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3.4 Conservation equations

Conservation of Tmn gives the standard general relativity energy and momentum conservation equations, in the general, nonlinear case:
m m mn r + Q(r + p) + \~/ qm + 2A qm + s pmn = 0, (105) 4- n n n n a q<m> + 3Qqm + \~/ mp + (r + p)Am + \~/ pmn + A pmn + smnq - emnaw q = 0. (106)
In these equations, an overdot denotes n u \~/ n, m Q = \~/ um is the volume expansion rate of the um worldlines, Am = um = A <m> is their 4-acceleration, smn = \~/ <mun> is their shear rate, and wm = - 12 curlum = w<m> is their vorticity rate.

On a Friedmann brane, we get

Am = wm = smn = 0, Q = 3H, (107)
where H = a/a is the Hubble rate. The covariant spatial curl is given by
curlVm = emab\ ~/ aV b, curlWmn = eab(m \~/ aW bn), (108)
where emab is the projection orthogonal to m u of the 4D brane alternating tensor, and \~/ m is the projected part of the brane covariant derivative, defined by
a... a... n a d g... \~/ mF ...b = ( \~/ mF ...b) _L u = hm h g ...hb \~/ nF ...d. (109)
In a local inertial frame at a point on the brane, with um = dm0, we have: 0 = A0 = w0 = s0m = e0ab = curlV0 = curlW0m, and
\~/ mF a......b = dmidaj ...dbk \~/ iF j......k (local inertial frame), (110)
where i,j,k = 1,2,3.

The absence of bulk source terms in the conservation equations is a consequence of having /\5 as the only 5D source in the bulk. For example, if there is a bulk scalar field, then there is energy-momentum exchange between the brane and bulk (in addition to the gravitational interaction) [225Jump To The Next Citation Point16Jump To The Next Citation Point236Jump To The Next Citation Point97Jump To The Next Citation Point194Jump To The Next Citation Point98Jump To The Next Citation Point35Jump To The Next Citation Point].

Equation (73View Equation) may be called the “nonlocal conservation equation”. Projecting along um gives the nonlocal energy conservation equation, which is a propagation equation for r E. In the general, nonlinear case, this gives

r + 4-Qr + \~/ mqE + 2AmqE + smnpE = E 3 E m m mn 1 [ mn mn m mn m n --- 6p pmn + 6(r + p)s pmn + 2Q (2q qm + p pmn) + 2A q pmn 4c ] -4qm\ ~/ mr + qm \~/ npmn + pmn \~/ mqn - 2smnpampna - 2smnqmqn . (111)
Projecting into the comoving rest space gives the nonlocal momentum conservation equation, which is a propagation equation for qEm:
4 1 4 qE&lt;m&gt; + --QqEm + -- \~/ mrE + -rE Am + \~/ npEmn + AnpEmn + smnqEn - emnawnqEa = 3 [ 3 3 -1- - 4(r + p) \~/ mr + 6(r + p) \~/ npmn + qnp&lt;mn&gt; + pmn\ ~/ n(2r + 5p) 4c ( ) 2- ab ab 28-n - 3 p \~/ mpab + 3 \~/ apbm - 3pma \~/ bp + 3 q \~/ mqn 8 +4rAnpmn - 3pmaAbpab + -Ampabpab - pmasabqb ab nab 3 a bn +smap qb + pmne waqb - emabw p qn + 4(r + p)Qqm n 14 n ab ] +6qmA qn + 3-Amq qn + 4qms pab . (112)
The 1 + 3-covariant decomposition shows two key features:

In particular cases, the Weyl anisotropic stress pE mn may drop out of the nonlocal conservation equations, i.e., when we can neglect mn E s pmn, n E \~/ pmn, and n E A p mn. This is the case when we consider linearized perturbations about an FRW background (which remove the first and last of these terms) and further when we can neglect gradient terms on large scales (which removes the second term). This case is discussed in Section 6. But in general, and especially in astrophysical contexts, the pE mn terms cannot be neglected. Even when we can neglect these terms, E pmn arises in the field equations on the brane.

All of the matter source terms on the right of these two equations, except for the first term on the right of Equation (112View Equation), are imperfect fluid terms, and most of these terms are quadratic in the imperfect quantities qm and pmn. For a single perfect fluid or scalar field, only the \~/ mr term on the right of Equation (112View Equation) survives, but in realistic cosmological and astrophysical models, further terms will survive. For example, terms linear in pmn will carry the photon quadrupole in cosmology or the shear viscous stress in stellar models. If there are two fluids (even if both fluids are perfect), then there will be a relative velocity v m generating a momentum density q = rv m m, which will serve to source nonlocal effects.

In general, the 4 independent equations in Equations (111View Equation) and (112View Equation) constrain 4 of the 9 independent components of Emn on the brane. What is missing is an evolution equation for pEmn, which has up to 5 independent components. These 5 degrees of freedom correspond to the 5 polarizations of the 5D graviton. Thus in general, the projection of the 5-dimensional field equations onto the brane does not lead to a closed system, as expected, since there are bulk degrees of freedom whose impact on the brane cannot be predicted by brane observers. The KK anisotropic stress pEmn encodes the nonlocality.

In special cases the missing equation does not matter. For example, if pE = 0 mn by symmetry, as in the case of an FRW brane, then the evolution of Emn is determined by Equations (111View Equation) and (112View Equation). If the brane is stationary (with Killing vector parallel to m u), then evolution equations are not needed for Emn, although in general pEmn will still be undetermined. However, small perturbations of these special cases will immediately restore the problem of missing information.

If the matter on the brane has a perfect-fluid or scalar-field energy-momentum tensor, the local conservation equations (105View Equation) and (106View Equation) reduce to

r + Q(r + p) = 0, (113) \~/ mp + (r + p)Am = 0, (114)
while the nonlocal conservation equations (111View Equation) and (112View Equation) reduce to
4 rE + -QrE + \~/ mqEm + 2AmqEm + smnpEmn = 0, (115) 3 qE + 4-QqE + 1 \~/ r + 4-r A + \~/ npE + AnpE + s nqE - e naw qE = - (r +-p) \~/ r. (116) &lt;m&gt; 3 m 3 m E 3 E m mn mn m n m n a c m
Equation (116View Equation) shows that [291]

A simple example of the latter point is the FRW case: Equation (116View Equation) is trivially satisfied, while Equation (115View Equation) becomes

rE + 4HrE = 0. (117)
This equation has the dark radiation solution
( ) r = r a0- 4. (118) E E0 a

If Emn = 0, then the field equations on the brane form a closed system. Thus for perfect fluid branes with homogeneous density and Emn = 0, the brane field equations form a consistent closed system. However, this is unstable to perturbations, and there is also no guarantee that the resulting brane metric can be embedded in a regular bulk.

It also follows as a corollary that inhomogeneous density requires nonzero E mn:

\~/ mr /= 0 ==&gt; Emn /= 0. (119)
For example, stellar solutions on the brane necessarily have Emn /= 0 in the stellar interior if it is non-uniform. Perturbed FRW models on the brane also must have Emn /= 0. Thus a nonzero Emn, and in particular a nonzero E pmn, is inevitable in realistic astrophysical and cosmological models.

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