General relativity — equivalence principle, curved spacetime (overview): Question bank

Before we start, three words we lean on constantly — pinned here so no line uses them cold:

And the symbols that appear in the formulas — each earned before use:

Definition Symbol glossary (with sign conventions)

$g$ = the local gravitational acceleration (near Earth's surface $\approx 9.8 m/s^{2}$ ), always taken as a positive magnitude here.
$c$ = the speed of light in vacuum ( $\approx 3 \times 1 0^{8} m/s$ ), the same for every observer.
$G$ = Newton's gravitational constant ( $\approx 6.67 \times 1 0^{- 11} N\cdotm^{2} / kg^{2}$ ), the strength dial of gravity in both Newton's law and the Einstein equations.
$h$ is overloaded on purpose in physics, so we disambiguate: in $g h / c^{2}$ the symbol $h$ means a height difference (metres); in $E = h f$ (photon energy) the symbol $h$ means Planck's constant. They are unrelated — we write "height $h$ " or "Planck's $h$ " wherever confusion is possible.
$Δ τ$ = the difference in elapsed proper time between the two clocks being compared. Sign convention: $Δ τ / τ = + g h / c^{2}$ compares the higher clock to the lower one — a positive value means the higher clock runs faster (the lower clock runs slower). $Δ f / f$ likewise compares received vs emitted frequency.

True or false — justify

Every "T/F" answer below must carry a reason, and starts with the letter. Guessing the letter earns nothing.

T/F — A uniformly accelerating rocket in deep space is exactly the same as sitting on Earth's surface.

False — only locally true. Inside a small box over a short time they are indistinguishable (Einstein EP), but over a large region Earth's field points toward its centre and weakens with height, while the rocket's is perfectly uniform — so real gravity has tidal effects the rocket lacks.

T/F — In true free-fall you feel gravity pulling on you strongly.

False. In free-fall you feel weightless — locally gravity is "switched off." You only feel weight when a floor or seat stops you from following your geodesic. (See Gravitational Time Dilation for what free-fallers do still detect: nothing locally, only tidal effects at a distance.)

T/F — Because inertial mass equals gravitational mass, all objects fall with the same acceleration.

True. Setting

m_{i} a = m_{g} g

gives

a = (m_{g} / m_{i}) g

; since

m_{g} / m_{i} = 1

for everything,

a = g

for everything and the mass cancels — a cancellation unique to gravity.

T/F — $m_{i} = m_{g}$ is obvious because we write both as one symbol $m$ .

False. They enter logically independent laws — inertia (

F = m_{i} a

) versus gravitational coupling (

F = m_{g} g

). Their equality to

\sim 1 0^{- 15}

is a deep experimental fact, not a definition; it's the seed of GR.

T/F — Light bends near the Sun because photons have mass and gravity pulls on them.

False. Photons are massless; light bends because it follows geodesics in curved spacetime. The naive "photon mass" Newtonian guess predicts only half the measured deflection (see Gravitational Lensing).

T/F — Time dilation depends only on how fast a clock moves.

False. That's the special-relativistic (velocity) part. There is also a gravitational part depending on potential; for GPS Corrections the gravitational effect is larger and opposite in sign to the velocity effect.

T/F — A clock at the bottom of a tall building runs slower than one at the top.

True. Clocks deeper in a gravitational potential (lower down) run slower; comparing higher to lower gives

Δ τ / τ = + g h / c^{2} > 0

, i.e. the top clock gains time. The Pound–Rebka experiment measured exactly this over

22.5

T/F — General relativity throws away special relativity.

False. GR contains Special Relativity as the local limit: in any small free-falling patch, spacetime looks flat (Minkowski) and SR holds exactly. GR only adds curvature over larger regions.

Spot the error

Each statement below hides one flawed step. Name it.

"The equivalence principle lets us cancel gravity everywhere just by free-falling, so gravity is a fiction."

The error is "everywhere." Free-fall cancels gravity only locally; the tidal relative acceleration of nearby free-fallers cannot be removed — that irreducible part is real curvature.

"Two balls dropped side by side in a falling elevator drift apart, which proves gravity is a normal force between them."

They drift because both fall toward Earth's centre, so their paths converge — this is geodesic convergence, not a force between the balls. Naming it "a force between them" is the mistake.

"In the Einstein Field Equations $G_{μν} = \frac{8 π G}{c ^{4}} T_{μν}$ , the left side is the matter and the right side is the geometry."

Swapped.

G_{μν}

(left) is the geometry/curvature;

T_{μν}

(right) is the matter–energy (stress-energy). Wheeler: "matter tells spacetime how to curve."

"Since the elevator thought-experiment works, a real gravitational field is identical to acceleration in every way."

"In every way" is wrong. It's identical only in a small, brief region. Extended real fields have curvature (tidal effects) that no acceleration of a rigid frame can reproduce.

"Gravitational redshift means the photon slows down as it climbs."

Light never changes speed (

c

is invariant). What changes is its frequency (energy) — it redshifts — not its speed. Confusing "loses energy" with "slows down" is the error.

"The heuristic $Δ f / f = - g h / c^{2}$ derived from a photon losing potential energy $m g h$ is the full, exact GR result."

It is only a weak-field, uniform-field heuristic — it treats the photon as if it had Newtonian potential energy

m g h

(with effective mass

E / c^{2}

). It gives the right leading answer but is not the exact GR derivation, which uses the full metric. Presenting it as exact is the error.

"A bowling ball on a trampoline shows marbles get grabbed and pulled inward by the dent."

Nothing "grabs" them. The marbles roll along the bent surface (geodesics); the analogy illustrates that curvature guides motion — replace "grabbed and pulled" with "follow the bent floor." (See Geodesics & Curvature.)

Why questions

Why does mass cancel in gravitational free-fall but not in electric acceleration?

Gravity's "charge" is the gravitational mass, which equals the inertial mass, so

m

cancels in

a = (m_{g} / m_{i}) g

. For the electric force

a = q E / m

— the charge-to-mass ratio

q / m

varies between objects, so it does not cancel.

Why did Einstein promote $m_{i} = m_{g}$ from a coincidence to a principle?

Because the cancellation is perfect for every material tested to extreme precision. Something that exact demands a structural reason — and treating it as a principle led directly to interpreting gravity as geometry.

Why can't a "force" picture of gravity work globally?

A force acts between objects, but you can make gravitational acceleration vanish for any single free-faller. What survives is the relative acceleration (tidal effect) — which is a property of the geometry, not a two-body force, so geometry, not force, is the correct global description.

Why does a climbing photon lose energy, giving redshift (in the weak-field heuristic)?

Assigning the photon an effective "mass"

E / c^{2} = h f / c^{2}

(here

h

is Planck's constant), lifting it through height

h

(here

h

is height) costs potential energy

(h f / c^{2}) g h

; that energy comes out of the photon, lowering its frequency by

Δ f / f = - g h / c^{2}

. This is a heuristic valid in a weak, roughly uniform field — the exact result comes from the metric (see Gravitational Time Dilation).

Why is the GPS gravitational correction positive (satellite clock runs fast)?

The satellite sits higher in Earth's potential (weaker gravity), and higher-potential clocks run faster: comparing satellite to ground,

Δ τ / τ = + g h / c^{2} > 0

. Its velocity (SR) effect slows it, but the gravitational effect wins, so the net satellite clock runs fast.

Why does spacetime need a metric $g_{μν}$ rather than plain Euclidean distance?

Because "distance" in spacetime mixes time and space with a minus sign and can vary from point to point when curved. The metric is the local rule for measuring intervals; in flat regions it reduces to Minkowski (see Spacetime Metric & Minkowski Diagram).

Edge cases

What happens to the equivalence-principle "elevator" as you shrink the box and shorten the experiment toward zero?

It becomes exact: in the limit of an infinitesimal region, gravity and acceleration are truly indistinguishable and spacetime is exactly flat. The principle is precisely a local (limiting) statement.

What happens to the equivalence-principle equivalence as the box grows large?

It breaks down: over a large box the field's direction and strength vary, tidal effects appear, and no single acceleration can mimic them. Curvature becomes measurable — this is the boundary where "geometry" replaces "acceleration."

In the weak-field, slow-speed limit, what does GR reduce to?

It reproduces Newtonian Gravity — Einstein's equations recover

F = G m_{1} m_{2} / r^{2}

and

g = GM / r^{2}

when curvature is tiny and speeds

≪ c

. GR must contain the classical answer it aims to generalise.

What happens to gravitational time dilation as $g$ or $h \to 0$ ?

Δ τ / τ = g h / c^{2} \to 0

: with no field or no height difference there is no relative slowing — clocks tick together, as required, recovering flat-spacetime SR.

What is the limiting case where curvature becomes so extreme that even light cannot escape?

A black hole: the geometry bends so sharply that all future-pointing geodesics lead inward past the event horizon. It's the extreme end of the same curvature that mildly bends starlight in Gravitational Lensing.

What does the equivalence principle say about a charged particle sitting still on a table versus one accelerating in free space — does it radiate?

This is a genuine subtlety: a charge held static on Earth is accelerated (the table pushes it off its geodesic), while a free-falling charge is not accelerating in the geometric sense. The EP is strictly local, so naive "accelerated charges radiate" arguments must be applied with care to the local free-falling frame.

A picture for the two hardest ideas

Text can hide the geometry. Two sketches make the concept traps visual — the first shows why free-fall cannot cancel tidal effects (geodesic deviation), the second shows what "curved spacetime" means as a bent surface guiding a geodesic.

Figure — General relativity — equivalence principle, curved spacetime (overview)

Recall One-line summary to lock in

Locally, gravity = acceleration and can be cancelled by free-fall; what cannot be cancelled — tidal effects — is curvature, and curvature is the whole of gravity.

Connections

Parent overview — the concepts these traps test.
Special Relativity — the local flat-space limit these edge cases return to.
Gravitational Time Dilation · GPS Corrections — sign and magnitude traps.
Gravitational Lensing · Black Holes — extreme-curvature edge cases.
Newtonian Gravity · Geodesics & Curvature · Spacetime Metric & Minkowski Diagram — limits and machinery.