Linearized gravity

Linearized gravity is an approximation scheme in general relativity in which the nonlinear contributions from the spacetime metric are ignored, simplifying the study of many problems while still producing useful approximate results.

The method

In linearized gravity the metric tensor, g, of spacetime is treated as a sum of an exact solution of Einstein's equations (often Minkowski spacetime) and a perturbation h.

g \, =\eta+h

where \eta is the nondynamical background metric that is being perturbed about, and h represents the deviation of the true metric (g) from flat spacetime.

The perturbation is treated using the methods of perturbation theory, "linearized" by ignoring all terms of order higher than one (quadratic in h, cubic in h etc...) in the perturbation.

Applications

The Einstein field equations (EFE), being nonlinear in the metric, are difficult to solve exactly and the above perturbation scheme allows linearised Einstein field equations to be obtained. These equations are linear in the metric, and the sum of two solutions of the linearized EFE is also a solution. The idea of 'ignoring the nonlinear part' is thus encapsulated in this linearization procedure.

The method is used to derive the Newtonian limit, including the first corrections, much like for a derivation of the existence of gravitational waves that led, after quantization, to gravitons. This is why the conceptual approach of linearized gravity is the canonical one in particle physics, string theory, and more generally quantum field theory where classical (bosonic) fields are expressed as coherent states of particles.

This approximation is also known as the weak-field approximation as it is only valid if the perturbation h is very small.

Weak-field approximation

In a weak-field approximation, the gauge symmetry is associated with diffeomorphisms with small "displacements" (diffeomorphisms with large displacements obviously violate the weak field approximation), which has the exact form (for infinitesimal transformations)

\delta_{\vec{\xi}}h=\delta_{\vec{\xi}}g-\delta_{\vec{\xi}}\eta=\mathcal{L}_{\vec{\xi}}g=\mathcal{L}_{\vec{\xi}}\eta+\mathcal{L}_{\vec{\xi}}h= \left[\xi_{\nu;\mu} + \xi_{\mu;\nu} + \xi^\alpha h_{\mu\nu;\alpha} + \xi^\alpha_{;\mu} h_{\alpha\nu} + \xi^\alpha_{;\nu} h_{\mu\alpha}\right]dx^\mu \otimes dx^\nu

Where \mathcal{L} is the Lie derivative and we used the fact that η does not transform (by definition). Note that we are raising and lowering the indices with respect to η and not g and taking the covariant derivatives (Levi-Civita connection) with respect to η. This is the standard practice in linearized gravity. The way of thinking in linearized gravity is this: the background metric η is the metric and h is a field propagating over the spacetime with this metric.

In the weak field limit, this gauge transformation simplifies to

\delta_{\vec{\xi}}h_{\mu\nu}\approx \left(\mathcal{L}_{\vec{\xi}}\eta\right)_{\mu\nu}=\xi_{\nu;\mu} + \xi_{\mu;\nu}

The weak-field approximation is useful in finding the values of certain constants, for example in the Einstein field equations and in the Schwarzschild metric.

Linearised Einstein field equations

The linearised Einstein field equations (linearised EFE) are an approximation to Einstein's field equations that is valid for a weak gravitational field and is used to simplify many problems in general relativity and to discuss the phenomena of gravitational radiation. The approximation can also be used to derive Newtonian gravity as the weak-field approximation of Einsteinian gravity.

The equations are obtained by assuming the spacetime metric is only slightly different from some baseline metric (usually a Minkowski metric). Then the difference in the metrics can be considered as a field on the baseline metric, whose behaviour is approximated by a set of linear equations.

Derivation for the Minkowski metric

Starting with the metric for a spacetime in the form

g_{ab} = \eta_{ab} + h_{ab}

where \, \eta_{ab} is the Minkowski metric and \, h_{ab} — sometimes written as \epsilon \, \gamma_{ab} — is the deviation of \, g_{ab} from it. h must be negligible compared to \eta: \left| h_{\mu \nu} \right| \ll 1 (and similarly for all derivatives of h). Then one ignores all products of h (or its derivatives) with h or its derivatives (equivalent to ignoring all terms of higher order than 1 in \epsilon). It is further assumed in this approximation scheme that all indices of h and its derivatives are raised and lowered with \eta.

The metric h is clearly symmetric, since g and η are. The consistency condition g_{ab}g^{bc}=\delta_a{}^c shows that

g^{ab} \, = \eta^{ab} - h^{ab}

The Christoffel symbols can be calculated as

2 \Gamma ^a_{bc} = (h^a{}_{b,c}+h^a{}_{c,b}-h_{bc,}{}^a)

where h_{bc,}{}^a \ \stackrel{\mathrm{def}}{=}\ \eta^{ar} h_{bc,r}, and this is used to calculate the Riemann tensor:

2R^a{}_{bcd} = 2(\Gamma^a_{bd,c}-\Gamma^a_{bc,d}) = \eta^{ae} (h_{eb,dc}+h_{ed,bc}-h_{bd,ec} - h_{eb,cd}-h_{ec,bd}+h_{bc,ed}) =
= \eta^{ae} (h_{ed,bc}-h_{bd,ec}-h_{ec,bd}+h_{bc,ed}) = h^a_{d,bc} - h_{bd,}{}^ a{}_c + h_{bc,}{}^a{}_d - h^a{}_{c,bd}

Using R_{bd}= \delta ^c{}_a R^a{}_{bcd} gives

2R_{bd}= h^r_{d,br} + h^r_{b,dr} -h_{,bd} - h_{bd, rs} \eta ^{rs}

For Ricci scalar we have:

R = R_{bd} \eta^{bd}= h^{a b}_{, a b} - \square h

Then the linearized Einstein equations are

8\pi T_{bd} \, = R_{bd} - R_{ac} \eta^{ac} \eta_{bd} / 2

or

8\pi T_{bd} = (h^r_{d,br} + h^r_{b,dr} -h_{,bd} - h_{bd, r}{}^r - h^r_{s,r}{}^s \eta_{bd})/2 + ( h_{,a}{}^a \eta_{bd} + h_{ac, r}{}^r \eta^{ac} \eta_{bd}) /4

Or, equivalently:

8\pi (T_{bd} - T_{ac} \eta^{ac} \eta_{bd}/2) \, = R_{bd}
16\pi (T_{bd} - T_{ac} \eta^{ac} \eta_{bd}/2) \, = h^r_{d,br} + h^r_{b,dr} -h_{,bd} - h_{bd, rs} \eta ^{rs}

With a coordinate condition

If one uses the Lorentz invariant harmonic coordinate condition

h_{\alpha \beta,\gamma} \eta^{\beta \gamma} = \frac12 h_{\beta \gamma,\alpha} \eta^{\beta \gamma} \,,

then the last form above of the linearized Einstein equation simplifies to

16\pi (T_{bd} - T_{ac} \eta^{ac} \eta_{bd}/2) \, = \, - h_{bd, rs} \eta ^{rs} \,.

To solve it, this can be rewritten as

\Delta h_{bd} = {-16\pi G \over c^4} (T_{bd} - T_{ac} \eta^{ac} \eta_{bd}/2) + \frac{ \partial^2 h_{bd} }{ c^2 {\partial t}^2 } \,

where ∆ is the Laplacian on a spatial slice. If the stress-energy changes slowly (velocities are low compared to c), then this gives

h_{bd} (r) = \frac{-1}{4\pi} \int \left( {-16\pi G \over c^4} (T_{bd} (s) - T_{ac} (s) \eta^{ac} \eta_{bd}/2) + \frac{ \partial^2 h_{bd} (s) }{ c^2 {\partial t}^2 } \right) \frac{1}{\vert r-s \vert} d^3 s \,

as a generalization of the Newtonian formula for gravitational potential. This is solved iteratively by first replacing the second time derivative by zero and then inserting the h so obtained repeatedly until convergence.

Applications

The linearised EFE are used primarily in the theory of gravitational radiation, where the gravitational field far from the source is approximated by these equations.

See also

References

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