Redshift

z = λ o b s v − λ e m i t λ e m i t {\displaystyle z={\frac {\lambda _{\mathrm {obsv} }-\lambda _{\mathrm {emit} }}{\lambda _{\mathrm {emit} }}}}

z = f e m i t − f o b s v f o b s v {\displaystyle z={\frac {f_{\mathrm {emit} }-f_{\mathrm {obsv} }}{f_{\mathrm {obsv} }}}}

1 + z = λ o b s v λ e m i t {\displaystyle 1+z={\frac {\lambda _{\mathrm {obsv} }}{\lambda _{\mathrm {emit} }}}}

1 + z = f e m i t f o b s v {\displaystyle 1+z={\frac {f_{\mathrm {emit} }}{f_{\mathrm {obsv} }}}}

Relativistic Doppler

Minkowski space(flat spacetime)

For motion completely in the radial orline-of-sight direction: 1 + z = γ ( 1 + v ∥ c ) = 1 + v ∥ c 1 − v ∥ c {\displaystyle 1+z=\gamma \left(1+{\frac {v_{\parallel }}{c}}\right)={\sqrt {\frac {1+{\frac {v_{\parallel }}{c}}}{1-{\frac {v_{\parallel }}{c}}}}}} z ≈ v ∥ c {\displaystyle z\approx {\frac {v_{\parallel }}{c}}} for small v ∥ {\displaystyle v_{\parallel }} For motion completely in the transverse direction: 1 + z = 1 1 − v ⊥ 2 c 2 {\displaystyle 1+z={\frac {1}{\sqrt {1-{\frac {v_{\perp }^{2}}{c^{2}}}}}}} z ≈ 1 2 ( v ⊥ c ) 2 {\displaystyle z\approx {\frac {1}{2}}\left({\frac {v_{\perp }}{c}}\right)^{2}} for small v ⊥ {\displaystyle v_{\perp }}

Cosmological redshift

FLRW spacetime(expanding Big Bang universe)

1 + z = a n o w a t h e n {\displaystyle 1+z={\frac {a_{\mathrm {now} }}{a_{\mathrm {then} }}}} Hubble's law: z ≈ H 0 D c {\displaystyle z\approx {\frac {H_{0}D}{c}}} for D ≪ c H 0 {\displaystyle D\ll {\frac {c}{H_{0}}}}

Gravitational redshift

Any stationary spacetime

1 + z = g t t ( receiver ) g t t ( source ) {\displaystyle 1+z={\sqrt {\frac {g_{tt}({\text{receiver}})}{g_{tt}({\text{source}})}}}} For the Schwarzschild geometry: 1 + z = 1 − r S r receiver 1 − r S r source = 1 − 2 G M c 2 r receiver 1 − 2 G M c 2 r source {\displaystyle 1+z={\sqrt {\frac {1-{\frac {r_{S}}{r_{\text{receiver}}}}}{1-{\frac {r_{S}}{r_{\text{source}}}}}}}={\sqrt {\frac {1-{\frac {2GM}{c^{2}r_{\text{receiver}}}}}{1-{\frac {2GM}{c^{2}r_{\text{source}}}}}}}} z ≈ 1 2 ( r S r source − r S r receiver ) {\displaystyle z\approx {\frac {1}{2}}\left({\frac {r_{S}}{r_{\text{source}}}}-{\frac {r_{S}}{r_{\text{receiver}}}}\right)} for r ≫ r S {\displaystyle r\gg r_{S}} In terms of escape velocity: z ≈ 1 2 ( v e c ) source 2 − 1 2 ( v e c ) receiver 2 {\displaystyle z\approx {\frac {1}{2}}\left({\frac {v_{\text{e}}}{c}}\right)_{\text{source}}^{2}-{\frac {1}{2}}\left({\frac {v_{\text{e}}}{c}}\right)_{\text{receiver}}^{2}} for v e ≪ c {\displaystyle v_{\text{e}}\ll c}