C.2. The Newtonian limit¶
In this and the next section, we consider the general theory of relativity in the limit of weak gravitational fields. In this section we focus on non-relativistic matter and demonstrate that Einstein’s field equations and the geodesic equation reduce to the Poisson equation and Newton’s second law, respectively. In the next section, we consider the trajectories of light in weak gravitational fields, which will allow us to discuss the way gravitational fields bend light and give rise to the phenomenon of gravitational lensing. To allow for these two applications, we start with a general discussion of GR in the limit of weak gravitational fields. For completeness, we’ll carry around the cosmological constant \(\Lambda\) for a while below, but the measured value of \(\Lambda\) is much smaller than the curvature induced by galactic gravitational fields and \(\Lambda\) is only relevant on cosmological scales (\(\Lambda \approx 1/\mathrm{Gpc}^2)\).
We will expand around the Minkowski metric of flat space, but the Universe that galaxies inhabit in GR is not static and is instead (in our Universe) expanding on large scales. Thus, technically, the proper background to expand around is the Friedmann–Lemaître–Robertson–Walker (FLRW) metric from Equation (C.115) below. This significantly complicates the required derivations (e.g., Bertschinger 1995), although they remain qualitatively similar. One of the main relevant results from the correct derivation is that the gravitational potential \(\Phi\) that appears below is sourced by the density contrast \(\rho - \bar{\rho}_m\) relative to the mean background matter density \(\bar{\rho}_m\) rather than the density itself. Thus, the density that enters the Poisson equation is \(\rho - \bar{\rho}_m\) rather than \(\rho\) (see Equation 17.17; there is then also no explicit \(\Lambda\) term in the Poisson equation, see Equation C.69 below). That expanding around the flat Minkowski metric is a reasonable approach for our two main applications of interest can be understood as follows. For the dynamics within galaxies, we have that galactic halos are decoupled from the overall expansion of the Universe through gravitational collapse; thus, no overall expansion of the metric occurs within galaxies and the background FLRW metric reduces to the Minkowski metric (also using that any large-scale curvature occurs on scales \(\gg\) that of galaxies). For gravitational lensing by galaxies, the relevant length and (crossing) time scales are also \(\ll\) the curvature and expansion time scales, so the FLRW metric can again be approximated by the Minkowski metric for a photon passing through a gravitational lens. The distinction between \(\rho\) and \(\rho -\bar{\rho}_m\) is also irrelevant for galaxies, because the density in all halos in our Universe is \(\gtrsim 100\,\bar{\rho}_m\) due to virialization (see Chapter 17.3.1); for example, at the Sun’s position in the Milky Way, \(\rho/\bar{\rho}_m \approx 10^6\).
C.2.1. Einstein’s field equations for weak gravitational fields¶
What we mean by a weak gravitational field in GR is that the metric \(g_{\mu\nu}\) is approximately the Minkowskian metric \(\eta_{\mu\nu}\) of flat space, with a first order correction \(h_{\mu\nu}\) whose elements satisfy \(|h_{\mu\nu}| \ll 1\) \begin{equation} g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}\,, \end{equation} and because \(|h_{\mu\nu}| \ll 1\), we can raise and lower indices simply by using the Minkowski metric, for example, \(h^{\mu\nu} = \eta^{\mu\delta}\eta^{\nu\epsilon} h_{\delta\epsilon}\) to first order in the small quantities \(h_{\mu\nu}\). We then need to work out Einstein’s field equations to first order in \(h_{\mu\nu}\) and therefore need to compute the Einstein tensor from Equation (C.39) to this order. The first thing we need to do is to compute the Christoffel connection from Equation (C.28). Because \(\partial_\lambda \eta_{\mu\nu} = 0\), the Christoffel connection in this case is simply \begin{equation}\label{eq-gr-christoffel-metric-weak} \Gamma^{\lambda}_{\mu\nu} = {1\over 2}\eta^{\lambda \epsilon}\,\left(\partial_\mu h_{\nu\epsilon} + \partial_\nu h_{\epsilon\mu}-\partial_\epsilon h_{\mu\nu}\right)+\mathcal{O}(h_{\mu\nu}^2)\,. \end{equation} As expected, the Christoffel connection is a first-order quantity, because it is zero for flat space. We then use the Christoffel connection to compute the Riemann tensor from Equation (C.36), which is similarly a first-order quantity. Because the Christoffel connection is a first-order quantity, the last two terms in the Riemann tensor of Equation (C.36), which involve products of two connections, vanish to first order and we have \begin{align} R^\delta_{\phantom{\delta}\epsilon\mu\nu} & = {1\over 2}\eta^{\delta \lambda}\,\left(\partial_\mu\partial_\epsilon h_{\lambda\nu}-\partial_\mu\partial_\lambda h_{\nu\epsilon} - \partial_\nu\partial_\epsilon h_{\lambda\mu}+\partial_\nu\partial_\lambda h_{\mu\epsilon}\right)+\mathcal{O}(h_{\mu\nu}^2)\,.\label{eq-gr-riemann-tensor-weak} \end{align} From this we find the Ricci tensor using Equation (C.37) \begin{equation}\label{eq-gr-ricci-tensor-weak} R_{\mu\nu} = {1\over 2}\,\left(\partial_\mu\partial_\lambda h^{\lambda}_{\phantom{\lambda}\nu}+\partial_\nu\partial_\lambda h^{\lambda}_{\phantom{\lambda}\mu}- \partial_\mu\partial_\nu h_{\lambda}^{\phantom{\lambda}\lambda}-\eta^{\delta \lambda}\partial_\delta\partial_\lambda h_{\mu\nu} \right)+\mathcal{O}(h_{\mu\nu}^2)\,. \end{equation} Finally, the Ricci scalar from Equation (C.38) \begin{equation}\label{eq-gr-ricci-scalar-weak} R =\partial_\mu\partial_\nu h^{\mu\nu}- \eta^{\mu\nu}\partial_\mu\partial_\nu h_{\lambda}^{\phantom{\lambda}\lambda}+\mathcal{O}(h_{\mu\nu}^2)\,. \end{equation} In deriving these, we have occasionally relabeled repeated indices to use \((\mu,\nu)\) as much as possible and we’ve made ample use of the symmetry of \(\eta_{\mu\nu}\) and \(h_{\mu\nu}\) and of the exchangeability of partial derivatives. Combining the Ricci tensor and scalar to form the Einstein tensor \(R_{\mu\nu}-{1\over 2}R\,g_{\mu\nu}\) and using the fact that the cosmological constant in our Universe itself is small such that \(\Lambda h_{\mu\nu} \ll \Lambda \eta_{\mu\nu}\), we finally find the linearized field equations \begin{align}\label{eq-gr-fieldeqs-linear} {1\over 2}\,& \left(\partial_\mu\partial_\lambda h^{\lambda}_{\phantom{\lambda}\nu} +\partial_\nu\partial_\lambda h^{\lambda}_{\phantom{\lambda}\mu}- \partial_\mu\partial_\nu h_{\lambda}^{\phantom{\lambda}\lambda}-\eta^{\delta \lambda}\partial_\delta\partial_\lambda h_{\mu\nu} \right. \\ &\qquad \qquad \left. -\eta_{\mu\nu}\partial_\delta\partial_\epsilon h^{\delta\epsilon}+\eta_{\mu\nu}\eta^{\delta\epsilon}\partial_\delta\partial_\epsilon h_{\lambda}^{\phantom{\lambda}\lambda}\right)+ \Lambda \eta_{\mu\nu}+\mathcal{O}(h_{\mu\nu}^2) = {8\pi G \over c^4}\,T_{\mu\nu}\,.\nonumber \end{align}
Before continuing, we will note without proof that even though there are ten equations relating \(h_{\mu\nu}\) to the stress-energy tensor, four of these equations are redundant, because the equations are invariant under the transformation \(h_{\mu\nu} \rightarrow h_{\mu\nu} + \varepsilon\partial_\mu\xi_\nu + \varepsilon\partial_\nu\xi_\mu\) for any four vector \(\xi^\mu\), where \(\epsilon \ll 1\). This is a form of gauge invariance that is similar to the fact that we can add a constant to the gravitational potential in Newtonian gravity and obtain the same forces and densities or to the gauge invariance in electromagnetism. We can use this gauge invariance to give \(h_{\mu\nu}\) convenient properties, by fixing the vector \(\xi^\mu\) (so-called “choosing a gauge”). Einstein’s general field equations (C.51) in fact have a similar gauge invariance that follows from \(g^{\mu\delta}\nabla_\delta\left(R_{\mu\nu}-{1\over 2}R\,g_{\mu\nu}+ \Lambda g_{\mu\nu}\right)=0\).
At this stage, it is standard to work in the rest frame in which the bulk velocity of the gravitating matter and energy is zero and write the components of \(h_{\mu\nu}\) suggestively as \begin{align}\label{eq-gr-weak-metric} h_{00} & = -2{\Phi \over c^2}\,;\quad h_{0i} = w_i\,;\quad h_{ij} = 2s_{ij} -2{\Psi \over c^2}\,\delta_{ij}\,, \end{align} where \(\delta_{ij}\) is the Kronecker delta. These correspond to the scalar, vector, and tensor parts of the transformation of the metric under spatial rotations of the rest frame. The tensor part is further split up into a trace-free part \(s_{ij}\) and half the trace of the spatial tensor \(\Psi/c^2 = -\sum_{ij} \delta^{ij}h_{ij}/6\) (as usual, latin indexing indicates that the indices only run over the spatial dimensions; we add the explicit summation to be very clear that we intend to sum over latin indices here). The line element is then given by \begin{equation} \mathrm{d} s^2 = -\left(1+2{\Phi \over c^2}\right)\,c^2\mathrm{d}t^2 + 2\sum_i w_i\,\mathrm{d}x^i\mathrm{d}t +\sum_{ij} \left[\left(1-2{\Psi \over c^2}\right)\delta_{ij} + 2s_{ij}\right]\mathrm{d}x^i\mathrm{d}x^j\,. \end{equation} It turns out that we can use the gauge invariance from the previous paragraph to fix \begin{align}\label{eq-gauge-poisson} \sum_i \partial_i w_i & = 0\,;\quad \sum_i \partial_i s_{ij} =0\,, \end{align} by fixing the gauge by requiring the four vector \(\xi^\mu\) to be the vector that leads to these constraints (this can be straightforwardly shown by demonstrating that these constraints lead to a well-defined \(\xi^\mu\) using the gauge condition). This gauge is known as the Poisson gauge (Bertschinger 1995). Plugging Equations (C.59) into the linearized field equations (C.58), using the gauge conditions from Equations (C.61), and dropping the explicit \(\mathcal{O}(h_{\mu\nu}^2)\) condition expression the weakness of the gravitational field, we find that the field equations become \begin{align}\label{eq-gr-fieldeqs-linear-components-1} {8\pi G \over c^4}\,T_{00} & = {2\over c^2}\nabla^2\Psi- \Lambda\,,\\ {8\pi G \over c^4}\,T_{0j} & = -{1\over 2} \nabla^2 w_j + {2\over c^2}\partial_0 \partial_j \Psi\,,\label{eq-gr-fieldeqs-linear-components-2}\\ {8\pi G \over c^4}\,T_{ij} & = \left(\delta_{ij}\nabla^2 - \partial_i\partial_j\right){\left(\Phi-\Psi\right)\over c^2} -\partial_0\partial_i w_j - \partial_0 \partial_j w_i +{2\over c^2}\delta_{ij}\partial^2_0 \Psi - \eta^{\delta \lambda}\partial_\delta\partial_\lambda s_{ij} + \Lambda\,\delta_{ij}\,.\label{eq-gr-fieldeqs-linear-components-3} \end{align} These are the general linearized field equations and they, for example, describe the behavior of gravitational waves. But we will use them here only to derive the Newtonian limit.
In the Newtonian limit, the source of gravity is non-relativistic matter (e.g., planets, gas, stars, galaxies, clusters of galaxies). In the limit of velocities that are small compared to the speed of light, the pressure and stress in a fluid is always much less than the energy density and the relevant stress-energy tensor is therefore that of pressureless dust from Equation (C.47). Because we are working in the frame where the bulk velocity of the fluid is zero, the only non-zero component of the stress-energy tensor is \(T^{00} = \rho c^2\) and \(T_{\mu\nu}\) is the same matrix (up to corrections of \(\mathcal{O}[h_{\mu\nu}]\)). Furthermore, for non-relativistic matter, derivatives \(\partial_0\) of the gravitational field with respect to \(ct\) are a fraction of order \(v/c \ll 1\) of the spatial derivatives, and we can therefore set all time derivatives to zero in the linearized field equations. The \(0j\) Equation (C.63) then becomes \begin{equation} \nabla^2 w_j = 0\,, \end{equation} which combined with the Poisson gauge condition from Equation (C.61) implies that \begin{equation} w_i = 0\,, \end{equation} for a well-defined solution at infinity. Similarly, the trace of the \(ij\) Equation (C.64) gives \begin{equation}\label{eq-gr-lensing-poisson-lambda} 2\nabla^2\left(\Phi-\Psi\right) = -3\Lambda c^2\,, \end{equation} and the \(00\) Equation (C.62) becomes \begin{equation} 2\nabla^2\Psi- \Lambda c^2 = 8\pi G\rho\,, \end{equation} Adding them together, these last two equations imply \begin{equation}\label{eq-gr-poisson-notquite} \nabla^2\Phi = 4\pi G\rho-\Lambda c^2\,, \end{equation} which looks a lot like the Poisson equation (2.2), except for the extra term involving \(\Lambda\)! We will soon see that this is indeed the equivalent of the Poisson equation, that is, that the potential \(\Phi\) that satisfies this equation is also the potential whose gradient sets the acceleration of a non-relativistic particle.
As discussed above, the measured value of the cosmological constant \(\Lambda \approx 1/\mathrm{Gpc}^2\) is such that it is only relevant on cosmological scales. Expressed as a fraction of the critical density, the cosmological constant is \(\Omega_\Lambda \approx 0.7\) today and smaller in the past. As we discuss in Chapter 17.3.1, virialized structures such as galaxy halos and clusters of galaxies form when the mean density in a region of the Universe exceeds about 100 times the critical density at \(z \approx 0\). Thus, at a galaxy or cluster’s virial radius, the equivalent density \(\Lambda c^2 \lesssim 1\,\%\) of the dark-matter halo’s mean density. If the halo forms earlier, the equivalent density is even smaller, because halos form at higher overdensity with respect to the critical density (\(\approx 200\)) and the critical density itself is higher in the past. Within galaxies, the density of matter is orders of magnitude larger near their centers than the mean density within their virial radius (e.g., the density profile of the Milky Way in Chapter 1.2.3). Therefore, on the scale of virialized halos and everything inside of them, \(\Lambda c^2/(4\pi G) \ll \rho\) and we can actually ignore the effect of the cosmological constant and solve for the remaining unknowns in the metric. For \(\Lambda =0\), we immediately find from Equation (C.67) that \begin{equation} \nabla^2\left(\Phi-\Psi\right) = 0 \end{equation} or \begin{equation}\label{eq-gr-lensing-equals-newtonian} \Phi = \Psi\,, \end{equation} for a well-defined solution at infinity (that is, they both approach zero at infinity). We then also have from Equation (C.63) that \(\nabla^2 s_{ij} = 0\), which together with the Poisson gauge condition implies \(s_{ij}=0\) for a well-defined solution at infinity. Finally, Equation (C.69) in the limit \(\Lambda =0\) becomes the actual equivalent of the Poisson equation \begin{equation}\label{eq-gr-poisson} \nabla^2\Phi = 4\pi G\rho\,. \end{equation} The metric for weak gravitational fields produced by non-relativistic matter in its rest frame is therefore \begin{equation}\label{eq-gr-metric-weak-solution} \mathrm{d} s^2 = -\left(1+2{\Phi \over c^2}\right)\,c^2\mathrm{d}t^2 +\left(1-2{\Phi \over c^2}\right)\,\left(\mathrm{d}x^2+\mathrm{d}y^2+\mathrm{d}z^2\right)\,. \end{equation} Note that below, to be able to clearly distinguish between the dynamics of non-relativistic and relativistic particles, we will use the metric \begin{equation}\label{eq-gr-metric-weak-solution-lensing-explicit} \mathrm{d} s^2 = -\left(1+2{\Phi \over c^2}\right)\,c^2\mathrm{d}t^2 +\left(1-2{\Psi \over c^2}\right)\,\left(\mathrm{d}x^2+\mathrm{d}y^2+\mathrm{d}z^2\right)\,, \end{equation} that is, we do not use the constraint from Equation (C.71). The potential \(\Phi\) is known as the Newtonian potential, while \(\Psi\) is known as the lensing potential or curvature potential—we’ll use curvature potential to avoid confusion with the two-dimensional lensing potential from Chapter 15. In GR, we have that \(\Phi = \Psi\) as we have just shown, but this may not hold in other theories of gravity and testing this equality is a strong test of GR, especially on cosmological scales (e.g., Reyes et al. 2010; Bertschinger 2011).
For completeness, we note that a similar derivation of the weak-field limit for a perturbed FLRW metric (Equation C.121 below) leads to the metric \begin{equation}\label{eq-gr-metric-weak-FLRW-solution} \mathrm{d} s^2 = -\left(1+2{\Phi \over c^2}\right)\,c^2\mathrm{d}t^2 +a^2(t)\,\left(1-2{\Phi \over c^2}\right)\,\left\{\mathrm{d}\chi^2 + f^2_K(\chi)\,\left[\mathrm{d}\theta^2 +\sin^2 \theta \mathrm{d}\phi^2\right]\right\}\,, \end{equation} where the gravitational potential \(\Phi\) is sourced by the density perturbation with respect to the mean cosmological matter density \(\bar{\rho}_m\): \(\nabla^2 \Phi = 4\pi G\,a^2\,\left(\rho-\bar{\rho}_m\right)\) (Bertschinger 1995). As discussed at the start of this section, this form is only relevant when considering perturbations or their effect on scales that are similar to the spatial curvature scale or to the Hubble time, for example, in cosmic shear.
C.2.2. Geodesics of non-relativistic matter in weak gravitational fields¶
We’ve seen how GR in the limit of weak gravitational fields and non-relativistic matter sources gives rise to Equation (C.72) that has the same form as the Poisson equation. To establish the equivalence of GR to Newtonian gravity and classical mechanics, we further need to demonstrate that the GR equation of motion reduces to Newton’s second law in the limit of \(v \ll c\), with the same gravitational potential appearing through its gradient. The equation of motion in GR is the geodesic Equation (C.34). Because we are interested in non-relativistic matter, which follows timelike paths, we can use the proper time \(\tau\) as the affine parameter and write this as \begin{equation}\label{eq-gr-geodesic-nonrelativistic} {\mathrm{d}^2 x^\mu \over \mathrm{d} \tau^2} +\Gamma^{\mu}_{\nu\epsilon}{\mathrm{d} x^\nu \over \mathrm{d} \tau} {\mathrm{d} x^\epsilon \over \mathrm{d} \tau} = 0\,. \end{equation} That the matter is non-relativistic means that \(|\mathrm{d} x^i/\mathrm{d} \tau|\ll |\mathrm{d} ct/\mathrm{d} \tau|\). Keeping in mind that the Christoffel connection itself is a first-order quantity in the weak gravitational field, to first order the geodesic equation becomes \begin{equation}\label{eq-gr-geodesic-nonrelativistic-2} {\mathrm{d}^2 x^\mu \over \mathrm{d} \tau^2} +c^2\Gamma^{\mu}_{00}\left({\mathrm{d} t \over \mathrm{d} \tau}\right)^2 = 0\,. \end{equation}
To compute the Christoffel connection, we use Equation (C.54) for the metric in Equation (C.73), again dropping time derivatives of the metric, because they are much smaller than spatial derivatives. Then we find \(c^2\Gamma^{\mu}_{00} = \eta^{\mu \epsilon}\partial_\epsilon \Phi\). The \(i\)-th component of Equation (C.77) is then \begin{equation}\label{eq-gr-newtoniangeodesic-int} {\mathrm{d}^2 x^i \over \mathrm{d} \tau^2}+ \partial_i \Phi\left({\mathrm{d} t \over \mathrm{d} \tau}\right)^2 = {\mathrm{d}^2 x^i \over \mathrm{d} t^2}\left({\mathrm{d} t \over \mathrm{d} \tau}\right)^2 + {\mathrm{d} x^i \over \mathrm{d} t}{\mathrm{d}^2 t \over \mathrm{d} \tau^2} + \partial_i \Phi\left({\mathrm{d} t \over \mathrm{d} \tau}\right)^2= 0 \,, \end{equation} while the 0-th component gives \begin{equation} {\mathrm{d}^2 t \over \mathrm{d} \tau^2} =\left({\mathrm{d} t \over \mathrm{d} \tau}\right)^2\,{\partial \Phi/c^2 \over \partial t}\,. \end{equation} Plugging this into Equation (C.78) and after dividing by \((\mathrm{d}t/\mathrm{d}\tau)^2\), we find \begin{equation} {\mathrm{d}^2 x^i \over \mathrm{d} t^2} = - {1 \over c}{\mathrm{d} x^i \over \mathrm{d} t}{\partial \Phi\over \partial [ct]} - \partial_i \Phi \,. \end{equation} For non-relativistic matter, the first term on the right-hand side is much smaller than the second term and in vector form we therefore get \begin{equation}\label{eq-gr-newtons2nd} {\mathrm{d}^2 \vec{x} \over \mathrm{d} t^2} = -\nabla \Phi \,, \end{equation} which is exactly Newton’s second law if we identify \(-\nabla \Phi\) with the gravitational force. Because \(\Phi\) obeys Equation (C.72), we therefore see that \(\Phi\) is identical to the Newtonian gravitational potential: \(\Phi\) obeys the Poisson equation (C.72) and its gradient acts like a force in Newton’s second law in Equation (C.81). We further see that Newton’s laws hold in any frame that moves slowly with respect to the rest frame of the matter: From the decomposition of the metric in Equation (C.59), \(\Phi\) acts like a scalar under spatial rotations and Lorentz transformations with \(v \ll c\) do not change \(\Phi\) or the manipulations of the geodesic equations either. Thus, we see that Newtonian gravity and classical mechanics is recovered in the limit of low velocities and weak gravitational fields. The gravitational potential in galaxies is \(|\Phi/c^2| \lesssim (300\,\mathrm{km\,s}^{-1} / 300,000\,\mathrm{km\,s}^{-1})^2 = 10^{-6}\) and even in large clusters, we have that \(|\Phi/c^2| \lesssim (3,000\,\mathrm{km\,s}^{-1} / 300,000\,\mathrm{km\,s}^{-1})^2 \approx 10^{-4}\), so the Newtonian limit applies to high accuracy within galaxy clusters and galaxies.
.
.
.