General relativity is best understood using the abstract but beautiful language of differential geometry. For a first approach of these topics, we refer for example to Baez & Muniain (1994), Coqueraux (2016) (for french speakers) and Nakahara (2003). Here is a condensed review of the key concepts that you will encounter over and over again when studying GR:

• A smooth manifold $M$ of dimension $n$ (here $n=4$): a space that locally “looks like” $\mathbb{R}^n$, in the sense that around every point $p\in M$ one can find an open neighborhood mapped bijectively, through a chart (a local coordinate system $x^\mu$, $\mu=0,1,2,3$), onto an open subset of $\mathbb{R}^4$, in such a way that the transition between any two overlapping charts is smooth ($C^\infty$). This is the minimal structure needed to talk about differentiable functions, curves, and tangent vectors on space-time without assuming it sits inside some larger, “flat” ambient space.

• A map $f:M\to \mathbb{R}$ is called a function on $M$. It associates a real number $f(p)$ to each point $p$ of the manifold. The space of all smooth functions on $M$ is denoted $C^\infty(M)$.

• A map $\gamma: I\to M$, where $I\subseteq\mathbb{R}$ is an interval, is called a curve on $M$. It associates to each parameter value $\lambda\in I$ a point $\gamma(\lambda)$ on $M$; in a chart it is described by $n$ functions $x^\mu(\lambda)$.

• At each point $p\in M$, the set of all tangent vectors to all possible curves passing through $p$ form a real vector space of dimension $n$, the tangent space $T_pM$. A tangent vector $\gamma’$ to a curve $\gamma$, parametrized by the parameter $\tau$, at a point $p$ can be defined abstractly using the derivation operation as the map acting on a function $f$ as: \(\gamma'(p)(f) := \frac{\text{d}}{\text{d}\tau}(f\circ\gamma)(p)\). Choosing a specific vector at each point of $M$ defines a vector field, understood as a section of the tangent bundle, $TM=\bigsqcup_{p\in M}T_pM$. $TM$ is itself a manifold of dimension $2n$. We write $\mathfrak{X}(M)$ the space of all possible vector fields. A vector field $v\in\mathfrak{X}(M)$ acts on a function $f\in C^\infty(M)$ to produce another function $v(f)\in C^\infty(M)$: in a chart $x^\mu$, the partial derivatives $\partial_\mu$ form a basis of $T_pM$ at every point, so $v=v^\mu\partial_\mu$ and $v(f)=v^\mu\partial_\mu f$. The tangent vector to a curve $\gamma$ of components $x^\mu(\tau)$ is \(\gamma'(\tau) = \frac{\text{d}x^\mu}{\text{d}\tau}\,\partial_\mu.\)

• The dual vector space of $T_pM$ is the cotangent space $T_p^{*}M$; its elements, called covectors (or one-forms), are linear maps $\omega:T_pM\to\mathbb{R}$. Collecting $T_p^{*}M$ at every point gives the cotangent bundle $T^{*}M=\bigsqcup_{p\in M}T_p^{*}M$, and a smooth choice of covector at each point is a one-form field $\omega$. In a chart $x^\mu$, the differentials $\text{d} x^\mu$, defined by $dx^\mu(\partial_\nu)=\delta^\mu_\nu$, form the basis dual to $\partial_\mu$, so that $\omega=\omega_\mu\, dx^\mu$ and $\omega(v)=\omega_\mu v^\mu$. The differential of a function $f\in C^\infty(M)$, $df=\partial_\mu f\,dx^\mu$, is the simplest example of a one-form.

• A tensor of type $(r,s)$ at $p$ is a multilinear map taking $r$ covectors and $s$ vectors to a real number, i.e. it is a map \(\underbrace{T^*_pM\otimes\cdots\otimes T^*_pM}_{r}\otimes\underbrace{T_pM\otimes\cdots\otimes T_pM}_{s}\to \mathbb{R}\). We write it as an element of \(\underbrace{T_pM\otimes\cdots\otimes T_pM}_{r}\otimes\underbrace{T_p^{*}M\otimes\cdots\otimes T_p^{*}M}_{s}\). A smooth assignment of such a tensor at every point of $M$ is a tensor field. In a chart it is written \(T=T^{\mu_1\cdots\mu_r}{}_{\nu_1\cdots\nu_s}\,\partial_{\mu_1}\otimes\cdots\otimes\partial_{\mu_r}\otimes dx^{\nu_1}\otimes\cdots\otimes dx^{\nu_s}\). Vectors are $(1,0)$-tensors, one-forms are $(0,1)$-tensors, and the metric $g$ (introduced below) is an example of a $(0,2)$-tensor.

• A metric tensor $g$: a symmetric, non-degenerate $(0,2)$-tensor field, i.e. at every point $p$ a bilinear symmetric form $g_p$ acting on pairs of vectors of the tangent space $T_pM$ (the vector space of all vectors “tangent to $M$” at $p$, e.g. velocities of curves through $p$). In a coordinate basis $e_\mu$ of the tangent space, it is represented by a symmetric matrix $g_{\mu\nu}(x)$ such that $g= g_{\mu\nu} e^\mu \otimes e^\nu$, with $e^\mu$ being the corresponding basis of $TM^{\star}$ such that $e^\nu(e_\mu)=\delta^\nu_\mu$. In a natural chart $e_\mu = \partial_\mu$ and $e^\mu = \text{d}x^\mu$. The inverse of the matrix $g_{\mu\nu}$ is $g^{\mu\nu}$ satisfying $g^{\mu\rho}g_{\rho\nu}=\delta^\mu_\nu$ (used to raise and lower indices). For two tangent vectors $u^\mu,v^\mu$, $g(u,v)=g_{\mu\nu}u^\mu v^\nu$ (Einstein summation convention: repeated up/down indices are summed) defines their scalar product, from which one recovers the “length” $g(v,v)$ of $v$ and the angle $\cos(\widehat{u,v})=g(u,v)/\sqrt{g(u,u)g(v,v)}$ between $u$ and $v$.

• A Lorentzian metric: a metric tensor whose signature (the numbers of positive and negative eigenvalues of $g_{\mu\nu}$ once diagonalized at a point) is $(-,+,+,+)$, as opposed to a Riemannian metric, of signature $(+,+,+,+)$, which only encodes distances. It is always possible to find locally (at each point of space-time), a frame (i.e. a basis for $TM$) in which $g$ is diagonal and equal to the Minkowski metric $\eta$. These frames are called inertial or free-falling frames. The Lorentzian signature is precisely what allows one to separate tangent vectors into timelike ($g(v,v)<0$), spacelike ($g(v,v)>0$) and null/lightlike ($g(v,v)=0$) vectors at every point, reproducing locally the light-cone structure and causal order of Minkowski space-time.

• A connection $\nabla$ provides a canonical way to differentiate a vector field along another vector field. The naive difference quotient $\big(v(\gamma(\lambda))-v(p)\big)/\lambda$ is not well defined: $v(\gamma(\lambda))\in T_{\gamma(\lambda)}M$ and $v(p)\in T_pM$ live in different vector spaces and cannot be subtracted directly, since a priori there is no way to compare vectors at different points of $M$. This is exactly what a connection supplies, through parallel transport: given a curve $\gamma$ with $\gamma(0)=p$, $\nabla$ defines a linear isomorphism $\Pi_{\gamma(\lambda)\to p}:T_{\gamma(\lambda)}M\to T_pM$ bringing a vector at $\gamma(\lambda)$ back into $T_pM$ before comparing it to $v(p)$. This gives a well-defined derivative of $v$ along $u=\gamma’(0)$ at $p$: \(\nabla_u v\big|_p = \lim_{\lambda\to 0}\frac{\Pi_{\gamma(\lambda)\to p}\,v(\gamma(\lambda)) - v(p)}{\lambda}.\) In a chart this reduces to the familiar $\nabla_uv=u^\mu\left(\partial_\mu v^\nu+\Gamma^\nu_{\mu\rho}v^\rho\right)\partial_\nu$, consistent with $\nabla_\mu v^\rho=\partial_\mu v^\rho+\Gamma^\rho_{\mu\nu}v^\nu$, where we introduced the Christoffel symbols, being the covariant derivative of the basis vector in a given chart \(\Gamma^{\lambda}_{\mu\nu}\partial_\lambda = \nabla_\mu \partial_\nu\).

  The connection is a very general and important concept; we present here only the special case relevant to general relativity, the so-called affine connection. For a fully general presentation, see our lecture on connections. Axiomatically, it is an $\mathbb{R}$-bilinear map $\nabla:\mathfrak{X}(M)\times\mathfrak{X}(M)\to\mathfrak{X}(M)$, $(u,v)\mapsto\nabla_uv$, such that for all $u,v\in\mathfrak{X}(M)$ and $f\in C^\infty(M)$:

  (i) $\nabla_{fu}v = f\,\nabla_uv$ ($C^\infty(M)$-linear, i.e. tensorial, in the direction $u$);

  (ii) $\nabla_u(fv) = u(f)\,v + f\,\nabla_uv$ (Leibniz rule in $v$, making it a derivative).

  Axiom (i) means $(\nabla_uv)(p)$ depends on $u$ only through $u_p\in T_pM$ — consistent with $u$ being merely a “direction” at $p$ — while (ii) shows $\nabla_uv$ genuinely differentiates $v$, so it depends on $v$ in a whole neighborhood of $p$, not just its value there.

• Metric compatibility (S3a): the connection is metric compatible when, for all vector fields $u,v,w\in\mathfrak{X}(M)$, \(u\big(g(v,w)\big) = g(\nabla_u v,w)+g(v,\nabla_u w),\). In components, this condition is also written \(\nabla_\rho g_{\mu\nu}=0\), where the action of $\nabla$ on a tensor is defined in another bullet point below. It guarantees that parallel transport preserves scalar products: if $u^\mu,v^\mu$ are parallel transported along a curve, $g(u,v)$ is constant along that curve. Without it, physical rulers and clocks carried along space-time would not stay consistent with the metric’s own notion of length and duration.

• Torsion free (S3b): the torsion tensor of the connection is defined, without reference to any chart, as \(T(u,v) \equiv \nabla_uv-\nabla_vu-[u,v],\) where $[u,v]$ is the Lie bracket of two vector fields — itself a vector field, defined by its action on functions, \([u,v](f)=u(v(f))-v(u(f))\). The connection is torsion free when $T\equiv0$, i.e. $\nabla_uv-\nabla_vu=[u,v]$ for all $u,v\in\mathfrak{X}(M)$. In a coordinate basis, $[\partial_\mu,\partial_\nu]=0$ (partial derivatives commute), so evaluating $T$ on $u=\partial_\mu$, $v=\partial_\nu$ gives \(T^\rho{}_{\mu\nu}\,\partial_\rho=\nabla_{\partial_\mu}\partial_\nu-\nabla_{\partial_\nu}\partial_\mu=\big(\Gamma^{\rho}{}_{\mu\nu}-\Gamma^{\rho}{}_{\nu\mu}\big)\partial_\rho\): torsion-freeness is thus equivalent to the symmetry of the connection coefficients in their lower indices, \(\Gamma^{\rho}{}_{\mu\nu}=\Gamma^{\rho}{}_{\nu\mu},\) i.e. to the vanishing of \(T^{\rho}{}_{\mu\nu}\equiv \Gamma^{\rho}{}_{\mu\nu}-\Gamma^{\rho}{}_{\nu\mu}.\) Geometrically, torsion measures the failure of an infinitesimal parallelogram, built by transporting one small displacement along another, to close.

Metric compatibility and torsion-freeness together (S3a+S3b) are precisely the two conditions of the fundamental theorem of Riemannian (here Lorentzian) geometry: they fix $\Gamma$ uniquely in terms of $g$ and its first coordinate derivatives, through

the Levi-Civita connection. This theorem is proven in this class. We thus see that, once (S1)-(S3) are assumed, no independent geometric degree of freedom is left besides $g$ itself: the connection (and hence curvature) is entirely determined by the metric.

• action of $\nabla$ on tensors: The action of the connection on a type $(r,s)$ tensor is give by asking that, for every $u\in\mathfrak{X}(M)$: (a) $\nabla_u f = u(f)$ on scalars $f\in C^\infty(M)$; (b) the Leibniz rule on tensor products, $\nabla_u(S\otimes T)=(\nabla_uS)\otimes T + S\otimes(\nabla_uT)$; and (c) that $\nabla_u$ commutes with every contraction (the pairing of an upper index against a lower one). Together with the original action on vector fields, these three requirements fix $\nabla_uT$ uniquely for any tensor field $T$. As a first consequence, take a one-form $\omega$ and a vector field $v$: $\omega(v)$ is exactly the contraction of $\omega\otimes v$, hence a scalar, so (a)-(c) give \(u\big(\omega(v)\big) = (\nabla_u\omega)(v) + \omega(\nabla_uv) \quad\Longrightarrow\quad (\nabla_u\omega)(v) = u(\omega(v)) - \omega(\nabla_uv).\) Writing this out in a chart with $u=\partial_\mu$, $v=\partial_\nu$, and comparing coefficients of $v^\nu$ using $\nabla_\mu v^\nu=\partial_\mu v^\nu+\Gamma^\nu_{\mu\lambda}v^\lambda$, gives \(\nabla_\mu\omega_\nu = \partial_\mu\omega_\nu - \Gamma^\lambda_{\mu\nu}\omega_\lambda:\) a minus sign for the lower (covariant) index, as opposed to the plus sign for the upper (contravariant) index of a vector.

  More generally, for a tensor field of type $(r,s)$, one $+\Gamma$ term appears for every upper index and one $-\Gamma$ term for every lower index:

  \[\nabla_\rho T^{\mu_1\cdots\mu_r}{}_{\nu_1\cdots\nu_s} = \partial_\rho T^{\mu_1\cdots\mu_r}{}_{\nu_1\cdots\nu_s} + \sum_{i=1}^r \Gamma^{\mu_i}_{\rho\lambda}\,T^{\mu_1\cdots\lambda\cdots\mu_r}{}_{\nu_1\cdots\nu_s} - \sum_{j=1}^s \Gamma^{\lambda}_{\rho\nu_j}\,T^{\mu_1\cdots\mu_r}{}_{\nu_1\cdots\lambda\cdots\nu_s}\]

  (the $i$-th upper or $j$-th lower index replaced by the dummy $\lambda$ in each summand).

The way to obtain the equation of motions from the action is a direct generalisation in four dimension of the standard approach used to derive the Euler-Lagrange equation.

We consider a region of space-time $\Omega$, and vary the action within that region. We consider variations of the action induced by small changes of the metric in that volume, which is the only dynamical variable of our theory of gravity. We consider small variations of the inverse metric $\delta g^{\mu\nu}$ instead of the metric ($\delta g_{\mu\nu}$) by convention, it allows to recover more immediately the D1 expression. Variations should cancel on the boundary $\partial \Omega$.

The action is stationary when $\delta S_{EH}=0$. That is, we ask for:

where $\delta$ indicates a variation when $g^{\mu\nu}$ is varied along the space-time trajectory.

From this, using the product rule one obtains:

Since this should hold for every $\delta g^{\mu\nu}$, and $\sqrt{-\vert g\vert}\neq0$, we obtain

Let’s now look at each term.

A useful formula first

Let us first find useful relations in order to switch freely from variations of the metric $\delta g_{\mu\nu}$ to variations of its inverse $\delta g^{\mu\nu}$, which we will need multiple times. The metric and its inverse obey the relation $g^{\mu\rho}g_{\rho\nu}=\delta^\mu_\nu$. Taking the variation of both side using the product rule, we get

Relabelling summation indices, we get:

hence, going from the metric to its inverse raise/lowers the indices and adds a minus sign.

Variation of the volume element

Here, we will assume the formula for the differential of the metric determinant, also known as Jacobi formula, giving:

We will not attempt at proving it, and more details can be found in most standard general relativity textbook. From this, we obtain

In term of the inverse metric, this gives:

and hence

Another useful formula

From the expression of $\Gamma^{\rho}_{\,\,\mu\nu}$ in terms of the metric and its derivative (found in the “complement” session above), we can easily derive the expression for its upper and lower index contraction:

Recalling the formula $\delta f/f=\delta(\ln(f))$, for a general function $f$, we found in the previous section on the volume element that:

where we used the first useful formula to lower the indices in the second step. Focusing on variations of the metric induced by a change in position $\delta(…) = \partial_\mu(…) \delta x^\mu$, we obtain:

Now, from the expression of the covariant derivative of a vector we have

From which, using the product rule, we infer another very useful formula, which we will call the divergence formula:

For any vector $v$.

The Riemann curvature term

Since $R=g^{\mu\nu}R_{\mu\nu}$, the product rule gives:

We now assume what is known as the Palatini identity:

which can be proven by looking at the full expression of the Riemann tensor in term of $\Gamma$ and its derivatives, and looking at $\delta R^{\rho}_{\,\,\sigma \mu \nu}$. See for example here.

Contracting with $g^{\mu\nu}$, which commutes freely with $\nabla$ by metric compatibility (S3a),

where we introduced \(v^\rho \equiv g^{\mu\nu}\delta\Gamma^\rho_{\mu\nu}-g^{\mu\rho}\delta\Gamma^\lambda_{\lambda\mu}\).

Hence, using the divergence formula, and Stoke’s theorem we find:

This term is thus a boundary term, which vanishes once $\delta g_{\mu\nu}$ is fixed (or compactly supported) on $\partial \Omega$. The cancellation of boundary terms is a subtle topic that we just gloss over here. For more, see e.g. York (2009).

Deleting this boundary term, we simply obtain:

Stress energy tensor

We assert that the stress-energy tensor of matter, introduced in S4, is defined such that:

A discussion of this choice and of the physical meaning of this quantity is given in one of the following complement.

Putting in all together:

By replacing each term one by one with the first expression we derived for $\delta S_{EH}$, we obtain immediately Einstein Equation (D1):

To prove D2, we must look at variations of $S_{EH}$ with respect to $\psi$. This is much easier, as the whole gravitational term does not depend on $\psi$: only $\mathcal{L}_m$ does.

First, make the replacement \(\mathcal{L}_m(\psi,\partial\psi,\eta)\to \mathcal{L}_m(\psi,\nabla\psi,g)\) required by the minimal coupling rule. Then, starting from \(S_m=\int\sqrt{-\vert g\vert}\,\mathcal{L}_m(\psi,\nabla\psi,g)\,\text{d}^4x\), vary \(\psi\to\psi+\delta\psi\) at fixed $g$ (so $\Gamma$ is fixed too – unlike the metric variation, there is no subtlety here about $\delta\Gamma$). Since $\nabla_\mu\psi$ is linear in $\psi$ for fixed $\Gamma$ (whatever the type of field, the derivative will be a sum of $\Gamma\psi$ contraction terms),

Hence

where, if $\psi$ carries indices, a sum over all its components is implied. Integrate the second term by parts using the Leibniz rule for $\nabla$:

In the first piece, all the indices of $\psi$ are contracted, leaving a genuine vector field \(v^\mu \equiv \frac{\partial\mathcal{L}_m}{\partial(\nabla_\mu\psi)}\,\delta\psi\); by the divergence formula, \(\sqrt{-\vert g\vert}\,\nabla_\mu v^\mu = \partial_\mu(\sqrt{-\vert g\vert}\,v^\mu)\) is a total derivative, which drops for $\delta\psi$ vanishing on $\partial\Omega$. So

for arbitrary $\delta\psi$, giving the covariant Euler-Lagrange equation for matter:

Two remarks on how to read it: (i) \(\partial\mathcal{L}_m/\partial\psi\) is taken at fixed $\nabla\psi$; (ii) if $\psi$ carries tensor indices, so does \(\partial\mathcal{L}_m/\partial(\nabla_\mu\psi)\), and $\nabla_\mu$ acts on all of them.

One can instead treat the Lagrangian density \(\sqrt{-\vert g\vert}\,\mathcal{L}_m\) as a function of the components of $\psi$ and their partial derivatives $\partial_\mu\psi$ (re-expressing $\nabla\psi=\partial\psi+\Gamma\,\psi$). The variational principle then yields the standard Euler-Lagrange equation, valid for any field:

This form is arguably the most fundamental, as it treats each component of $\psi$ as an independent dynamical variable and makes no reference to its tensorial nature. It is strictly equivalent to the boxed equation: the $\Gamma$ terms hidden in \(\partial(\sqrt{-\vert g \vert}\mathcal{L}_m)/\partial\psi\) (through the $\psi$-dependence of $\nabla\psi$) exactly reproduce those generated by $\nabla_\mu$ acting on the indices of \(\partial\mathcal{L}_m/\partial(\nabla_\mu\psi)\). Beware of the tempting hybrid \(\frac{1}{\sqrt{-\vert g \vert}}\partial_\mu\big(\sqrt{-\vert g\vert}\,\partial\mathcal{L}_m/\partial(\nabla_\mu\psi)\big) = \partial\mathcal{L}_m/\partial\psi\): it is correct only for scalar $\psi$, for which $\nabla\psi=\partial\psi$ and the two forms coincide via the divergence formula.

Finally, in a local inertial frame, $g\to\eta$ and $\nabla\to\partial$, so the boxed equation reduces to the standard special-relativistic Euler-Lagrange equation: matter fields locally obey their special-relativistic dynamics, which is exactly the content of D2.

In the derivation of D1 from the Lagrangian, we connected the stress-energy tensor (introduced in S4) to the matter Lagrangian $\mathcal{L}_m$ through:

Expanding the variation with the product rule and using the formula for the variation of the metric determinant established before,

we find

so that, whenever \(\mathcal{L}_m\) depends on the metric but not on its derivatives (which is the case for all standard matter fields, since \(\partial_\mu g_{\alpha\beta}\) only enters through covariant derivatives of the metric itself, which vanish), the functional derivative reduces to an ordinary partial derivative and:

This is often the quickest way to compute $T_{\mu\nu}$ in practice, without having to vary the full action $S_m$.

Physically, $T_{\mu\nu}$ collects the densities and fluxes of energy and momentum: $T^{00}$ is the energy density, $T^{0i}$ the momentum density (equivalently the energy flux), and the spatial block $T^{ij}$ contains the stresses — pressure on the diagonal, shear off the diagonal.

In flat-spacetime classical field theory, the stress-energy tensor arises instead as the Noether current associated with invariance under spacetime translations (the corresponding conserved charges are the total energy and momentum, $P^\nu = \int T^{0\nu}\,\mathrm{d}^3x$). For a field theory with Lagrangian \(\mathcal{L}_m(\psi, \partial_\mu\psi)\), it reads:

(The overall sign is a convention; this choice matches $T^0{}_0 = -\rho$ below. Many quantum field theory textbooks use the opposite sign.) This definition is the one familiar to particle physicists. The two definitions — the “metric” one and the canonical one — agree for simple models as scalar fields (discussed later in this class), but not always in general: the canonical tensor need not be symmetric (e.g. for the electromagnetic field). It can, however, always be brought to agree with the metric definition by adding suitable total-derivative terms that change neither the conservation law nor the total charges (the Belinfante–Rosenfeld procedure). In this sense the metric definition is the more fundamental one.

For a single point particle of rest mass $m$, the action is simply (minus) the mass times the proper time elapsed along its worldline: $S_m = -m\int \mathrm{d}\tau$. Dust is a cloud of many such particles, described by a rest-mass density $\rho$ and a common 4-velocity field $u^\mu$, normalized so that $g(u,u) \equiv g_{\mu\nu}u^\mu u^\nu = -1$. The corresponding Lagrangian density is

One word of caution: a naive application of the boxed formula, holding $\rho$ and $u^\mu$ fixed while varying the metric, gives a wrong extra term. The density $\rho$ itself responds to a change of metric, because the number of particles in a volume is conserved; taking this constraint into account, one obtains

which in the local rest frame is simply $\mathrm{diag}(-\rho, 0, 0, 0)$: energy density and nothing else — dust carries no pressure.

For a perfect fluid with energy density $\rho$ and isotropic pressure $P$ (both measured in the local rest frame):

which in the local rest frame reads $\mathrm{diag}(-\rho, P, P, P)$. Dust is recovered as the special case $P = 0$.

Its conservation law $\nabla_\mu T^{\mu\nu}=0$ reduces, in the non-relativistic limit, to the continuity equation and the (inviscid) Euler equation of fluid dynamics — not Navier–Stokes, since a perfect fluid has no viscosity by construction; the viscous stress term needed for Navier–Stokes simply is not part of this $T^{\mu\nu}$.

The boxed formula pays off immediately for the simplest possible Lagrangian, a constant: \(\mathcal{L}_m = -\rho_\Lambda\). Then \(\partial\mathcal{L}_m/\partial g^{\mu\nu} = 0\) and

which is exactly a perfect fluid with $\rho = \rho_\Lambda$ and $P = -\rho_\Lambda$. A constant vacuum energy behaves as a fluid with negative pressure $P = -\rho$ — the equation of state of the cosmological constant.

Working with the expression of $G_{\mu\nu}$, it is possible to show that:

This is a contracted form of the so-called Bianchi identity, which is a purely geometric identity, valid for the Riemann tensor of any torsion-free, metric-compatible connection, and reads It holds identically, independently of the field equations: it is a property of geometry alone, valid for any metric $g$. It even has a geometric interpretation related to the concept of “holonomies”, which will not discuss further here (see e.g. Baez & Muniain (1994)). Now, taking the covariant divergence $\nabla^\mu$ of both sides of the Einstein equation (D1), the geometric side vanishes by the Bianchi identity, and since $\nabla^\mu(\Lambda g_{\mu\nu})=0$ by metric compatibility ($\nabla g = 0$) and constancy of $\Lambda$, we are left with \(\nabla^\mu T_{\mu\nu}=0.\) This equation can be understood as the local conservation of energy-momentum of matter, and is as we saw, deeply encoded in the relationship between matter and geometry (D1).

Consider a pressureless fluid of free point particles (“dust”), of stress-energy tensor $T^{\mu\nu}=\rho\, u^\mu u^\nu$, where $\rho$ is the rest-mass density measured in the local rest frame and $u^\mu$ the 4-velocity field, normalized as $u^\mu u_\mu = -c^2 = -1$ (using the signature convention introduced above). Plugging into the continuity equation:

Contracting this equation with $u_\nu$, and using that $u^\nu u_\nu=-1$, we obtain \(-\nabla_\mu(\rho u^\mu) + \rho u^\mu u_\nu \nabla_\mu u^\nu=0\). Now, the second terms disapears by noticing that \(\nabla_\mu(u_\nu u^\nu) =\nabla_\mu(-1)=0\) and that, on the other hand, by the product rule \(\nabla_\mu(u_\nu u^\nu) = u_\nu \nabla_\mu u^\nu + u^\nu \nabla_\mu(u_\nu) = u_\nu \nabla_\mu u^\nu + u^\nu \nabla_\mu(g_{\nu \alpha}u^\alpha) = u_\nu \nabla_\mu u^\nu + u^\nu g_{\nu \alpha}\nabla_\mu(u^\alpha) = 2 u_\nu \nabla_\mu u^\nu\) (using metricity condition $\nabla g=0$ and relabelling summation indices). We are thus left with:

i.e. rest mass is conserved along the flow, the curved-spacetime analogue of the continuity equation of fluid mechanics. Substituting this back into the first expression we had for $\nabla_\mu(\rho u^\mu u^\nu)$, and since $\rho\neq0$ for a physical fluid, we are left with

the geodesic equation. Notice that this equation contains no reference whatsoever to the mass, composition, or internal structure of the particle: any free test body, regardless of what it is made of, follows the same worldline once its initial position and 4-velocity are fixed. This mass-independence is exactly the mathematical translation of the UFF, and we will use it again below to show how (S1)-(S3)+(D2) encode the EEP.

The geodesic equation can also be recovered from a least-action principle when looking for the curve $\gamma(\tau)$ extremizing the total space-time length of the particle, that is:

where $\tau$ is, for now, an arbitrary parameter along the curve (the action is invariant under reparametrization $\tau\to\tau’(\tau)$, so no particular choice of $\tau$ is assumed yet). Writing $-\vert u \vert^2 \equiv -g_{\alpha\beta}\dot x^\alpha\dot x^\beta$, the Lagrangian is $L=-m\sqrt{-\vert u \vert^2}$, and since this is an ordinary mechanics-type action (one independent variable $\tau$, first derivatives only), the standard Euler-Lagrange equation applies directly:

Computing the two derivatives gives:

and:

such that the Euler-Lagrange equation is

Reparametrization invariance means we are free to now specialize to proper time, i.e. the parametrization for which $(\sqrt{-\vert u \vert^2})^2=1$ is constant along the curve (equivalently $u^\mu u_\mu=-1$, with $u^\mu\equiv\dot x^\mu$); this simplifies the equation without loss of generality, since any solution curve can always be reparametrized this way after the fact. With $\vert u \vert^2=-1$ constant, $\text{d}\mathcal{\sqrt{-\vert u \vert^2}}/\text{d}\tau=0$, and the equation becomes, after dividing through by the (now manifestly cancelling) mass $m$ – a first, direct illustration of the UFF/WEP already discussed above –

We shall not forget that $g$ depends also on $\tau$ as the particle evolves through space time and hence we can use the chain rule as:

Such that the total equation becomes:

Since \(\partial_\lambda g_{\beta\alpha}\,u^\lambda u^\alpha\) is contracted with the symmetric object $u^\lambda u^\alpha$, it may be symmetrized in $\lambda,\alpha$ at no cost, giving

The bracketed combination is exactly \(g_{\beta\lambda}\Gamma^\lambda{}_{\mu\nu}\), from the boxed Levi-Civita formula given above. So

That is, raising the free index:

which is precisely $u^\lambda\nabla_\lambda u^\beta=0$, since

This is exactly the geodesic equation, matching the derivation obtained above from the continuity equation.

The Newtonian regime is defined by three independent assumptions, which we will invoke by name:

1. Weak field: We assume that the metric can be written as a small correction to Minkowski: $g_{\mu\nu}=\eta_{\mu\nu}+\epsilon h_{\mu\nu}$ with \(\epsilon \in \mathbb{R}\) and \(\vert\epsilon\vert\ll1\). Every term in $\epsilon^2$ will be considered as null.
2. Static field: all time derivatives of the metric vanish, $\partial_0 h_{\mu\nu}=0$.
3. Slow motion: the test particle and the source move with $v\ll c$.

We use Cartesian coordinates $x^0=ct,x^1=x,x^2=y,x^3=z$. Let $\tau$ be the proper time of a test particle. In such a frame, the four velocity $u^\mu= \text{d}x^\mu/\text{d}\tau = (c\text{d}t/\text{d}\tau, \text{d}x^i/\text{d}\tau)$.

i) geodesic equation

The geodesic equation is

with the Christoffel symbols:

Using the small velocity approximation $u^0 = c\text{d}t/\text{d}\tau \gg u^i = \text{d} x^i/\text{d}\tau$, because of the factor of $c$.

Simplifying all the terms in $\text{d}x^i/\text{d}\tau$, one gets simply:

From the expression of the christoffel symbol, we have

where we simplified the time derivatives (because of the static field approximation) and then cancelled all the terms in $\epsilon^2$ (weak field assumption).

For $\lambda=0$, we get \(\Gamma^{0}{}_{00}=1(\eta^{00}\partial_0h_{00} - \eta^{0i}\partial_i h_00)/2=0\) (because of static field and \(\eta^{0i}=0\) is diagonal). The zeroth component of the geodesic equation thus becomes simply \(c^2 \text{d}^2 t/ \text{d}\tau^2 =0\) and hence \(\text{d}t/\text{d}\tau = {\rm cst}\).

For $\lambda=i$, using the chain rule \(\frac{\text{d}x}{\text{d}t} = \frac{\text{d}x}{\text{d}\tau}\frac{\text{d}\tau}{\text{d}t}\), we get:

Now, since we know that in Newtonian mechanics

where $\Phi$ is the gravitational potential, we introduce the suggestive notation:

So far we can simply consider this as a definition or a renaming. We will see in the second part that $\Phi$ can really be identified with the Newtonian potential through Einstein equations giving the Poisson equation for $\Phi$ in the Newtonian limit.

Using the time signature $\eta_{00}=-1$, we can then write the metric component in term of the Newtonian gravitational potential:

Hence, Newtonian gravitation is the limit of general relativity in which only the geometry of time matters (hence clocks ticks at different rates in a gravitational field) and the geometry of space can be neglected.

i) Einstein equation

Taking the Einstein equation (D1) and contracting with $g^{\mu\nu}$ on both sides one obtains:

where we used: $g^{\mu\nu}g_{\mu\nu}=4$ and $\mathcal{T}= g^{\mu\nu}T_{\mu\nu}$. Re-inserting back this expression of the Ricci scalar $R$ in the original Einstein equation, we get:

Now, let’s compute the left-hand side. Using the expression of the Riemann tensor in term of $\Gamma$, we have:

all the $\Gamma\Gamma$ term are second order (they will give $\epsilon^2$ terms only), and they can thus be neglected. We are then left with \(R_{\mu\nu}= \partial_\rho \Gamma^{\rho}{}_{\nu\mu} - \partial_\nu \Gamma^{\rho}{}_{\rho\mu}\). Focusing on the $\mu=\nu=0$ component \(R_{00} = \partial_\rho \Gamma^{\rho}{}_{00} - \partial_0 \Gamma^{\rho}{}_{00}\). Using the static approximation, \(R_{00} = \partial_\rho \Gamma^{\rho}{}_{00}\) and using our expression for \(\Gamma^{\rho}{}_{00}\) obtained above:

Now, since only the spatial derivative are non-vanishing (static approximation), and using the expression for $\Phi$ in term of $h_{00}$, we simply get:

where we introduced the notation \(\nabla^2=\partial_i\partial^i\).

Now, we are ready to put everything back in our Einstein Equation ! Considering a fluid of mass density $\rho$, the stress energy tensor has an only non negligable component: $T_{00}=\rho c^2$. The trace is $\mathcal{T}\simeq \eta^{00}T_{00}=-\rho c^2$. For a solar system type solution, we can easily neglect the cosmological constant $\Lambda \simeq 0$. Hence \(T_{00}-\frac{1}{2}\mathcal{T}g_{00} = \rho c^2 - \rho c^2/2 = \rho c^2/2\) and the zero-zero component of the Einstein equation is:

that is, the Poisson equation:

Confirming that the $\Phi$ field driving the motion in the limit of the geodesic equation really corresponds to the Newtonian potential.

Consider a single point mass $m_1$ sitting at the origin. Its mass density is a distribution, not a function:

where $\delta$ is the Dirac delta. The Poisson equation becomes

In order to solve this equation, we can use the divergence theorem that tells us that the integral of the divergence of a vector field in a volume is equal to the integral of that vector over its surface. Considering a sphere of volume $V$, and surface $S$ surounding the point mass:

inserting Poisson equation on the left and integrating

which we can integrate again to get:

Rather than solving this by brute force, we could also have called on a friendly mathematician, who tells us that

where $f$ is called the Green function of the operator $\nabla^2$: the response of the field to a unit point (Dirac) source. Equivalently, $\nabla^2 (1/r) = -4\pi\delta^3(\vec{x})$. Green functions are a general tool which can help us solving this equation with general $\rho$ or more complicated derivation operators (which we will encounter later): once we know the response to a point source, linearity gives the response to any source $\rho$ by superposition, $\Phi(\vec{x}) = 4\pi G \int f(\vec{x}-\vec{x}\,’)\,\rho(\vec{x}\,’)\,d^3x’$. We will use green functions again.

From Newtonian physics, if a second mass $m_2$ is placed at distance $r$ and couples to this field, the interaction potential energy of the pair is