (16pG/c²)T_0i

(9.2.1)

We recall that T is the invariant total mass density r _Tot, and that T⁰ⁱ has the interpretation of momentum density which we will call cJⁱ, and that T_ii has the interperetation of a pressure which we will call p_i. The potential's equations becomes

Ñ ²f '₀₀ - ¶ ²f '₀₀/¶ct² = (8pG/c²)[T₀₀ - (1/2)r _Totc²]

Ñ ²f '_ii - ¶ ²f '_ii/¶ct² = - (8pG/c²)[p_i + (1/2)r _Totc²]

Ñ ²f _i - ¶ ²f _i/¶ct² = - (16pG/c)J_i

(9.2.2)

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

T ' ⁰⁰ » r₀c² » r _Totc².

Likewise if we consider only gravitational "statics" it becomes

Ñ ²f '₀₀ = - Ñ ²f '_ii = 4pGr₀

f '₀₀, and - f '_ii 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 untill the problems section we will choose boundary conditions for the potentials that result in f '₀₀ = - f '_ii = f. This way only the Newtonian potential f 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

Ñ ²f - ¶ ²f/¶ct² = (8pG/c²)[T₀₀ - (1/2)T]

Ñ ²f _i - ¶ ²f _i/¶ct² = - (16pG/c²)T_0i

In a low speed limit T ⁰⁰ = r₀c² resulting in

Ñ ²f - ¶ ²f/¶ct² = 8pG[r₀ - (1/2)r_Tot]

Ñ ²f _i - ¶ ²f _i/¶ct² = - (16pG/c)J_i

(9.2.5)

Refer again to equation 9.1.8

h^lsh'_mn,_ls = (16pG/c⁴)[T '_mn - (1/2)h_mnT ' ]

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

Ñ ²f - ¶ ²f /¶ct² = - r/e₀

Ñ ²f_i - ¶ ²f_i/¶ct² = - cm₀J_i

(9.2.6)

105 9.2 Newtonian Limit Vs Gravitomagnetism

Neglecting pressure, the invariant interval so far is

ds² = (1 + 2f/c²)dct² - (2f_x/c²)dctdx - (2f_y/c²)dctdy - (2f_z/c²)dctdz - (1 - 2f/c²)(dx² + dy² + dz²)

(9.2.7a)

or more simply expressed

ds² = (1 + 2f/c²)dct² - (2/c²)dct(f_idxⁱ) - (1 - 2f/c²)d_ijdxⁱdx^j

(9.2.7b)

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 f '₀₀ = - f '_ii, we obtain a more general linearized weak field limit which covers a wider variety of spacetime geometries:

ds² = (1+2f '₀₀/c²)dct² - (2/c²)dct(f_idxⁱ) - (1-2f '_xx/c²)dx² - (1-2f '_yy/c²)dy² - (1-2f '_zz/c²)dz²

(9.2.7c)

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

h^sm[h' _mn - (1/2)d_mnh' ^l_l],_s = 0.

Equation 9.1.13 implies

[h' _0n - (1/2)d_0nh' ^l_l],₀ - [h' _1n - (1/2)d_1nh' ^l_l],₁ - [h' _2n - (1/2)d_2nh' ^l_l],₂ - [h' _3n - (1/2)d_3nh' ^l_l],₃ = 0.

This implies for n = 0,

[h' ₀₀ - (1/2)h' ^l_l],₀ - h' ₁₀,₁ - h' ₂₀,₂ - h' ₃₀,₃ = 0

[h' ₀₀ - (1/2)h^mlh' _ml],₀ - h' ₁₀,₁ - h' ₂₀,₂ - h' ₃₀,₃ = 0

[h' ₀₀ - (1/2)(h' ₀₀ - h' ₁₁ - h' ₂₂ - h'₃₃)],₀ - h' ₁₀,₁ - h' ₂₀,₂ - h' ₃₀,₃ = 0

And in the case of 9.2.7a,b solutions:

[2f/c² - (1/2)( - 4f/c²)],₀ + (f₁/c²),₁ + (f₂/c²),₂ + (f₃/c²),₃ = 0

f,₀ + (1/4)f₁,₁ + (1/4)f₂,₂ + (1/4)f₃,₃ = 0

f º S_if_ie_i

(¶f/¶ct) + (1/4)Ñ×f = 0

(9.2.8)

This is the Lorentz gauge condition for gravitation.

106 Chapter 9 Linearized Weak Field Gravitation

Where in this text we define E_g and B_g by

E_g = - Ñf - ¶f/¶ct

(9.2.9a)

B_g = Ñ´f/c

 (9.2.9b) 

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

Ñ´E_g = - ¶B/¶t

(9.2.10a)

Ñ×B_g = 0

(9.2.10b)

107 9.2 Newtonian Limit Vs Gravitomagnetism

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

e₀ Û -1/8pG

m₀ Û 16pG/c².

(9.2.11)

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

c_g = c.

(9.2.12)

In a slow speed week field limit one can go on to calculate the acceleration or a from geodesic motion which for a results in

a = E_g + u´B_g

(9.2.13)

108 Chapter 9 Linearized Weak Field Gravitation

Refer back section 6.2 on Mach's principle. 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

 f_Cor = mu´(2w)

(9.2.14)

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

f_B = qu´B

(9.2.15)

From the perspective of this accelerated frame 2w corresponds to a gravitational magnetic field of B_g = 2w. 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.

Problem 9.2.1

Show Ñ×B_g = 0. 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

 and

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

ds² = (1 + az/c²)²dct² - dx² - dy² - dz²

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

ds² = (1 + 2az/c²)dct² - dx² - dy² - dz²/(1 + 2az/c²)

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

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

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

c. What would be a local transformation that takes one of these to equation 9.2.7a for f = az?

Problem 9.2.5

Show that the spacetime interval given by

ds² = [1 + 2F/c²]dct² - dr²/[1 - 2r(dF/dr)/c²] - r²dq² - r²sin²qdf²

, where F is a function of r and corresponds to the Newtonian gravitational potential is, a weak field solution for the interior of a localized spherically symmetric matter distribution by finding the stress energy tensor for this spacetime to first order in the potential and its derivatives. Side note - if it is used to represent a uniformly dense matter distribution, then

F = (1/2)(GM_totr²/R³) - (3/2)(GM_tot/R)

and this weak field solution becomes

ds² = [1 + (GM_totr²/R³c²) - (3GM_tot/Rc²)]dct² - dr²/[1 - 2GM_totr²/R³c²] - r²dq² - r²sin²qdf²

This matches the Schwarzschild solution at the surface. Also, note that the gravitational length contraction or "spatial curvature" vanishes at the center while the gravitational time dilation does not.

Problem 9.2.6

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.

Problem 9.2.7

Show that equation 9.2.16 for linearized weak field gravitation implies

f_grav = -mÑf + mu´B_g - m¶f/¶ct

and that therefor it is when one can both consider f constant in time and neglect gravi-magnetism that this reduces to the Newtonian form

f_grav = -mÑf

Problem 9.2.8

a. Equation 6.3.23 is ds² for an observer that spins on his z axis. Even though equation 6.3.23 is an exact solution, verify that it is in the linearized weak field solution "form" of 9.2.7c.

b. The Coriolis force that comes from an exact calculation of geodesic motion for this spacetime is equation 6.3.33a. The gravi-magnetic field that exactly comes from that expression is

B_g = 2w

Show that the same result comes from equation 9.2.9b.

c. Show that equation 9.2.9a applied to 6.3.23 also yields the same centrifugal and transverse force terms as 6.3.33b and 6.3.33c.