Gravitational Redshift (GR)

Compact Object

Mass M (M_sun) 1.0 Radius R (km) 695700 Emitter r (R units) 1.5 Receiver r (R units) 5.0

Redshift Calculation

Schwarzschild r_s—

r_s / R (compactness)—

z (gravitational)—

Δλ/λ₀—

Freq ratio f_obs/f_emit—

Time dilation factor—

Schwarzschild Metric

ds² = −(1−r_s/r)c²dt²
+ (1−r_s/r)⁻¹dr²
+ r²dΩ²

r_s = 2GM/c² (Schwarzschild radius)

Gravitational redshift:
z = f_emit/f_obs − 1

1+z = √[(1−r_s/r₁)/(1−r_s/r₂)]

Pound-Rebka 1959: Δν/ν = gh/c²
= 2.5×10⁻¹⁵ (22m on Earth)

GPS correction: +45.9 μs/day (GR)
−7.1 μs/day (SR time dilation)