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

(b).

• (a) and (c) are the weak equivalence principle — they only talk about falling and masses.
• (b) is the Einstein version: it upgrades the statement to all of physics inside the box, not just falling. That "you can't tell the difference with any experiment" is the extra bite that lets Einstein turn gravity into geometry.

(b). A geodesic is the generalisation of "straight line" to a curved surface — it is the path you follow when no force pushes you sideways. A free-faller travels a geodesic. See Geodesics & Curvature.

WHAT: Newton's 2nd law $F=m_{i}a$; gravity $F=m_{g}g$. Set equal. $m_{i}a=m_{g}g\Rightarrow a=\frac{m_g}{m_i}g.$ WHY the masses cancel: the equivalence principle says $m_g/m_i=1$ for everything, so $a=g$ regardless of mass. $a_1=a_2=9.8{\text{m/s}}^2.$ The marble and the shot-put accelerate identically. Mass never enters the answer — that cancellation is the seed of GR.

WHY this formula: a photon climbing out of a gravity well loses "effective energy," so its frequency drops — this is Gravitational Time Dilation read as a colour shift. $\frac{\Delta f}{f}=\frac{gh}{c^2}=\frac{9.8\times 22.5}{(3\times 10^8{)}^2}=\frac{220.5}{9\times 10^{16}}\approx 2.45\times 10^{-15}.$ Tiny — but Pound and Rebka measured it in 1959.

WHAT (look at the figure): the beam travels straight in space, but the box rises under it, so from inside the beam curves down.

• Time to cross: $t=L/c=3.0/(3\times 10^8)=1.0\times 10^{-8}\text{s}$.
• Box rises like any accelerating thing: $\Delta y=\frac{1}{2}g{t}^2$. $\Delta y=\frac{1}{2}(9.8)(1.0\times 10^{-8}{)}^2=4.9\times 10^{-16}\text{m}.$ WHY it matters: By the Einstein EP, the same bend must happen in real gravity — gravity bends light. See Gravitational Lensing.

WHICH is faster: the top clock — higher gravitational potential means faster ticking. $\frac{\Delta \tau }{\tau }=\frac{gh}{c^2}=\frac{9.8\times 100}{9\times 10^{16}}=1.089\times 10^{-14}.$ So the top clock gains about $1.09\times 10^{-14}$ seconds per second — over one full day ($86400$ s) that is roughly $9.4\times 10^{-10}$ s $\approx 0.94$ ns.

WHAT (see figure): each ball falls toward Earth's centre, not "straight down parallel." Two lines aimed at the same centre must converge. WHY this kills the force picture: you can cancel gravity in one small box by free-falling — but you cannot make both balls' paths straight at once. This irreducible convergence of free-fallers is the tidal effect, and it is exactly what mathematicians call spacetime curvature. See Geodesics & Curvature.

WHAT: the two effects have opposite signs, so add them: $\Delta t_{\text{net}}=+45+(-7)=+38\mu \text{s/day}.$ WHY both: velocity (SR, Special Relativity) always slows a moving clock; altitude (GR) speeds a high clock. GPS is high and fast, so both matter. Ignore either and positions drift kilometres per day. See GPS Corrections.

WHAT: $\text{GR}/\text{Newton}=1.75/0.875=2.00$ exactly. WHY the factor 2: Newtonian gravity only curves the time part of spacetime (energy in a potential). GR curves both space and time, and the space-curvature contributes an equal second half. The photon "falls" through curved time and rides curved space — two equal contributions. The lesson: matching Newton isn't enough; the extra factor of 2 (measured by Eddington, 1919) is a direct fingerprint of space itself being curved, not just a force acting.

Step 1 — photon's effective mass. Energy $E=hf$ (Planck), and mass-energy $E=m{c}^2$, so effective mass $m_{\text{eff}}=hf/c^2$. WHY: to talk about a photon "climbing" against gravity we need something for gravity to grab — its energy plays the role of mass. Step 2 — energy lost climbing height $h$. Potential energy gained $=m_{\text{eff}}gh=\frac{hf}{c^2}gh$. WHY: lifting mass $m$ by height $h$ in field $g$ costs $mgh$ — standard mechanics. Step 3 — the photon pays with its own energy. It cannot slow down (light is always $c$), so it pays by losing frequency: $\Delta E=-\frac{hf}{c^2}gh,\Delta E=h\Delta f.$ Divide by $E=hf$: $\frac{\Delta f}{f}=\frac{\Delta E}{E}=-\frac{gh}{c^2}.$ WHAT it means: the minus sign = the frequency drops climbing up (redshift). The receiver's clock therefore ticks faster by the same $+gh/c^2$.

WHY this formula: the Newtonian potential is $\Phi =-GM/r$. A clock higher up (less negative $\Phi$) runs faster; the fractional rate difference is $\Delta \Phi /c^2$. We want (satellite − ground), and since ${\Phi }_{\text{sat}}-{\Phi }_{\text{ground}}=-\frac{GM}{r}+\frac{GM}{R}=GM\left(\frac{1}{R}-\frac{1}{r}\right)>0$, the satellite gains. Numbers: $\frac{1}{R}-\frac{1}{r}=\frac{1}{6.37\times 10^6}-\frac{1}{2.66\times 10^7}=1.5699\times 10^{-7}-3.759\times 10^{-8}=1.194\times 10^{-7}{\text{m}}^{-1}.$ $\frac{GM}{c^2}=\frac{6.674\times 10^{-11}\times 5.97\times 10^{24}}{(3\times 10^8{)}^2}=\frac{3.984\times 10^{14}}{9\times 10^{16}}=4.427\times 10^{-3}\text{m}.$ $\frac{\Delta \tau }{\tau }=4.427\times 10^{-3}\times 1.194\times 10^{-7}=5.285\times 10^{-10}.$ Over one day: $\Delta \tau =5.285\times 10^{-10}\times 86400=4.57\times 10^{-5}\text{s}\approx 45.7\mu \text{s/day}.$ WHAT it means: this recovers the famous "$\approx 45\mu$s/day" gravitational speed-up — the number your phone corrects for every day. See GPS Corrections.

The chain: Experiment shows $m_i=m_g$ to $\sim 10^{-15}$, so all objects fall identically — mass cancels. Einstein promoted this to the equivalence principle: a free-falling lab is locally indistinguishable from no gravity, and an accelerating box mimics a gravitational field. From the accelerating box we derive that light bends (the floor rises under the beam) and clocks lower down run slow (climbing photons redshift). But the box trick only works locally — two distant free-fallers still converge toward Earth's centre, a tidal effect no free-fall frame can erase. That irreducible convergence is spacetime curvature, so gravity is not a force but geometry: matter curves spacetime ($G_{\mu \nu }=\frac{8\pi G}{c^4}{T}_{\mu \nu }$), and free objects glide along geodesics — "matter tells spacetime how to curve; spacetime tells matter how to move."

Einstein EP in one line ::: A freely-falling lab is locally indistinguishable from one with no gravity. Why does mass cancel in free fall ::: Because $m_g/m_i=1$ for everything, so $a=g$ independent of mass. Sign of GPS net drift ::: Positive — satellite clock runs $\approx +38\mu$s/day fast. What can't free-fall remove ::: Tidal effects — the convergence of nearby geodesics = curvature. Newtonian light bending vs GR ::: Newton gives exactly half; GR adds the equal space-curvature part.