Gravitational interaction · R.A.V.E.N. Living Encyclopedia

The gravitational interaction is one of the four fundamental interactions, described by general relativity as the curvature of a four-dimensional spacetime and carried by a mediator of spin 2, and its strength for two electrons is set by the coupling α_G = Gm_e²/ℏc = 1.751 81(4) × 10⁻⁴⁵ (computed from Mohr et al. 2025). It is carried by the metric tensor g_μν, whose observable content is the curvature tensor formed from second derivatives of the metric. The classical limit of that field obeys the Einstein field equations, no quantum theory of the field is established, and the connection coefficients from which the curvature derives vanish at any point in a freely falling frame, so the curvature and not the connection is the measured quantity. The stress-energy tensor of matter and radiation sources the field, and the field acts on every form of energy including light, with no species exempt (Touboul et al. 2022). Tests of the theory give the parametrised post-Newtonian parameter γ = 1 + (2.1 ± 2.3) × 10⁻⁵ (Bertotti et al. 2003), an Eötvös parameter η(Ti, Pt) = [−1.5 ± 2.3 (stat) ± 1.5 (syst)] × 10⁻¹⁵ (Touboul et al. 2022), no Yukawa deviation from the inverse-square law for ranges above 38.6 μm at 95% confidence (Lee et al. 2020), an orbital decay in PSR B1913+16 of 0.9983 ± 0.0016 of the radiative rate of general relativity (Weisberg & Huang 2016), and a wave speed satisfying −3 × 10⁻¹⁵ ≤ (v_gw − c)/c ≤ +7 × 10⁻¹⁶ (Abbott et al. 2017). The graviton mass is below 2.42 × 10⁻²³ eV/c² at 90% credibility (Abbott et al. 2025), so the range exceeds ℏ/m_gc = 8 × 10¹⁵ m and is consistent with an unbounded range (computed from Abbott et al. 2025). The interaction binds planetary systems, stars and galaxies, drives the collapse of stellar cores to neutron stars and black holes, and carries the radiation by which compact binaries are observed (Abbott et al. 2016). The inverse-square law of the gravitational force was established by Newton (1687), the geometric field theory by Einstein (1916), and the deflection of starlight at the solar limb by Dyson, Eddington & Davidson (1920).

02 · Equations

Governing equations

The gravitational interaction is described by general relativity, in which the metric of spacetime is the field and matter is its source. Relations are written in SI units with the Newtonian constant G and the speed of light c; numerical values are from the CODATA 2022 adjustment (Mohr et al. 2025) and the cited measurements.

Master · Einstein–Hilbert action

Action and field equation of the metric

$$S=\frac{c^4}{16\pi G}\int R\sqrt{-g}\,d^4x+S_{\rm m},\qquad G_{\mu\nu}+\Lambda g_{\mu\nu}=\frac{8\pi G}{c^4}T_{\mu\nu}$$

Variation with respect to the metric g_μν gives the Einstein field equations, in which the Einstein tensor G_μν is built from second derivatives of the metric and T_μν is the stress-energy tensor of matter and radiation (Einstein 1916). The constant of proportionality is 8πG/c⁴ with G = 6.674 30(15) × 10⁻¹¹ m³ kg⁻¹ s⁻² (Mohr et al. 2025). The relations below follow from this action as weak-field limits, as vacuum solutions, or as expansions in v/c.

01 · Force

Newton's law of gravitation

$$\begin{gathered}\mathbf F_{12}=-\frac{Gm_1m_2}{r^2}\,\hat{\mathbf r}_{12},\\ G=6.674\,30(15)\times10^{-11}\ \mathrm{m^3\,kg^{-1}\,s^{-2}}\end{gathered}$$

Force between two masses at rest, attractive for all pairs, recovered from the field equations in the limit of weak fields and low speeds (Newton 1687). The constant is measured with a relative standard uncertainty of 2.2 × 10⁻⁵ (Mohr et al. 2025). A Yukawa deviation of gravitational strength is excluded for ranges above 38.6 μm at 95% confidence (Lee et al. 2020).

02 · Field

Metric, interval and connection

$$\begin{gathered}ds^2=g_{\mu\nu}\,dx^\mu dx^\nu,\\ \Gamma^{\lambda}_{\ \mu\nu}=\tfrac12 g^{\lambda\sigma}\left(\partial_\mu g_{\sigma\nu}+\partial_\nu g_{\sigma\mu}-\partial_\sigma g_{\mu\nu}\right)\end{gathered}$$

The metric fixes the interval between neighbouring events and the proper time along a worldline, and the connection coefficients are its first derivatives (Einstein 1916). At any point a coordinate system exists in which the connection vanishes and the metric takes its flat form, so the connection carries no local information and the curvature built from its derivatives is the field strength.

03 · Potential

Newtonian limit and time dilation

$$\begin{gathered}g_{00}=-\left(1+\frac{2\Phi}{c^2}\right),\qquad \Phi=-\frac{GM}{r},\\ \frac{d\tau}{dt}=\left(1-\frac{r_s}{r}\right)^{1/2}\end{gathered}$$

In a static weak field the time-time component of the metric reduces to the Newtonian potential Φ, and the geodesic equation reduces to Newton's law (Einstein 1916). The same component fixes the rate of a static clock relative to coordinate time, so the potential is measured through the redshift while the metric remains the field. At the solar surface the fractional rate difference is 2.12 × 10⁻⁶ (computed from IAU 2015).

04 · Equation of motion

Geodesic equation and geodesic deviation

$$\begin{gathered}\frac{d^2x^\lambda}{d\tau^2}+\Gamma^{\lambda}_{\ \mu\nu}\frac{dx^\mu}{d\tau}\frac{dx^\nu}{d\tau}=0,\\ \frac{D^2\xi^\lambda}{d\tau^2}=-R^{\lambda}_{\ \mu\nu\sigma}u^\mu\xi^\nu u^\sigma\end{gathered}$$

A free test body follows a geodesic of the metric, and the path is independent of its mass and composition (Einstein 1916). The separation ξ between two neighbouring geodesics obeys the second relation, in which the Riemann tensor appears directly, so relative acceleration and not acceleration is the observable. The composition independence is measured as η(Ti, Pt) = [−1.5 ± 2.3 (stat) ± 1.5 (syst)] × 10⁻¹⁵ (Touboul et al. 2022).

05 · Field equations

Einstein field equations

$$\begin{gathered}R_{\mu\nu}-\tfrac12 R\,g_{\mu\nu}+\Lambda g_{\mu\nu}=\frac{8\pi G}{c^4}T_{\mu\nu},\\ \nabla_\mu T^{\mu\nu}=0\end{gathered}$$

Ten coupled nonlinear equations relating the curvature of spacetime to the stress-energy of its content (Einstein 1916). The contracted Bianchi identity makes the left side divergence-free, so local conservation of energy and momentum follows from the field equations rather than being imposed. The cosmological term Λ enters with the dimensions of an inverse squared length.

06 · Vacuum solution

Schwarzschild metric

$$\begin{gathered}ds^2=-\left(1-\frac{r_s}{r}\right)c^2dt^2+\left(1-\frac{r_s}{r}\right)^{-1}dr^2+r^2d\Omega^2,\\ r_s=\frac{2GM}{c^2}\end{gathered}$$

The spherically symmetric vacuum solution of the field equations, fixed by the mass alone (Schwarzschild 1916). The Schwarzschild radius of the Sun is 2.953 km (computed from IAU 2015), the circular photon orbit lies at 3r_s/2 and the innermost stable circular orbit of a massive body at 3r_s. The surface r = r_s is a coordinate singularity of these coordinates and a null surface of the geometry.

07 · Radiation

Quadrupole formula and orbital decay

$$\begin{gathered}P=\frac{G}{5c^5}\left\langle\dddot{Q}_{ij}\dddot{Q}_{ij}\right\rangle,\\ \dot P_b=-\frac{192\pi}{5}\left(\frac{2\pi G\mathcal{M}}{P_bc^3}\right)^{5/3}f(e)\end{gathered}$$

Power radiated by a time-varying mass quadrupole, the lowest radiative multipole of the field (Einstein 1918). Applied to a Keplerian binary of chirp mass ℳ and eccentricity e the formula gives the orbital period derivative, with f(e) the enhancement factor of Peters & Mathews (1963). For PSR B1913+16 the predicted rate is (−2.402 63 ± 0.000 05) × 10⁻¹² and the corrected measurement is 0.9983 ± 0.0016 of it (Weisberg & Huang 2016).

08 · Coupling

Gravitational coupling and the Planck scale

$$\begin{gathered}\alpha_G=\frac{Gm_e^2}{\hbar c}=1.751\,81(4)\times10^{-45},\\ m_P=\left(\frac{\hbar c}{G}\right)^{1/2},\qquad \ell_P=\left(\frac{\hbar G}{c^3}\right)^{1/2}\end{gathered}$$

The dimensionless coupling formed from G and a particle mass, quoted here for the electron mass (computed from Mohr et al. 2025). The coupling carries a mass and is therefore not a constant of the interaction alone, and the scale at which it reaches unity is the Planck mass m_P = 2.176 434(24) × 10⁻⁸ kg, with Planck length ℓ_P = 1.616 255(18) × 10⁻³⁵ m (Mohr et al. 2025).

03 · Numbers

Measured properties

Recommended values from the CODATA 2022 adjustment (Mohr et al. 2025), the exact 2019 SI definition of the metre, the nominal solar conversion constants of IAU (2015), and the cited tests of general relativity. Parenthesised digits give the standard uncertainty in the final digits shown.

α_G = 1.751 81 × 10⁻⁴⁵

Coupling at the electron mass · 2.2 × 10⁻⁵ relative · Mohr et al. 2025

6.674 30 × 10⁻¹¹ m³ kg⁻¹ s⁻²

Newtonian constant G · 2.2 × 10⁻⁵ relative · Mohr et al. 2025

299 792 458 m s⁻¹

Speed of the free field c · exact · SI 2019

m_g < 2.42 × 10⁻²³ eV

Graviton mass · 90% credibility · Abbott et al. 2025

Quantity	Value	Status	Meaning & convention
Gravitational coupling α_G	1.751 81(4) × 10⁻⁴⁵	derived · Gm_e²/ℏc	Dimensionless coupling formed with the electron mass; it carries a mass and changes with the particle chosen (computed from Mohr et al. 2025).
Newtonian constant G	6.674 30(15) × 10⁻¹¹ m³ kg⁻¹ s⁻²	measured · 2.2 × 10⁻⁵	Constant of the field equations and of Newton's law; determined by laboratory torsion measurements (Mohr et al. 2025).
G/ℏc	6.708 83(15) × 10⁻³⁹ (GeV/c²)⁻²	measured · 2.2 × 10⁻⁵	The constant in natural units, in which it is an inverse squared mass (Mohr et al. 2025).
Speed of light c	299 792 458 m s⁻¹	exact · SI 2019	Propagation speed of the free field and the conversion between the time and space parts of the metric; exact by definition of the metre (Mohr et al. 2025).
Planck mass m_P	2.176 434(24) × 10⁻⁸ kg	derived · (ℏc/G)^½	Mass at which the coupling α_G reaches unity, equivalent to 1.220 890(14) × 10¹⁹ GeV (Mohr et al. 2025).
Planck length ℓ_P	1.616 255(18) × 10⁻³⁵ m	derived · (ℏG/c³)^½	Length formed from G, ℏ and c; the corresponding Planck time is 5.391 247(60) × 10⁻⁴⁴ s (Mohr et al. 2025).
Mediator	graviton · J^P = 2⁺ · Q = 0	exact · representation	Massless tensor boson with two helicity states ±2 in the linearised theory; not observed as a quantum (Navas et al. 2024).
Graviton mass	< 2.42 × 10⁻²³ eV/c²	limit · 90% credibility	From the absence of dispersion in the gravitational-wave signals of the third transient catalogue (Abbott et al. 2025).
Range	> 8 × 10¹⁵ m	derived · ℏ/m_gc	Reduced Compton wavelength at the graviton mass limit; consistent with an unbounded range (computed from Abbott et al. 2025).
Wave speed	−3 × 10⁻¹⁵ ≤ (v_gw − c)/c ≤ +7 × 10⁻¹⁶	measured · Abbott et al. 2017	From the arrival-time difference between GW170817 and GRB 170817A over a propagation distance of 40 Mpc (Abbott et al. 2017).
PPN parameter γ	1 + (2.1 ± 2.3) × 10⁻⁵	measured · Bertotti et al. 2003	Space curvature produced by unit rest mass, measured through the Shapiro delay of the Cassini radio link at solar conjunction (Bertotti et al. 2003).
Eötvös parameter η(Ti, Pt)	[−1.5 ± 2.3 (stat) ± 1.5 (syst)] × 10⁻¹⁵	measured · Touboul et al. 2022	Fractional difference in free-fall acceleration of titanium and platinum test masses in orbit, testing composition independence of the geodesic (Touboul et al. 2022).
Inverse-square law range	λ < 38.6 μm at \|α\| = 1	limit · 95% CL	Excluded range of a Yukawa addition of gravitational strength, from a torsion measurement at separations down to 52 μm (Lee et al. 2020).
Quadrupole-formula test	0.9983 ± 0.0016	measured · Weisberg & Huang 2016	Ratio of the galactic-corrected orbital decay of PSR B1913+16 to the rate predicted by the quadrupole formula (Weisberg & Huang 2016).

Sources · NIST CODATA 2022 (physics.nist.gov/constants) · IAU 2015 Resolution B3 nominal solar conversion constants · Bertotti, Iess & Tortora Nature 425, 374 (2003) · Weisberg & Huang ApJ 829, 55 (2016) · Touboul et al. PRL 129, 121102 (2022) · Lee et al. PRL 124, 101101 (2020) · Abbott et al. PRD 112, 084080 (2025)

04 · Situations

Representative configurations

Six configurations computed from CODATA 2022 constants, the IAU 2015 nominal solar values and the cited measurements: the binary pulsar PSR B1913+16, the MICROSCOPE free-fall test, the Shapiro delay of the Cassini radio link, the binary black hole merger GW150914, the eclipse deflection measurement of 1919, and the orbit of the star S2 around Sgr A*. Each scene states any scale factor applied to the rendering.

05 · Latest research

Recent literature

Preprints and papers retrieved at page load, ordered by submission date. arXiv: categories gr-qc, astro-ph.HE, astro-ph.CO, hep-th and physics.class-ph, abstracts matching general relativity, gravitational waves, the equivalence principle, the Newtonian constant, black hole spacetimes or tests of gravity. INSPIRE-HEP: the same terms in titles of high-energy-physics records. Dates are arXiv submission dates and INSPIRE record dates; no publisher issue dates are used.

Querying arXiv · INSPIRE-HEP …

Date	Source	Title and authors	Reference
Retrieving …

06 · References

References

Declared representations

Vertical axis of the heroThe hero draws a 2+1 spacetime: the horizontal plane is the equatorial slice of the mass and the vertical axis is Schwarzschild coordinate time. Heights are times, not distances, and the ratio of the two scales is a display choice.

Light conesA cone is drawn at a finite number of lattice points with a finite opening; the null structure is defined at every event, and the cone size marks the coordinate light speed dr/dt = ±(1 − r_s/r) and r dφ/dt = ±(1 − r_s/r)^½.

Force layerThe force layer draws, on the equatorial plane, the force per unit mass on a body held at rest, GM/r² (1 − r_s/r)^−½, directed toward the centre; the arrow length is the value at a display scale, capped so that no arrow reaches the horizon. No arrow is drawn on a free worldline, on which the proper acceleration is zero.

Curvature layerThe curvature layer draws two neighbouring radial geodesics with rungs marking their separation, and rings of free test masses at fixed events on the infalling worldline with semi-axes 1 + 2kGM/r³ and 1 − kGM/r³ in units of the undeformed radius, in the ratio of the radial and tangential tidal eigenvalues of the Schwarzschild field; k = 4 is a display factor.

Radiation layerThe radiation layer draws a plane transverse-traceless wave through a ring of free test masses, superposed on the static spacetime as an external field: the central mass is static and radiates nothing. The strain is drawn at a display amplitude, and the amplitude is stated in the scene.

Declared scalingWhere a physical separation, rate or angle is below display resolution (the orbital decay per orbit, the deflection at the solar limb, the perihelion advance, the strain), the applied scale factor is stated in the scene.

Sources

I. Newton (1687). Philosophiae Naturalis Principia Mathematica. Jussu Societatis Regiae ac Typis J. Streater, London.

A. Einstein (1916). Die Grundlage der allgemeinen Relativitätstheorie. Ann. Phys. 354, 769.

K. Schwarzschild (1916). Über das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie. Sitzungsber. Preuss. Akad. Wiss. 1916, 189.

A. Einstein (1918). Über Gravitationswellen. Sitzungsber. Preuss. Akad. Wiss. 1918, 154.

F. W. Dyson, A. S. Eddington & C. Davidson (1920). A determination of the deflection of light by the Sun's gravitational field, from observations made at the total eclipse of May 29, 1919. Phil. Trans. R. Soc. A 220, 291.

P. C. Peters & J. Mathews (1963). Gravitational radiation from point masses in a Keplerian orbit. Phys. Rev. 131, 435.

I. I. Shapiro (1964). Fourth test of general relativity. Phys. Rev. Lett. 13, 789.

B. Bertotti, L. Iess & P. Tortora (2003). A test of general relativity using radio links with the Cassini spacecraft. Nature 425, 374.

IAU (2015). Resolution B3 on recommended nominal conversion constants for selected solar and planetary properties. XXIX IAU General Assembly, Honolulu.

B. P. Abbott et al. (2016). Observation of gravitational waves from a binary black hole merger. Phys. Rev. Lett. 116, 061102.

J. M. Weisberg & Y. Huang (2016). Relativistic measurements from timing the binary pulsar PSR B1913+16. Astrophys. J. 829, 55.

B. P. Abbott et al. (2017). Gravitational waves and gamma-rays from a binary neutron star merger: GW170817 and GRB 170817A. Astrophys. J. Lett. 848, L13.

GRAVITY Collaboration (2020). Detection of the Schwarzschild precession in the orbit of the star S2 near the Galactic centre massive black hole. Astron. Astrophys. 636, L5.

J. G. Lee et al. (2020). New test of the gravitational 1/r² law at separations down to 52 μm. Phys. Rev. Lett. 124, 101101.

P. Touboul et al. (2022). MICROSCOPE mission: final results of the test of the equivalence principle. Phys. Rev. Lett. 129, 121102.

S. Navas et al., Particle Data Group (2024). Review of particle physics. Phys. Rev. D 110, 030001 · PDG graviton listing.

P. J. Mohr et al. (2025). CODATA recommended values of the fundamental physical constants: 2022. Rev. Mod. Phys. 97, 025002 · physics.nist.gov/constants.

R. Abbott et al. (2025). Tests of general relativity with GWTC-3. Phys. Rev. D 112, 084080.