Equivalence Principle

Einstein's equivalence principle states that a gravitational field is locally indistinguishable from acceleration. This deep insight implies that gravity curves spacetime: clocks run slower in stronger gravitational fields, and light bends in gravitational potentials.

Key Concepts

Weak EP: gravitational mass = inertial mass
Einstein EP: no local experiment distinguishes gravity from acceleration
Gravitational time dilation: Δt_high = Δt_low √(1 + 2Φ/c²)
Gravitational redshift: Δν/ν = -ΔΦ/c²
Light deflection by gravity: 2GM/(c²b) (Newtonian), 4GM/(c²b) (GR)

Key Equations

Gravitational time dilation

\frac{d\tau}{dt} = \sqrt{1 + \frac{2\Phi}{c^2}} \approx 1 + \frac{\Phi}{c^2}

Gravitational redshift

z = \frac{\Delta\nu}{\nu} \approx -\frac{\Delta\Phi}{c^2} = \frac{GM}{rc^2}

GPS time correction

\Delta t_{grav} = \frac{GM_E}{c^2}\left(\frac{1}{R_E} - \frac{1}{r}\right)\times\text{period}

Light deflection (GR)

\delta\theta = \frac{4GM}{c^2 b}

Worked Example

Example Problem

Problem

Find the gravitational redshift z for a photon emitted from the solar surface (R_sun=7×10⁸ m, M_sun=2×10³⁰ kg) to infinity.

Solution

z = GM/(Rc²) = 6.674×10⁻¹¹×2×10³⁰/(7×10⁸×9×10¹⁶) = 1.335×10²⁰/6.3×10²⁵ = 2.12×10⁻⁶.

Practice

Exercises

7 problems

1 of 7

Find the gravitational time dilation factor Δτ/Δt on Earth's surface (R=6.371×10⁶ m, M=5.97×10²⁴ kg). Compute 1 + Φ/c² where Φ=-GM/R.

Your answer

2 of 7

GPS satellites orbit at r=2.66×10⁷ m (Φ_sat=-GM/r). The gravitational time dilation makes GPS clocks run fast by ~45.9 μs/day. An SR effect makes them run slow by 7.2 μs/day. Find net gain in μs/day.

Your answer

μs/day

3 of 7

The gravitational redshift of the Sun: z = GM_sun/(R_sun c²). M_sun=2×10³⁰ kg, R_sun=7×10⁸ m. Find z.

Unlock Exercise 3