5 When Gravity is Weak

The elegance of the Einstein field equations ensures that they hold a special place in the hearts of many physicists. However, any fondness you may feel for these equations will be severely tested if you ever try to solve them. The Einstein equations comprise ten, coupled partial differential equations. While a number of important solutions which exhibit large amounts symmetry are known, the general solution remains a formidable challenge.

We can make progress by considering situations in which the metric is almost flat. We work with $\Lambda =0$ and consider metrics which, in so-called almost-inertial coordinates $x^{\mu }$ , takes the form

$$g_{\mu \nu }={\eta }_{\mu \nu }+h_{\mu \nu }\text{\hspace{1em}(5.1)}$$

Here ${\eta }_{\mu \nu }=\mathrm{diag}(-1,+1,+1,+1)$ is the Minkowski metric. The components $h_{\mu \nu }$ are assumed to be small perturbation of this metric: $h_{\mu \nu }\ll 1$

Our strategy is to expand the Einstein equations to linear order in the small perturbation $h_{\mu \nu }$ . At this order, we can think of gravity as a symmetric “spin $2^{\mathfrak{g}}$ field $h_{\mu \nu }$ propagating in flat Minkowski space ${\eta }_{\mu \nu }.$ . To this end, all indices will now be raised and lowered with ${\eta }_{\mu \nu }$ rather than $g_{\mu \nu }$ . For example, we have

$$h^{\mu \nu }={\eta }^{\mu \rho }{\eta }^{\nu \sigma }{h}_{\rho \sigma }$$

Our theory will exhibit a Lorentz invariance, under which $x^{\mu }\to {\Lambda }^{\mu }{}_{\nu }{x}^{\nu }$ and the gravitational field transforms as

$$h^{\mu \nu }(x)\to {\Lambda }_{\rho }^{\mu }{\Lambda }_{\sigma }^{\nu }{h}^{\rho \sigma }({\Lambda }^{-1}x)$$

In this way, we construct a theory around flat space that starts to look very much like the other field theories that we meet in physics.

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