Gravitational Redshift

Equivalent forms

\frac{\nu_{rec}}{\nu_{em}} = \sqrt{\frac{1 - r_s/r_1}{1 - r_s/r_2}}

z \approx \frac{\Delta\Phi}{c^2}

A direct consequence of equivalence: gravity slows time, so light from below arrives redshifted with no need for the full machinery of GR.

Symbol	Name	SI unit	Dimension	Range
$z$	Redshiftoutput Fractional wavelength shift Δλ/λ	dimensionless	1	0 – 1
$M$	Mass Mass of gravitating body	kg	M	100000000000000000000 – 1e+40
$r$	Emission Radius Radius at which photon is emitted	m	L	1 – 1000000000000
$G$	Gravitational Constant Newton's gravitational constant	m^3·kg^-1·s^-2	M^-1L^3T^-2	6.674e-11 – 6.674e-11
$c$	Speed of Light Speed of light in vacuum	m/s	LT^-1	299792458 – 299792458

Unit systems

Symbol	SI	CGS	Imperial
$z$	1	1	1
$M$	kg	g	lb
$r$	m	cm	ft
$G$	m^3·kg^-1·s^-2	cm^3·g^-1·s^-2	ft^3·lb^-1·s^-2
$c$	m/s	cm/s	ft/s

Where it holds

Static spherical masses outside r_s. For strong fields, use exact Schwarzschild formula with square roots; weak-field formula valid for GM/(rc^2) ≪ 1.

Dimensional analysis

z \to [1]

[M^-1L^3T^-2][M] / ([L][LT^-1]^2) = [1]

Dimensionless.

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Discovery

Albert Einstein · 1907

Predicted in Einstein's 1907 equivalence-principle paper, eight years before GR was complete. Pound-Rebka confirmed it in 1959 at Harvard using Mössbauer spectroscopy over a 22.5 m tower.

Try this

How much does light reddened climbing out of the Sun's gravity?

Light leaving the Sun loses energy climbing out of its gravitational well: Δλ/λ ≈ GM/(Rc^2) ≈ 2.12e-6. Solar absorption lines shift by ~0.6 km/s — measured to confirm GR.

Research status: stable

Real-world applications

GPS frequency standards
Tests of equivalence principle
Black hole and neutron star spectroscopy
Cosmological redshift contribution (alongside expansion)

Common misconceptions

Gravitational redshift is NOT a Doppler effect — emitter and observer can be stationary
It's the same effect that makes lower clocks tick slower
Photons don't 'fight gravity' — spacetime geometry stretches their wavelength

Experimental verification

Pound-Rebka 1959 (Mössbauer Fe-57 over 22.5 m, agreement to 10%). Pound-Snider 1965 (1% agreement). Solar limb darkening shifts. White dwarf Sirius

B (z \approx 3e-4)

. Stars near Sgr A* show

z \approx 10^-4

from S2 orbit.

Derivation

From gravitational time dilation, frequency emitted at r_1 and received at r_2 satisfies

\nu _2/\nu _1 = \sqrt((1 - r_s/r_1)/(1 - r_s/r_2))

.

For

r_2 \to \infty

:

\nu _\infty /\nu

_emit

= \sqrt{1 - r_s/r}

.

Expanding to first order in r_s/r gives

z \approx GM/(rc^2)

.

Equivalently, photon energy

E = h\nu

loses GMmh/(rc^2 h) climbing out of the potential.

Limiting cases

weak field⟶

z \approx GM/(rc^2)

First-order Newtonian potential expansion.

r \to r_s

⟶

z \to \infty

Light emitted at the event horizon is infinitely redshifted.

r ≫ r_s⟶

z \to 0

Far from any mass, photons travel without shift.

What if…

What if light fell INTO a gravity well?

It blueshifts symmetrically: $\Delta \lambda /\lambda = -GM/(rc^2)$ . Photons gain energy climbing in.

What if you observed light from

r = 1.5\,\mathrm{r}_s

?

$z = \sqrt(1/(1 - 2/3)) - 1 \approx 0.73$ — significant non-perturbative redshift.

What if gravity didn't redshift light?

Energy conservation would fail: a photon could climb a tower, convert to mass, fall, convert back, and gain energy — a perpetual motion machine.

1

Sun's surface

Given ·

M:: 1.989e+30
r:: 696000000
G:: 6.674e-11

Find · z

Steps

$GM = 1.327e20$
$rc^2 = 6.96e_{8} * 8.99e16 = 6.26e25$
$z = 1.327e20 / 6.26e25 \approx 2.12e-6$

Result ·

z = GM/(rc^2) = 6.674e-11 * 1.989e30 / (6.96e_{8} * 8.99e16) \approx 2.12e-6

— corresponds to 636 m/s velocity shift.

2

White dwarf Sirius B

Given ·

M:: 2e+30
r:: 5840000
G:: 6.674e-11

Find · z

Steps

$GM = 1.335e20$
$rc^2 = 5.84e_{6} * 8.99e16 = 5.25e23$
$z = 1.335e20 / 5.25e23 \approx 2.54e-4$

Result ·

z \approx GM/(rc^2) \approx 2.54e-4

— observed value 2.7e-4 confirms GR.

Gravitational Time Dilation→Schwarzschild Radius→Spacetime Interval→relativistic doppler effect