Appearance
6.8 — The Einstein Field Equations
\boxed{R_{\mu\nu} - \tfrac{1}{2}Rg_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4}T_{\mu\nu}}
That is the whole of general relativity. John Wheeler's summary of what it says has never been improved on:
Spacetime tells matter how to move; matter tells spacetime how to curve.
Chapter 6.7 built everything on the left except one term. This chapter reads the equation piece by piece, explains why that particular combination and no other could have worked, derives Newton's law of gravity from it as a special case, and takes apart the term Einstein called his greatest blunder.
Reading the right-hand side
\frac{8\pi G}{c^4}T_{\mu\nu}
T_{\mu\nu} is the stress–energy tensor, and it is the complete inventory of everything that gravitates. It has two downstairs indices, so it is a 4\times4 table at every point, and symmetry reduces its independent entries to ten.
Each entry has a physical meaning:
| Component | Meaning |
|---|---|
| T_{00} | Energy density |
| T_{0i} | Momentum density / energy flow |
| T_{ii} | Pressure in direction i |
| T_{ij}, i\neq j | Shear stress |
In Newton's theory, mass alone gravitates. Here, energy, momentum, pressure and stress all do. That is not a refinement; it changes the physics.
Why must pressure gravitate? Because E = mc^2 from Chapter 6.4. Energy has mass, and pressure is energy per unit volume. A box of hot gas weighs more than the same box cold, and the extra weight comes from the thermal energy and the pressure it exerts.
Usually the effect is negligible. Air's pressure contributes P/c^2 = 10^5/9\times10^{16} = 10^{-12} kg/m³ against its density of 1.2 kg/m³ — a part in 10^{12}.
Inside a neutron star it is not negligible at all. Pressures there approach \rho c^2, so pressure contributes as much gravity as mass does. This has a vicious consequence: compressing the star raises the pressure, which increases the gravity, which compresses it further. In Newtonian physics pressure only ever resists collapse. In general relativity it eventually assists it, and that is why a neutron star above about 2.2 solar masses cannot exist at all — no equation of state, however stiff, can hold it up. Chapter 12.2 works this through.
The constant 8\pi G/c^4 is the coupling — how much curvature a given amount of energy produces. Its value:
\frac{8\pi G}{c^4} = \frac{8\pi(6.674\times10^{-11})}{(3\times10^{8})^4} = \frac{1.678\times10^{-9}}{8.1\times10^{33}} = 2.07\times10^{-43}
in units of inverse newtons. That is a very small number, and it is why gravity is weak. To produce one unit of curvature you need 5\times10^{42} newtons. The whole Earth bends spacetime enough to make things fall at 9.8 m/s² and no more.
The 8\pi and the c^4 are fixed by requiring that Newton's law comes out correctly in the weak-field limit, which is derived below.
Reading the left-hand side
G_{\mu\nu} = R_{\mu\nu} - \tfrac{1}{2}Rg_{\mu\nu}
This combination is called the Einstein tensor, and the question worth asking is why this combination.
The obvious first guess is R_{\mu\nu} = kT_{\mu\nu} — curvature proportional to matter, index for index. Einstein tried it in 1913 and it is wrong.
Why it fails. Energy and momentum are conserved, and in tensor language that statement is:
\nabla^\mu T_{\mu\nu} = 0
where \nabla^\mu is the covariant derivative — the derivative that accounts for the coordinate twisting via the Christoffel symbols of Chapter 6.7. This equation is not optional; it is conservation of energy and momentum, and it must hold everywhere.
So the left-hand side must satisfy the same identity. And \nabla^\mu R_{\mu\nu} \neq 0 in general.
What does satisfy it is a mathematical fact discovered by Luigi Bianchi in 1902, before anyone wanted it for physics. The contracted Bianchi identity says:
\nabla^\mu\left(R_{\mu\nu} - \tfrac{1}{2}Rg_{\mu\nu}\right) = 0
Identically. For any metric whatsoever. Not as a condition to be imposed — as a theorem of differential geometry.
So the combination R_{\mu\nu} - \frac{1}{2}Rg_{\mu\nu} is not a lucky guess. It is the unique object built from the metric and its first two derivatives that is automatically conserved. Given that the source must be conserved, the field equations could hardly have taken any other form. Einstein reached this in November 1915 after two years of wrong turns, and Hilbert derived the same equations independently within days by a different route.
The ten equations
Both sides are symmetric 4\times4 tensors, so there are ten independent equations. They are coupled, nonlinear partial differential equations for the ten components of the metric.
Nonlinear, because the curvature terms contain products of Christoffel symbols, which themselves contain the metric. Physically: gravitational field energy gravitates. A region of strong field curves spacetime even with no matter present, which is why two black holes can orbit and merge with nothing between them but geometry.
This nonlinearity is why general relativity is hard. Maxwell's equations are linear, so solutions add: the field of two charges is the sum of their individual fields. Einstein's do not. You cannot solve for two masses by adding two one-mass solutions, and the two-body problem in general relativity has no closed-form solution at all — merging black holes are simulated numerically on supercomputers, and getting those simulations to work took forty years.
Exact solutions are correspondingly rare. The known ones almost all assume a large symmetry: spherical (Schwarzschild, Chapter 6.9), axial (Kerr, for rotation), or homogeneous and isotropic (Friedmann, Chapter 12.5).
The Newtonian limit
A new theory must reproduce the old one where the old one worked. Derive Newton's gravity from the field equations.
Three assumptions, each corresponding to a regime where Newton was tested:
- Weak field. The metric is nearly flat: g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu} with |h| \ll 1, so all terms quadratic in h can be dropped.
- Slow motion. Velocities are much less than c, so time derivatives are negligible compared with space derivatives.
- Non-relativistic matter. Energy density dominates everything else: T_{00} = \rho c^2 and all other components are negligible, since pressure is small compared with \rho c^2.
Step 1: the geodesic equation becomes Newton's second law.
For a slow particle, dx^i/d\tau \approx 0 and dx^0/d\tau \approx c, so in the geodesic equation of Chapter 6.7 only the \alpha = \beta = 0 term survives:
\frac{d^2x^i}{d\tau^2} = -\Gamma^i_{00}\left(\frac{dx^0}{d\tau}\right)^2 = -c^2\Gamma^i_{00}
For a static weak field, computing \Gamma^i_{00} from the metric gives:
\Gamma^i_{00} = -\frac{1}{2}\frac{\partial h_{00}}{\partial x^i}
So:
\frac{d^2x^i}{dt^2} = \frac{c^2}{2}\frac{\partial h_{00}}{\partial x^i}
Compare with Newton, \ddot{x}^i = -\partial\phi/\partial x^i where \phi is the gravitational potential:
\boxed{h_{00} = -\frac{2\phi}{c^2} \quad\Longrightarrow\quad g_{00} = 1 + \frac{2\phi}{c^2}}
Read what that says. The time–time component of the metric is the Newtonian gravitational potential. Gravity, in the Newtonian regime, is entirely a distortion of the rate of time.
Check it against Chapter 6.6. Proper time relates to coordinate time by d\tau = \sqrt{g_{00}}\,dt:
d\tau = \sqrt{1+\frac{2\phi}{c^2}}\,dt \approx \left(1+\frac{\phi}{c^2}\right)dt
With \phi = -GM/r negative, d\tau < dt: clocks deeper in the well run slow, by exactly the factor derived from the equivalence principle. The two arguments agree.
Step 2: the field equation becomes Poisson's equation.
Take the trace of the field equations (with \Lambda = 0 for now). Contracting with g^{\mu\nu} and using g^{\mu\nu}g_{\mu\nu} = 4 in four dimensions:
R - 2R = \frac{8\pi G}{c^4}T \quad\Longrightarrow\quad R = -\frac{8\pi G}{c^4}T
Substituting back gives the equivalent form:
R_{\mu\nu} = \frac{8\pi G}{c^4}\left(T_{\mu\nu}-\tfrac{1}{2}Tg_{\mu\nu}\right)
Take the 00 component. For non-relativistic matter T = T^\mu_{\ \mu} = \rho c^2, so:
R_{00} = \frac{8\pi G}{c^4}\left(\rho c^2 - \tfrac{1}{2}\rho c^2\right) = \frac{4\pi G\rho}{c^2}
And in the weak static field, computing R_{00} from h_{00} gives:
R_{00} = -\frac{1}{2}\nabla^2h_{00} = \frac{1}{c^2}\nabla^2\phi
Setting them equal:
\boxed{\nabla^2\phi = 4\pi G\rho}
That is Poisson's equation for gravity, which is Newton's inverse-square law written as a field equation. For a point mass it has the solution \phi = -GM/r, which gives F = -m\nabla\phi = -GMm/r^2.
Newton's law of gravitation has been derived from the Einstein field equations, as the limit for weak fields, slow motion and ordinary matter. Everything Newton got right is contained in Einstein's theory, and the departures appear exactly where those three assumptions fail.
Where the assumptions fail, and by how much
The relevant dimensionless number is:
\epsilon = \frac{2GM}{rc^2}
which is how deep the gravitational well is compared with c^2. Corrections to Newton are of order \epsilon.
| Situation | \epsilon | Newton's error |
|---|---|---|
| Earth's surface | 1.4\times10^{-9} | 1 part in 10^{9} |
| Sun's surface | 4.2\times10^{-6} | 4 parts in 10^{6} |
| Mercury's orbit | 5.1\times10^{-8} | enough for 43″/century |
| White dwarf surface | 10^{-4} | measurable redshift |
| Neutron star surface | 0.4 | Newton useless |
| Black hole horizon | 1 | Newton meaningless |
This table is the honest statement of Newton's status. He is not wrong. He is exact to nine decimal places on Earth, and the places he fails are the places where \epsilon stops being tiny.
The cosmological constant
\Lambda g_{\mu\nu}
\Lambda (capital lambda) is the cosmological constant, and it has the strangest history of any term in physics.
Why it is allowed. The Bianchi identity requires the left-hand side to be divergence-free. The metric itself satisfies \nabla^\mu g_{\mu\nu} = 0 identically — that is a defining property of the covariant derivative. So any constant multiple of g_{\mu\nu} can be added without breaking anything. Mathematically, \Lambda is free.
Why Einstein added it, in 1917. He applied his equations to the whole universe and found they had no static solution. Matter attracts, so a static universe collapses. Everyone in 1917 knew the universe was static — the galaxies had not yet been recognised as galaxies. So Einstein added a term producing a repulsion that grows with distance, tuned to exactly balance gravity.
Why it was a blunder. The balance is unstable, like a pencil on its point: a slight expansion weakens gravity and strengthens the \Lambda repulsion, and it runs away. Worse, had he trusted the equations he would have predicted the expanding universe, which Hubble found in 1929. Einstein removed the term and called it his greatest blunder.
Why it came back, in 1998. Two independent teams measuring distant Type Ia supernovae found that the expansion of the universe is accelerating. Nothing in a matter-filled universe does that. What does it is a \Lambda term. Both teams shared the 2011 Nobel Prize.
What it means physically. Move the term to the right-hand side:
R_{\mu\nu}-\tfrac{1}{2}Rg_{\mu\nu} = \frac{8\pi G}{c^4}T_{\mu\nu} - \Lambda g_{\mu\nu}
Now it looks like a contribution to the stress–energy — an energy density of empty space, with:
\rho_\Lambda = \frac{\Lambda c^2}{8\pi G}, \qquad P_\Lambda = -\rho_\Lambda c^2
Negative pressure, equal in magnitude to the energy density. And in general relativity, pressure gravitates — so negative pressure anti-gravitates, which is what drives acceleration.
The measured value:
\Lambda \approx 1.1\times10^{-52}\ \text{m}^{-2}, \qquad \rho_\Lambda \approx 6\times10^{-27}\ \text{kg/m}^3
About six hydrogen atoms per cubic metre — and because it fills all of space, including the vast emptiness between galaxies, it amounts to 68 % of the total energy of the universe.
The worst prediction in physics
Quantum field theory (Chapter 7.10) says empty space is not empty: it contains zero-point energy from every field. That vacuum energy should act exactly like \Lambda.
Compute it, summing zero-point energies up to the Planck scale, and you get:
\rho_{\text{vacuum}} \approx 10^{96}\ \text{kg/m}^3
Against the measured 10^{-26}.
\frac{\text{predicted}}{\text{observed}} \approx 10^{122}
A discrepancy of 122 orders of magnitude. It is routinely described as the worst quantitative prediction in the history of science, and it is unresolved. Something must cancel the vacuum energy to 122 decimal places and then stop, and nobody knows what. Chapter 12.7 covers the attempts.
What the equations predict
The field equations were not fitted to data. Every one of these came out of them and was then tested.
Mercury's perihelion. A discrepancy of 43 arcseconds per century, known since Le Verrier in 1859 and unexplained for fifty-six years. Einstein computed it in November 1915 and got 43. He wrote that he was so excited he had heart palpitations. Chapter 6.9 does the calculation.
Light bending: 1.75″. Confirmed 1919, now to 0.02 %.
Gravitational redshift. Confirmed by Pound and Rebka, and now measured over 33 cm.
Frame dragging. A rotating mass drags spacetime around with it.
Gravity Probe B, launched in 2004 after forty years of development, carried four gyroscopes so perfectly spherical that if scaled to the size of the Earth their highest mountain would be 2.4 m. It measured a geodetic precession of -6601.8 milliarcseconds/year against a prediction of -6606.1, and a frame-dragging precession of -37.2 \pm 7.2 against a prediction of -39.2.
Gravitational waves. Predicted 1916, detected 2015. Chapter 6.10.
Black holes. Predicted as a solution in 1916, believed to be a mathematical artefact for decades, imaged in 2019.
The expanding universe. Predicted by Friedmann in 1922 from these equations, found by Hubble in 1929.
Gravitational lensing. Predicted 1936, now a standard tool for mapping dark matter.
Not one confirmed failure in a hundred and ten years, across a range of field strengths from a laboratory bench to merging black holes.
Why it is still not the final word
Two problems, and they are large.
It is incompatible with quantum mechanics. General relativity treats spacetime as a smooth continuous manifold. Quantum field theory requires fields to fluctuate. Attempting to quantise gravity the way electromagnetism was quantised produces infinities that cannot be removed by the usual methods. Chapter 8.7 covers the candidate solutions.
It predicts its own failure. The equations produce singularities — points of infinite curvature and infinite density — at the centres of black holes and at the Big Bang. Penrose and Hawking proved in the 1960s that these are not artefacts of assumed symmetry; they are unavoidable given reasonable conditions. A theory that predicts infinite physical quantities is telling you it has left its domain of validity, and what replaces it there must be a quantum theory of gravity.
Where this shows up in your life
GPS, computed in Chapter 6.9.
Gravitational lensing is the main tool for weighing galaxy clusters and mapping dark matter, and Chapter 12.4 uses it heavily.
The expansion of the universe and everything in Part 12 is the Friedmann solution of these equations.
And a small point about time. Because g_{00} = 1 + 2\phi/c^2, the rate of your own clock depends on where you are in the Earth's gravitational field. That is not a metaphor and it is measured routinely. The equation at the top of this page is what governs it.
What the next chapter fixes
Ten coupled nonlinear equations are not much use without a solution. Chapter 6.9 solves them for the simplest case that matters — the spacetime around a single spherical non-rotating mass — a solution Karl Schwarzschild found in a matter of weeks while serving on the Russian front in 1916. From it come the event horizon, the orbits of planets, Mercury's 43 arcseconds computed in full, and the GPS clock correction worked out to the microsecond.