Equivalence Principle Forces a Curved Metric

Derivation

Equivalence Principle Forces a Curved Metric

D-384 Home PU-403 Threads force · symmetry · fields Depends on minkowski-metric-from-lorentz-invariance

Statement

Because all test bodies fall with the same acceleration in a gravitational field (the weak equivalence principle, WEP), gravity cannot be a genuine force field acting on particles that live on a fixed flat spacetime. Instead the effect of gravity on free particles must be encoded in the geometry itself: the flat Minkowski metric \(\eta_{\mu\nu}\) is promoted to a position-dependent metric \(g_{\mu\nu}(x)\), and free fall is geodesic motion \(\frac{d^2x^\mu}{d\tau^2}+\Gamma^{\mu}{}_{\alpha\beta}\frac{dx^\alpha}{d\tau}\frac{dx^\beta}{d\tau}=0\) with \(\Gamma^{\mu}{}_{\alpha\beta}\) built from \(g_{\mu\nu}\). We derive this necessity: universality of free fall forces a curved metric, and the connection coefficients are the gravitational field.

Why it matters

This is the conceptual pivot from Newtonian gravity to general relativity. Every other force in physics (electromagnetic, weak, strong) acts differently on different particles because the coupling — charge, isospin — varies from body to body. Gravity is the unique exception: the "gravitational charge" is the inertial mass itself, so it cancels out of the equation of motion. The moment a coupling cancels universally, it stops being a property of the particle and becomes a property of spacetime.

Recognising this lets us replace an infinite family of gravitational trajectories (one per initial condition, all identical) with a single geometric statement: particles follow the straightest available paths, and the paths bend because spacetime itself is not flat. This is what makes gravity geometrical and everything else not.

Assumptions

Weak equivalence principle (universality of free fall).If inertial and gravitational mass differed between bodies, trajectories would depend on composition; gravity would be an ordinary force and no single geometry could describe all worldlines. The whole derivation collapses.

Locality: gravitational fields are smooth, so over a small enough region the field is uniform.If the field could not be approximated as uniform on small scales, no local inertial frame would exist and the transition to free fall would be impossible even instantaneously.

Special relativity holds locally; freely falling frames are Lorentz frames with metric \(\eta_{\mu\nu}=\mathrm{diag}(-1,1,1,1)\).Without local Lorentz structure there is no fixed light-cone geometry to glue together, and the object we build need not be a Lorentzian metric at all.

Equivalence of a uniform gravitational field and a uniformly accelerated frame (Einstein's elevator).If acceleration and gravity were locally distinguishable by a mechanical experiment, one could define a preferred flat background and keep gravity as a force; the geometric reinterpretation would not be forced, only permitted.

Free-fall worldlines are the extremal (geodesic) curves of whatever geometric structure we build.If free particles did not extremise proper time, the connection extracted below would not be metric-compatible and "straightest line" would lose meaning, breaking the identification of \(\Gamma\) with the field.

Derivation

\[ m_{\mathrm{I}}\,\vec{a} = m_{\mathrm{G}}\,\vec{g}, \qquad \vec{a} = \frac{m_{\mathrm{G}}}{m_{\mathrm{I}}}\,\vec{g}. \]

Newton's second law with a gravitational force \(\vec F = m_{\mathrm G}\vec g\); \(m_{\mathrm I}\) is inertial mass, \(m_{\mathrm G}\) gravitational charge. This is the setup, not yet a claim. A

\[ \frac{m_{\mathrm{G}}}{m_{\mathrm{I}}}=1 \ \text{(all bodies)} \quad\Longrightarrow\quad \vec{a}=\vec{g}\ \ \text{independent of the body}. \]

Impose the WEP: the ratio is universal (measured to \(\lesssim 10^{-15}\)). The acceleration no longer references any property of the particle. A

\[ \vec{x}\,'(t)=\vec{x}(t)-\tfrac{1}{2}\vec{g}\,t^{2},\qquad \vec{a}\,'=\vec{a}-\vec{g}=\vec 0 . \]

Because \(\vec a=\vec g\) for every body, one single coordinate change (a uniform acceleration of the frame) removes the gravitational acceleration for all of them at once. A composition-dependent \(\vec a\) could not be cancelled by one frame change. B

\[ \text{In the falling frame } S':\quad \frac{d^{2}\vec{x}\,'}{dt^{2}}=0 \quad\text{for every free particle.} \]

The free-fall frame is (locally, instantaneously) an inertial frame: no gravity appears in it. This is the equivalence principle in operational form — free fall abolishes gravity locally. B

\[ ds^{2}=\eta_{\mu\nu}\,d\xi^{\mu}d\xi^{\nu},\qquad \eta_{\mu\nu}=\mathrm{diag}(-1,1,1,1), \]

In that local inertial frame, coordinates \(\xi^\mu\), special relativity holds exactly and free particles move on straight lines \(\frac{d^{2}\xi^{\mu}}{d\tau^{2}}=0\), with proper time \(d\tau^2=-\eta_{\mu\nu}d\xi^\mu d\xi^\nu/c^2\). This uses the prior result that Lorentz invariance fixes \(\eta_{\mu\nu}\). B

\[ \xi^{\mu}=\xi^{\mu}(x^{0},x^{1},x^{2},x^{3}),\qquad d\xi^{\mu}=\frac{\partial \xi^{\mu}}{\partial x^{\alpha}}\,dx^{\alpha}. \]

The local inertial frame exists only in a neighbourhood of one event; to describe an extended region we must patch these frames together with general (curvilinear, possibly accelerating) laboratory coordinates \(x^\alpha\). The Jacobian \(\partial\xi^\mu/\partial x^\alpha\) varies from event to event. C

\[ ds^{2}=\eta_{\mu\nu}\frac{\partial \xi^{\mu}}{\partial x^{\alpha}}\frac{\partial \xi^{\nu}}{\partial x^{\beta}}\,dx^{\alpha}dx^{\beta} \equiv g_{\alpha\beta}(x)\,dx^{\alpha}dx^{\beta}. \]

Substitute the differentials of Step 6 into the invariant interval of Step 5. The interval is coordinate-independent, but its components are now a position-dependent symmetric tensor. This defines the metric. B

\[ g_{\alpha\beta}(x)=\eta_{\mu\nu}\,\frac{\partial \xi^{\mu}}{\partial x^{\alpha}}\,\frac{\partial \xi^{\nu}}{\partial x^{\beta}} . \]

If the map \(\xi(x)\) were a global Lorentz/Poincaré transformation, \(g_{\alpha\beta}\) would just be \(\eta_{\alpha\beta}\) in disguise (flat). Gravity is precisely the case where no global \(\xi\) linearises the whole field, so \(g_{\alpha\beta}\) is genuinely \(x\)-dependent and cannot be transformed to \(\eta\) everywhere at once. C

\[ \frac{d^{2}\xi^{\mu}}{d\tau^{2}}=0 \;\xrightarrow{\ \xi=\xi(x)\ }\; \frac{d^{2}x^{\lambda}}{d\tau^{2}}+\underbrace{\frac{\partial x^{\lambda}}{\partial \xi^{\mu}}\frac{\partial^{2}\xi^{\mu}}{\partial x^{\alpha}\partial x^{\beta}}}_{\displaystyle \Gamma^{\lambda}{}_{\alpha\beta}}\frac{dx^{\alpha}}{d\tau}\frac{dx^{\beta}}{d\tau}=0 . \]

Apply the chain rule to the flat-frame free equation \(d^2\xi^\mu/d\tau^2=0\). The second derivative of the coordinate map cannot be removed globally and appears as a new term. Rename it \(\Gamma^{\lambda}{}_{\alpha\beta}\): the affine connection. C

\[ \Gamma^{\lambda}{}_{\alpha\beta} =\tfrac{1}{2}g^{\lambda\rho}\!\left(\partial_{\alpha}g_{\rho\beta}+\partial_{\beta}g_{\rho\alpha}-\partial_{\rho}g_{\alpha\beta}\right). \]

Compute \(\partial_\alpha g_{\rho\beta}\) from Step 8, symmetrise the three cyclic permutations; the second-derivative-of-\(\xi\) terms combine so the connection is expressed purely through first derivatives of the metric (metric compatibility \(\nabla_\lambda g_{\alpha\beta}=0\) and \(\Gamma^\lambda{}_{[\alpha\beta]}=0\)). The gravitational field is now \(\partial g\). C

\[ \Gamma^{i}{}_{00}\approx \frac{1}{c^{2}}\,\partial_{i}\Phi,\qquad g_{00}\approx-\Big(1+\frac{2\Phi}{c^{2}}\Big) \quad (\text{weak, static, slow}). \]

In the Newtonian limit (\(|\Phi|\ll c^2\), velocities \(\ll c\)) the geodesic equation reduces to \(\ddot{x}^i=-\partial_i\Phi\), recovering \(\vec a=-\nabla\Phi=\vec g\) of Steps 1–2. The metric carries Newtonian gravity, confirming the identification. C

Result

\[ \boxed{\; \eta_{\mu\nu}\ \longrightarrow\ g_{\mu\nu}(x),\qquad \frac{d^{2}x^{\lambda}}{d\tau^{2}}+\Gamma^{\lambda}{}_{\alpha\beta}\frac{dx^{\alpha}}{d\tau}\frac{dx^{\beta}}{d\tau}=0,\qquad \Gamma^{\lambda}{}_{\alpha\beta}=\tfrac{1}{2}g^{\lambda\rho}\big(\partial_\alpha g_{\rho\beta}+\partial_\beta g_{\rho\alpha}-\partial_\rho g_{\alpha\beta}\big)\;} \]

Reading. Universality of free fall means the "force of gravity" can be transformed away locally by falling with the field. What cannot be transformed away globally is the way neighbouring free-fall frames fail to align — encoded in a position-dependent metric \(g_{\mu\nu}(x)\). Free particles then move on geodesics, and the connection \(\Gamma^\lambda{}_{\alpha\beta}\), built from first derivatives of \(g\), plays the role of the gravitational field. Gravity is not a force on flat spacetime; it is the curvature of the metric. (Genuine curvature — irremovable at second order — is the tidal part; see Discussion.)

Units check. \(g_{\mu\nu}\) is dimensionless when coordinates carry length (with \(x^0=ct\)); \(ds^2\) then has units of \(\mathrm{m^2}\). \(\Gamma^\lambda{}_{\alpha\beta}=\tfrac12 g^{\lambda\rho}\partial g\) has units \(\mathrm{m^{-1}}\), so \(\Gamma\,\dot x\,\dot x\) with \(\dot x=dx/d\tau\ [\mathrm{m\,s^{-1}}]\) gives \(\mathrm{m^{-1}(m\,s^{-1})^2}=\mathrm{m\,s^{-2}}\), matching \(d^2x/d\tau^2\). In the Newtonian term \(\Gamma^i{}_{00}=c^{-2}\partial_i\Phi\): \([\Phi]=\mathrm{m^2\,s^{-2}}\), so \([c^{-2}\partial_i\Phi]=\mathrm{s^2\,m^{-2}\cdot m\,s^{-2}}=\mathrm{m^{-1}}\). Consistent.

Limiting cases

No gravity: \(\xi(x)\) is a global Poincaré transformation, \(g_{\mu\nu}\to\eta_{\mu\nu}\), all \(\Gamma\to 0\), and geodesics are straight lines — special relativity restored.
Uniform field: a single accelerated frame (Rindler coordinates) makes \(\Gamma\neq0\) but the Riemann tensor \(R^\lambda{}_{\mu\alpha\beta}=0\); spacetime is flat, gravity is entirely a "fictitious" inertial effect, removable globally.
Weak/static/slow (Newtonian): \(g_{00}=-(1+2\Phi/c^2)\) and geodesics reproduce \(\ddot{\vec x}=-\nabla\Phi\).
Strong field: full nonlinear \(g_{\mu\nu}(x)\) with \(R^\lambda{}_{\mu\alpha\beta}\neq0\) (Schwarzschild, Kerr); irreducible tidal curvature remains.

Breaks when

Tidal (non-uniform) fields over extended regions. The equivalence principle is only local. A single freely falling frame cancels \(\vec g\) at one event; over a finite region the field varies and residual tidal accelerations \(\sim \partial_i\partial_j\Phi\) survive. These are genuine curvature (\(R^\lambda{}_{\mu\alpha\beta}\neq0\)) and cannot be transformed away — that is precisely what distinguishes real gravity from a mere accelerated frame.
Composition-dependent forces (WEP violation). If \(m_{\mathrm G}/m_{\mathrm I}\) varied between bodies — or a fifth force coupled to baryon number — different particles would fall differently and no single metric could describe all worldlines. The geometric picture fails; one recovers a force on a fixed background. Current Eötvös/MICROSCOPE bounds \(\eta\lesssim10^{-15}\) protect the assumption.
Spinning or extended bodies. Bodies with intrinsic spin or finite size feel curvature-coupling forces (Mathisson–Papapetrou–Dixon terms \(\sim R\,S\)) and do not follow geodesics; the clean "everything falls the same" statement holds only for structureless test particles.
Quantum / non-classical regimes. For a delocalised wavepacket the notion of a single trajectory (and hence a single local inertial frame along it) is ambiguous; the classical WEP statement no longer directly applies.

Failure modes

"Curvature = a nonzero \(\Gamma\)." False. \(\Gamma\) can be nonzero in flat spacetime (accelerated/curvilinear coordinates). Curvature is the tensor \(R^\lambda{}_{\mu\alpha\beta}\), built from \(\partial\Gamma+\Gamma\Gamma\); it is the coordinate-independent, irremovable part.
Treating the equivalence principle as global. Students write "gravity can always be transformed away." It can be transformed away only locally (one event, to first order). Tidal effects are the obstruction.
Confusing \(m_{\mathrm G}=m_{\mathrm I}\) with \(m_{\mathrm G}/m_{\mathrm I}=\text{const}\). Only the universality of the ratio matters; a global constant could be absorbed into \(G\). The physics is that it is the same ratio for all bodies.
Forgetting proper time. Parameterising geodesics by coordinate time instead of \(\tau\) (or an affine parameter for null geodesics) produces spurious extra terms and a non-geodesic-looking equation.
Index errors in \(\Gamma\). Dropping the symmetry \(\Gamma^\lambda{}_{\alpha\beta}=\Gamma^\lambda{}_{\beta\alpha}\), or mis-signing the \(-\partial_\rho g_{\alpha\beta}\) term, yields a non-metric connection with wrong Newtonian limit.
Believing the derivation gives the field equations. The equivalence principle fixes how matter moves in a given \(g_{\mu\nu}\) (geodesics) but not how matter sources \(g_{\mu\nu}\); that requires the Einstein equations, an independent input.

Discussion

The deep point is a cancellation. In electromagnetism the trajectory of a charge depends on \(q/m\), which differs from particle to particle, so the electromagnetic field is unambiguously a field acting on matter that lives on a separate spacetime stage. Gravity's coupling is the inertial mass itself, so \(q_{\mathrm{grav}}/m = m_{\mathrm I}/m_{\mathrm I}=1\) universally, and the coupling disappears from the equation of motion. A property that cancels for all matter is not a property of matter — it is a property of the arena. That is the logical hinge that turns gravity into geometry.

Locally, gravity is indistinguishable from acceleration (the elevator), which is why it can always be removed at a point: the metric can be brought to \(g_{\mu\nu}(p)=\eta_{\mu\nu}\), \(\partial_\lambda g_{\mu\nu}(p)=0\) in a local inertial (Riemann normal) frame. What cannot be removed is the second derivative: \(\partial^2 g\) contains the Riemann tensor \(R^\lambda{}_{\mu\alpha\beta}\), and \(R\neq0\) is the invariant signature of real gravity. Geodesic deviation, \(\frac{D^2\xi^\mu}{d\tau^2}=-R^\mu{}_{\alpha\nu\beta}u^\alpha\xi^\nu u^\beta\), shows that neighbouring free-fallers accelerate relative to each other in proportion to curvature — the operational meaning of tidal gravity.

Rigorously, the equivalence principle selects the connection but not uniquely the metric one. Any symmetric affine connection defines geodesics and parallel transport; demanding in addition that lengths and angles measured by ideal rods and clocks are preserved along transport (local Lorentz structure, \(\nabla_\lambda g_{\mu\nu}=0\)) plus zero torsion forces the connection to be the unique metric (Levi-Civita) connection of Step 10. Drop metric compatibility and one has metric-affine or teleparallel formulations; drop the symmetry and torsion appears (Einstein–Cartan). The observed universality of free fall for structureless bodies is what empirically privileges the torsion-free metric connection, but the equivalence principle alone is strictly weaker than "spacetime is pseudo-Riemannian" — this is a genuine and often-glossed subtlety.

Common misconceptions. (i) "Curved spacetime means space is bent like a rubber sheet." The rubber-sheet picture is a two-dimensional spatial cartoon; the physically dominant curvature for slow motion is in the time component \(g_{00}\), i.e. gravitational time dilation, not spatial bending. (ii) "The equivalence principle proves general relativity." It proves gravity must be metric/geometric and fixes free-fall as geodesic motion; it does not, by itself, give the dynamics \(G_{\mu\nu}=8\pi G T_{\mu\nu}/c^4\). (iii) "Because gravity can be transformed away, it isn't real." Only the uniform part is a coordinate artefact; tidal curvature is coordinate-invariant and physical.

Worked examples

\[ \text{Elevator: does light bend? Uniform } g,\ \text{frame accelerates up at } a=g. \]

By the equivalence principle a uniform gravitational field \(g\) is locally identical to an upward acceleration \(a=g\). In the accelerating frame a horizontal light ray traces a parabola; hence it must also bend in the field. A

\[ \text{Crossing time } t=\frac{L}{c},\qquad \text{transverse drop } \delta=\tfrac{1}{2}g\,t^{2}=\frac{gL^{2}}{2c^{2}}. \]

Kinematics of the accelerated frame: floor rises \(\tfrac12 a t^2\), so the beam appears to fall by \(\delta\). Deflection angle \(\theta\approx d\delta/dx = gL/c^2\). B

\[ L=10\ \mathrm{m},\ g=9.81\ \mathrm{m\,s^{-2}},\ c=3.00\times10^{8}\ \mathrm{m\,s^{-1}}:\quad \delta=\frac{9.81\times10^{2}}{2(3.00\times10^{8})^{2}}. \]

Insert numbers. \(\delta = 981/(1.8\times10^{17})=5.45\times10^{-15}\ \mathrm{m}\). A

\[ \boxed{\ \delta \approx 5.5\times10^{-15}\ \mathrm{m},\qquad \theta=\frac{gL}{c^2}\approx1.1\times10^{-15}\ \mathrm{rad}\ } \]

Reading. Even the equivalence-principle "uniform field" argument predicts light must fall — tiny in a lab, but the same effect, integrated along a ray grazing the Sun and doubled by spatial curvature, gives the famous \(1.75''\) deflection. Light bending is forced the instant gravity becomes geometry.

\[ \text{Gravitational time dilation from } g_{00}=-\Big(1+\tfrac{2\Phi}{c^{2}}\Big),\quad d\tau=\sqrt{-g_{00}}\,\frac{dx^0}{c}. \]

Two static clocks at potentials \(\Phi_1,\Phi_2\). Proper time relates to coordinate time through the metric component derived in Step 11. B

\[ \frac{d\tau_2}{d\tau_1}=\sqrt{\frac{1+2\Phi_2/c^2}{1+2\Phi_1/c^2}}\approx 1+\frac{\Phi_2-\Phi_1}{c^{2}} =1+\frac{g\,\Delta h}{c^{2}}. \]

Expand to first order in \(\Phi/c^2\); near Earth \(\Phi_2-\Phi_1=g\,\Delta h\) for height difference \(\Delta h\). Higher clock runs faster. B

\[ \Delta h=22.5\ \mathrm{m},\ g=9.81\ \mathrm{m\,s^{-2}},\ c=3.00\times10^{8}\ \mathrm{m\,s^{-1}}:\quad \frac{\Delta\tau}{\tau}=\frac{9.81\times22.5}{(3.00\times10^{8})^{2}}. \]

Pound–Rebka geometry. \(\Delta\tau/\tau = 220.7/9.0\times10^{16}=2.45\times10^{-15}\). A

\[ \boxed{\ \frac{\Delta\tau}{\tau}=\frac{g\,\Delta h}{c^2}\approx 2.5\times10^{-15}\ } \]

Reading. A clock 22.5 m higher ticks faster by \(2.5\times10^{-15}\) — the fractional blueshift Pound and Rebka measured in 1959. This is curvature in \(g_{00}\): the "bending of spacetime" that governs everyday free fall is overwhelmingly time curvature, and it is measurable with a tower and a Mössbauer source.

Problems

A MICROSCOPE-type experiment compares the free-fall of platinum and titanium test masses and finds accelerations differing by no more than \(|\Delta a/a|\le 1\times10^{-15}\). Express this as a bound on the Eötvös parameter \(\eta=2\frac{|a_1-a_2|}{a_1+a_2}\) and state what it protects.
Solution
The Eötvös parameter is \(\eta=2\frac{|a_1-a_2|}{a_1+a_2}\approx |\Delta a/a|\le 1\times10^{-15}\). This bounds any composition dependence of \(m_{\mathrm G}/m_{\mathrm I}\) to one part in \(10^{15}\), protecting the WEP (Assumption 1). Any larger violation would mean different materials trace different worldlines, so no single metric \(g_{\mu\nu}\) could describe all free-fall — the geometric picture would break and gravity would revert to a composition-dependent force.
Show explicitly that in flat spacetime described in rotating coordinates the connection \(\Gamma\) is nonzero yet the Riemann tensor vanishes. Use this to argue that "\(\Gamma\neq0\)" is not curvature. (Qualitative reasoning acceptable.)
Solution
Transform Minkowski coordinates \((t,x,y)\) to rotating ones \(x=x'\cos\omega t-y'\sin\omega t\), etc. The metric acquires off-diagonal, position-dependent terms (e.g. \(g_{0i}\sim\omega\)), so \(\partial g\neq0\) and \(\Gamma\neq0\) — these produce the centrifugal and Coriolis "forces." But the map from the rotating frame back to \((t,x,y)\) is a global diffeomorphism to genuine Minkowski space, whose Riemann tensor is identically zero; \(R^\lambda{}_{\mu\alpha\beta}\) is a tensor, so it vanishes in every frame. Hence \(R=0\) despite \(\Gamma\neq0\): the connection can be entirely inertial. Curvature is the coordinate-independent obstruction to global flatness, not the mere presence of \(\Gamma\).
Starting from \(g_{00}=-(1+2\Phi/c^2)\), derive the geodesic equation's Newtonian limit and show \(\Gamma^i{}_{00}\approx c^{-2}\partial_i\Phi\) leads to \(\ddot{x}^i=-\partial_i\Phi\). State the approximations used.
Solution
Slow motion: \(dx^i/d\tau\ll dx^0/d\tau\approx c\), so in \(\ddot x^\lambda+\Gamma^\lambda{}_{\alpha\beta}\dot x^\alpha\dot x^\beta=0\) keep only \(\alpha=\beta=0\): \(\ddot x^i\approx-\Gamma^i{}_{00}(\dot x^0)^2\). Static field: \(\partial_0 g=0\), so \(\Gamma^i{}_{00}=-\tfrac12 g^{ij}\partial_j g_{00}=\tfrac12\partial_i(2\Phi/c^2)=c^{-2}\partial_i\Phi\) (using \(g^{ij}\approx\delta^{ij}\), \(g_{00}\approx-1\)). With \(\dot x^0=c\,dt/d\tau\approx c\), \(\ddot x^i\approx -c^{-2}\partial_i\Phi\cdot c^2=-\partial_i\Phi\). Thus \(\vec a=-\nabla\Phi=\vec g\), recovering Newton. Approximations: weak field \(|\Phi|\ll c^2\), static, non-relativistic velocities.
An astronaut in a windowless capsule measures no gravitational acceleration and no tidal force over the capsule's size \(\ell=2\ \mathrm{m}\), with instruments sensitive to relative acceleration \(\ge 10^{-9}\ \mathrm{m\,s^{-2}}\). If the capsule is in free fall near Earth (\(r=6.4\times10^{6}\ \mathrm{m}\), \(M=6.0\times10^{24}\ \mathrm{kg}\)), can she detect that she is in a real gravitational field rather than deep space? Tidal acceleration across \(\ell\) is \(\Delta a=\frac{2GM}{r^3}\ell\).
Solution
\(\Delta a=\dfrac{2GM\ell}{r^3}=\dfrac{2(6.67\times10^{-11})(6.0\times10^{24})(2)}{(6.4\times10^{6})^3}\). Numerator \(=2\times6.67\times10^{-11}\times6.0\times10^{24}\times2=1.60\times10^{15}\). Denominator \(=(6.4\times10^{6})^3=2.62\times10^{20}\). So \(\Delta a=6.1\times10^{-6}\ \mathrm{m\,s^{-2}}\), far above the \(10^{-9}\) threshold. Yes — she detects the tidal (curvature) signal, which cannot be transformed away. This is exactly why the equivalence principle is only local: a big enough or sensitive enough capsule reveals \(R\neq0\).
The equivalence principle is sometimes stated as "spacetime is pseudo-Riemannian." Explain, with reference to Step 10 and the Discussion, why this is stronger than what free-fall universality strictly implies, and name one alternative geometry consistent with universal free fall.
Solution
Universality of free fall implies free particles follow the geodesics of some symmetric affine connection \(\Gamma^\lambda{}_{\alpha\beta}\) (Step 9) — parallel transport and "straightest lines" are defined without any metric. Pseudo-Riemannian geometry additionally requires a metric \(g_{\mu\nu}\) with \(\nabla_\lambda g_{\mu\nu}=0\) (metric compatibility) and zero torsion, which uniquely fixes \(\Gamma\) to Levi-Civita (Step 10). Free-fall universality alone does not force metric compatibility; it is an extra empirical input (local Lorentz invariance / clock behaviour, the Einstein equivalence principle). An alternative consistent with universal free fall is teleparallel gravity (curvature zero, torsion carries gravity) or, allowing torsion, Einstein–Cartan geometry. Both reproduce the same test-particle geodesics for structureless bodies, showing the geometric requirement is weaker than "pseudo-Riemannian."