General Relativity: Gravity Is Not a Force
In 1907, Einstein had what he later called “the happiest thought of my life.”
He was sitting in his chair at the patent office and imagined a man falling freely from a roof.
The falling man feels no gravity. He is weightless. He drops his keys, they fall with him, hovering beside him. From his perspective, gravity has vanished.
Einstein realized: a freely falling reference frame is locally equivalent to an inertial frame.
Gravity can be transformed away, locally and temporarily, by choosing the right reference frame.
This is the equivalence principle.
From this single thought, from a man falling off a roof, Einstein spent eight years constructing the most beautiful physical theory ever written.
General Relativity.
I. The Failures of Special Relativity
Special relativity is incomplete.
It describes physics in inertial frames, frames moving at constant velocity. It says nothing about accelerated frames. It says nothing about gravity.
More precisely: Newtonian gravity is incompatible with special relativity.
Newton’s gravitational force:
\[\mathbf{F} = -\frac{Gm_1m_2}{r^2}\hat{r}\]acts instantaneously across any distance.
Special relativity forbids instantaneous action. No influence propagates faster than \(c\).
Gravity must be modified. The modification must reduce to Newton’s law in the appropriate limit. It must be consistent with special relativity.
Three candidates for relativistic gravity were proposed. None worked correctly.
Einstein’s approach was different. He did not try to make gravity fit special relativity. He rebuilt the framework entirely.
II. The Equivalence Principle, Precisely
The equivalence principle exists in several forms of increasing strength.
Weak Equivalence Principle (WEP):
The trajectory of a freely falling test body is independent of its internal structure and composition.
Equivalently: inertial mass equals gravitational mass.
\[m_i \mathbf{a} = m_g \mathbf{g}\]If \(m_i = m_g\): all bodies fall the same way. Galileo’s experiment from the Tower of Pisa. Eötvös experiment (1889): confirmed to \(10^{-9}\). Modern tests (MICROSCOPE, 2017): confirmed to \(10^{-15}\).
Einstein Equivalence Principle (EEP):
In any freely falling frame, the laws of special relativity hold locally.
No local experiment, mechanical or electromagnetic, can distinguish a freely falling frame from an inertial frame in empty space.
Strong Equivalence Principle (SEP):
The same holds for all physics, including gravitational physics. Self-gravitating bodies fall the same way as test particles.
The SEP is violated by some alternative gravity theories. GR satisfies it exactly.
III. Differential Geometry, the Mathematical Language
General relativity is written in the language of differential geometry.
You cannot understand GR without this language. Let me build it carefully.
Manifolds:
A manifold \(\mathcal{M}\) is a topological space that locally looks like \(\mathbb{R}^n\).
Spacetime is a 4-dimensional manifold. Locally, in a small enough region, it looks like flat Minkowski space. Globally, it can be curved.
Tangent spaces:
At each point \(p \in \mathcal{M}\), there is a tangent space \(T_p\mathcal{M}\), the space of all tangent vectors at \(p\).
A tangent vector \(V^\mu\) at \(p\) represents a direction and magnitude in which you can move from \(p\).
The metric tensor:
The fundamental object in Riemannian geometry is the metric tensor \(g_{\mu\nu}\), a symmetric (0,2) tensor field that assigns an inner product to tangent vectors:
\[ds^2 = g_{\mu\nu}dx^\mu dx^\nu\]This defines distances, angles, and volumes on the manifold.
In flat Minkowski spacetime:
\[g_{\mu\nu} = \eta_{\mu\nu} = \text{diag}(-1, +1, +1, +1)\] \[ds^2 = -c^2dt^2 + dx^2 + dy^2 + dz^2\]In curved spacetime, \(g_{\mu\nu}\) varies from point to point. The metric carries all information about the geometry.
Covariant derivative:
On a curved manifold, ordinary partial derivatives do not transform as tensors.
The covariant derivative \(\nabla_\mu\) corrects for the curvature of the manifold:
\[\nabla_\mu V^\nu = \partial_\mu V^\nu + \Gamma^\nu_{\mu\lambda}V^\lambda\]where \(\Gamma^\nu_{\mu\lambda}\) are the Christoffel symbols:
\[\Gamma^\lambda_{\mu\nu} = \frac{1}{2}g^{\lambda\sigma} \left(\partial_\mu g_{\nu\sigma} + \partial_\nu g_{\mu\sigma} - \partial_\sigma g_{\mu\nu}\right)\]The Christoffel symbols encode how basis vectors change from point to point. They are not tensors. They vanish in local inertial frames.
Geodesics:
A geodesic is the straightest possible path on a curved manifold, the path of zero covariant acceleration:
\[\frac{d^2x^\mu}{d\tau^2} + \Gamma^\mu_{\alpha\beta} \frac{dx^\alpha}{d\tau}\frac{dx^\beta}{d\tau} = 0\]where \(\tau\) is proper time.
This is the equation of motion for a freely falling particle.
In flat spacetime: \(\Gamma = 0\), straight lines. In curved spacetime: geodesics are curved, but they are the straightest possible paths.
The Riemann curvature tensor:
The Riemann tensor \(R^\rho{}_{\sigma\mu\nu}\) measures the curvature of the manifold:
\[R^\rho{}_{\sigma\mu\nu} = \partial_\mu\Gamma^\rho_{\nu\sigma} - \partial_\nu\Gamma^\rho_{\mu\sigma} + \Gamma^\rho_{\mu\lambda}\Gamma^\lambda_{\nu\sigma} - \Gamma^\rho_{\nu\lambda}\Gamma^\lambda_{\mu\sigma}\]It has 20 independent components in 4 dimensions.
Physical interpretation: if you parallel-transport a vector around a closed loop, the vector returns rotated. The rotation is proportional to \(R^\rho{}_{\sigma\mu\nu}\) times the area of the loop.
Zero Riemann tensor: flat spacetime. Nonzero Riemann tensor: curved spacetime.
The Ricci tensor and Ricci scalar:
Contracting the Riemann tensor:
\[R_{\mu\nu} = R^\lambda{}_{\mu\lambda\nu}\] \[R = g^{\mu\nu}R_{\mu\nu}\]The Einstein tensor:
\[G_{\mu\nu} = R_{\mu\nu} - \frac{1}{2}Rg_{\mu\nu}\]The Einstein tensor is symmetric, (0,2), and satisfies the contracted Bianchi identity:
\[\nabla^\mu G_{\mu\nu} = 0\]This is crucial. It guarantees that the Einstein equations are consistent with energy-momentum conservation.
IV. The Einstein Field Equations
After eight years of struggle (involving wrong turns, false starts, a race against Hilbert, and the development of entirely new mathematics), Einstein wrote down his field equations in November 1915:
\[G_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4}T_{\mu\nu}\]Ten coupled nonlinear partial differential equations.
The left side: the geometry of spacetime. The right side: the matter and energy content.
John Wheeler’s summary: “Spacetime tells matter how to move; matter tells spacetime how to curve.”
The stress-energy tensor \(T_{\mu\nu}\):
It encodes all matter and energy content:
- \(T_{00}\): energy density
- \(T_{0i}\): energy flux and momentum density
- \(T_{ij}\): stress tensor (pressure and shear)
For a perfect fluid: \(T_{\mu\nu} = (\rho + p/c^2)u_\mu u_\nu + pg_{\mu\nu}\)
The cosmological constant \(\Lambda\):
Einstein introduced \(\Lambda\) in 1917 to produce a static universe, his “greatest blunder.”
After Hubble’s discovery of cosmic expansion (1929), he abandoned it.
In 1998, the discovery of accelerating expansion rehabilitated \(\Lambda\) as dark energy.
Current value: \(\Lambda \approx 1.1 \times 10^{-52} \text{ m}^{-2}\)
The Newtonian limit:
For weak fields and slow motion: \(g_{\mu\nu} \approx \eta_{\mu\nu} + h_{\mu\nu}\), \(|h_{\mu\nu}| \ll 1\).
The Einstein equations reduce to: \(\nabla^2\Phi = 4\pi G\rho\)
Newton’s law of gravity. GR contains Newtonian gravity as a weak-field, slow-motion approximation.
V. The Schwarzschild Solution
The Einstein equations are nonlinear. Finding exact solutions is extraordinarily difficult.
The first exact solution was found by Karl Schwarzschild in 1916, while serving on the Russian front in World War I. He died four months later.
The Schwarzschild metric:
For a spherically symmetric, static, vacuum spacetime:
\[ds^2 = -\left(1 - \frac{r_s}{r}\right)c^2dt^2 + \left(1 - \frac{r_s}{r}\right)^{-1}dr^2 + r^2d\Omega^2\]where: \(r_s = \frac{2GM}{c^2}\) is the Schwarzschild radius,
\(d\Omega^2 = d\theta^2 + \sin^2\theta\,d\phi^2\).
This describes spacetime outside a spherical mass \(M\): a planet, a star, a black hole.
Gravitational time dilation:
A clock at radius \(r\) runs at rate:
\[d\tau = \sqrt{1 - \frac{r_s}{r}}\,dt\]At \(r \to \infty\): \(d\tau \to dt\), coordinate time. Near \(r_s\): time slows dramatically. At \(r = r_s\): time stops.
The Schwarzschild radius:
For the Sun: \(r_s = 2.95\) km. For the Earth: \(r_s = 8.87\) mm. For a proton: \(r_s \sim 10^{-54}\) m.
If you could compress the Sun to a sphere of radius 2.95 km, it would become a black hole.
Geodesics in Schwarzschild spacetime:
The geodesic equation gives planetary orbits.
For nearly circular orbits, the perihelion precesses at rate:
\[\Delta\phi = \frac{6\pi GM}{c^2 a(1-e^2)}\]per orbit, where \(a\) is the semi-major axis and \(e\) is the eccentricity.
For Mercury: \(\Delta\phi = 43''\) per century.
This had been observed and was unexplained for decades. GR explained it exactly.
VI. Black Holes
At \(r = r_s\): the metric component \(g_{tt} = 0\) and \(g_{rr} \to \infty\).
This is the event horizon.
What happens at the event horizon:
For an infalling observer, nothing special. They cross smoothly. The Schwarzschild coordinates are singular there but the physics is not.
In Kruskal-Szekeres coordinates, which cover the full Schwarzschild manifold, the metric is:
\[ds^2 = \frac{32G^3M^3}{r}e^{-r/2GM}(-dT^2 + dX^2) + r^2d\Omega^2\]No coordinate singularity at \(r = r_s\).
The event horizon is a null hypersurface. It moves outward at the speed of light. Once inside, nothing, not even light, can escape.
The singularity:
At \(r = 0\): a genuine curvature singularity. The Kretschmann scalar:
\[K = R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma} = \frac{48G^2M^2}{c^4r^6} \to \infty\]as \(r \to 0\).
The singularity is not a point in space. It is a moment in time: for any infalling observer, it lies in the future.
The Penrose-Hawking singularity theorems:
Roger Penrose (1965) proved that if a trapped surface forms (a surface from which all light rays converge), then a singularity must form.
Stephen Hawking extended this to cosmology: the Big Bang is a singularity.
These theorems show that singularities are generic. They are not artifacts of the special symmetry of Schwarzschild. They are inevitable consequences of GR.
They also show where GR breaks down. At singularities, the curvature diverges, and quantum effects must become important. GR is incomplete.
VII. Gravitational Waves
The linearized Einstein equations, GR in the weak-field limit, have wave solutions.
Write \(g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}\) with \(|h_{\mu\nu}| \ll 1\).
In the transverse-traceless (TT) gauge:
\[\Box \bar{h}_{\mu\nu} = -\frac{16\pi G}{c^4}T_{\mu\nu}\]In vacuum:
\[\Box h_{\mu\nu}^{TT} = 0\]Wave equation. Solutions: gravitational waves propagating at speed \(c\).
Properties of gravitational waves:
Gravitational waves are transverse. They oscillate perpendicular to their direction of travel.
They have two polarizations: \(+\) (plus) and \(\times\) (cross).
A plus-polarized wave stretches space in the x-direction while compressing in the y-direction, then vice versa.
The strain, fractional change in length:
\[h = \frac{\Delta L}{L}\]For the first detected gravitational wave (GW150914): \(h \sim 10^{-21}\).
A fractional change of \(10^{-21}\). For a detector 4 km long: \(\Delta L \sim 4 \times 10^{-18}\) m.
One thousandth the diameter of a proton.
The quadrupole formula:
Gravitational wave luminosity:
\[P = \frac{G}{5c^5}\langle\dddot{Q}_{ij}\dddot{Q}^{ij}\rangle\]where \(Q_{ij}\) is the quadrupole moment of the source.
No monopole radiation (mass conservation). No dipole radiation (momentum conservation). Only quadrupole and higher.
Binary pulsar, indirect detection:
In 1974, Hulse and Taylor discovered the first binary pulsar: PSR B1913+16.
Two neutron stars orbiting each other. The orbital period decreases over time. The system is losing energy.
The rate of energy loss exactly matches the GR prediction for gravitational wave emission.
Hulse and Taylor received the Nobel Prize in 1993 for this indirect confirmation.
LIGO, direct detection:
On September 14, 2015, at 09:50:45 UTC, the LIGO Livingston and Hanford detectors recorded a signal:
GW150914, the merger of two black holes.
Masses: \(36 M_\odot\) and \(29 M_\odot\). Final mass: \(62 M_\odot\). Energy radiated: \(3 M_\odot c^2 \approx 5 \times 10^{47}\) J, more power than all stars in the observable universe combined, for 0.2 seconds.
Distance: 1.3 billion light-years.
The signal lasted 0.2 seconds. The frequency swept from 35 Hz to 150 Hz. The peak strain: \(10^{-21}\).
Einstein predicted gravitational waves in 1916. We detected them in 2015.
99 years.
VIII. The Tests of GR
GR makes specific predictions that differ from Newton. All have been confirmed.
1. Perihelion precession of Mercury: Predicted: 43’’ per century. Observed: 43.1 ± 0.5’’ per century. ✓
2. Gravitational deflection of light: Light bends in a gravitational field. GR predicts: \(\delta\phi = \frac{4GM}{c^2b}\), where \(b\) is the impact parameter.
For light grazing the Sun: \(\delta\phi = 1.75''\).
Eddington (1919) confirmed the deflection within the limits of his ~30% uncertainty. This was the measurement that made Einstein world-famous overnight, despite its modest precision. Modern VLBI radio measurements now confirm the deflection to better than 0.01%. ✓
3. Gravitational redshift: Light climbing out of a gravitational field loses energy. \(\frac{\Delta\nu}{\nu} = \frac{\Delta\Phi}{c^2}\)
Pound-Rebka experiment (1959): measured redshift over 22.5 m height. Confirmed to 1%. ✓
GPS satellites: gravitational redshift (clocks run faster at altitude) combined with special relativistic time dilation (clocks run slower due to orbital velocity) gives a net correction of +38.4 microseconds/day. Without correction, GPS error accumulates at ~11 km/day. ✓
4. Gravitational time delay (Shapiro delay): Radar signals passing near the Sun are delayed by the curved spacetime.
\[\Delta t = \frac{4GM}{c^3}\ln\left(\frac{4r_E r_P}{b^2}\right)\]Measured to 0.1% precision (Cassini 2003, to better still). ✓
5. Frame dragging (Lense-Thirring effect): A rotating mass drags spacetime around it, like a ball rotating in honey.
Gravity Probe B (2004-2005): measured frame dragging by Earth’s rotation. Confirmed to 19%. ✓
6. Gravitational waves: Predicted 1916. Confirmed 2015. ✓
7. Black holes: Event Horizon Telescope (2019): first image of a black hole shadow, M87*, mass \(6.5 \times 10^9 M_\odot\).
Shadow size: consistent with GR prediction. ✓
8. Strong-field tests, binary pulsars: Post-Newtonian parameters measured to parts per million. Consistent with GR. ✓
Every prediction of GR has been confirmed. Not one failure in over a century.
IX. The Cosmological Solutions
Apply GR to the universe as a whole.
Assume: homogeneity and isotropy (the cosmological principle).
The most general metric consistent with this: the Friedmann-Lemaître-Robertson-Walker (FLRW) metric:
\[ds^2 = -c^2dt^2 + a(t)^2\left[\frac{dr^2}{1-kr^2} + r^2d\Omega^2\right]\]where \(a(t)\) is the scale factor, the size of the universe relative to today, and \(k = -1, 0, +1\) is the spatial curvature.
The Einstein equations for this metric give the Friedmann equations:
\[H^2 = \left(\frac{\dot{a}}{a}\right)^2 = \frac{8\pi G}{3}\rho - \frac{kc^2}{a^2} + \frac{\Lambda c^2}{3}\] \[\frac{\ddot{a}}{a} = -\frac{4\pi G}{3}\left(\rho + \frac{3p}{c^2}\right) + \frac{\Lambda c^2}{3}\]where \(H\) is the Hubble parameter, \(\rho\) is the energy density, \(p\) is pressure.
These equations govern the expansion of the universe.
The Hubble parameter today: \(H_0 \approx 67.4 \text{ km/s/Mpc}\) (Planck 2018, early universe). Local distance-ladder measurements give \(\sim 73\) km/s/Mpc. The discrepancy between the two methods, above \(5\sigma\) in current data, is the Hubble tension: unresolved.
The density parameters:
\(\Omega_m \approx 0.315\) (matter, dark + baryonic). \(\Omega_\Lambda \approx 0.685\) (dark energy). \(\Omega_k \approx 0\) (spatial curvature, flat). \(\Omega_r \approx 9 \times 10^{-5}\) (radiation).
The universe is flat, dominated by dark energy, and expanding at an accelerating rate.
The Big Bang:
Extrapolating backward: \(a \to 0\) as \(t \to 0\).
At \(t \approx 10^{-43}\) s (Planck time), the energy density exceeds Planck density. GR breaks down.
The Big Bang singularity is where GR ends. Quantum gravity must take over.
X. Where GR Fails
GR is, empirically, the most successful theory in physics. It is also, mathematically, incomplete.
Singularities:
The Penrose-Hawking theorems guarantee singularities inside black holes and at the Big Bang. At singularities, curvature diverges. GR breaks down. Physical quantities become infinite.
A complete theory must resolve singularities.
Quantum incompatibility:
GR is a classical theory. Quantum mechanics is fundamentally probabilistic.
In GR, the spacetime metric is a smooth classical field. In quantum mechanics, matter fields are quantized operators.
The Einstein equations equate a classical geometric quantity (left side) with the expectation value of a quantum operator (right side):
\[G_{\mu\nu} = \frac{8\pi G}{c^4}\langle\hat{T}_{\mu\nu}\rangle\]This is the semiclassical approximation, used by Hawking to derive Hawking radiation.
It is not fundamental. A proper quantum theory of gravity must quantize the metric itself.
The cosmological constant problem:
As discussed in the Planck scale essay, the quantum field theory prediction for \(\Lambda\) exceeds the observed value by \(10^{122}\).
This is the worst prediction in physics. It involves GR and quantum field theory simultaneously, and we cannot reconcile them.
Dark matter and dark energy:
The Friedmann equations fit cosmological data only if we include:
- \(\Omega_{DM} \approx 0.265\): unknown matter.
- \(\Omega_\Lambda \approx 0.685\): unknown energy.
95% of the universe’s content is unknown.
GR tells us it must be there. It does not tell us what it is.
XI. The Beauty of the Theory
I want to say something about beauty.
The Einstein equations are beautiful.
Not in a superficial sense, not because they are short (ten coupled nonlinear PDEs is not short), but because of what they express:
The geometry of spacetime is determined by its matter content. The matter moves along geodesics of that geometry. The geometry and the matter are one system.
There is no background. No fixed stage on which physics plays out. The stage is part of the play.
Spacetime is not a container for physics. Spacetime is physics.
This is what general relativity discovered: not just a theory of gravity but a new conception of what spacetime is. Not absolute, not fixed, not prior to physics, but dynamic, curved, responsive, inseparable from the matter and energy that inhabit it.
The universe is not a box. It is a conversation: between geometry and matter, between curvature and motion, between the shape of space and the paths of things through it.
“Space is not a rigid, invariable background. It is a physical object, which can wave, distort, and curve.” — Carlo Rovelli
Einstein understood this in 1915.
We are still learning what it means.
Falling
There are physics ideas you understand, and there are physics ideas you carry. General relativity is the second kind for me.
I learned it as an undergraduate in İzmir, on a chalkboard, from a man who is no longer alive. What got me wasn’t the field equations. Those came later, and with effort. What got me was something quieter: the equivalence principle.
A man falling from a roof feels no gravity. I had heard this many times. But the day I understood that this is not a thought experiment, not a trick, not a metaphor, that this is actually what gravity is, something in the world rearranged itself. Gravity is not a force pulling on the falling man. The falling man is moving along the straightest possible path through curved spacetime. There is no force. There is only geometry, and a body following it. Falling is the most natural thing in the universe.
I have thought about this many times since, in contexts that have nothing to do with physics. People who shape you bend the geometry of the world you move through. The trajectory you take afterward is not the trajectory you would have taken without them. It is, in the most literal sense the geometry allows, theirs as well as yours.
This is not consolation, exactly. It is just true. The universe is a manifold. The metric depends on what is in it. Spacetime tells matter how to move; matter tells spacetime how to curve. The people who teach you the framework are part of the framework.
The happiest thought of Einstein’s life was about a man falling. I think about this often. It is a tender image, not a tragic one. He is weightless. His keys hover beside him. For one moment, before he hits the ground, the universe has lifted the weight of itself off him. Maybe that is what good teaching does. Not protection from gravity, exactly. The brief discovery that there is no gravity, only geometry, and you are inside it.
Essential reading:
Einstein, “The Foundation of the General Theory of Relativity” (1916). Annalen der Physik 49, 769. The original paper.
Misner, Thorne & Wheeler, Gravitation (1973). The bible. 1279 pages. Essential.
Carroll, Spacetime and Geometry (2004). The best modern textbook.
Wald, General Relativity (1984). Mathematically rigorous. Essential for serious study.
Penrose, The Road to Reality (2004). The most ambitious book in physics.
For the historically inclined:
Pais, Subtle is the Lord (1982). The definitive Einstein biography.
Bana GR öğreten ve şu anda hayatta olmayan hocam, Durmuş Ali Demir’e saygıyla. Hayatta değilken bile beni inspire ettiğiniz için sonsuz teşekkürler. Sizi tanıdığım için, sizin öğrenciniz olduğum için sonsuz gurur duyuyorum.
Enjoy Reading This Article?
Here are some more articles you might like to read next: