Redshift and Blueshift

Point a spectrograph at almost anything beyond the Sun — a nearby star, a glowing gas cloud, a galaxy ten billion light-years away — and the same fingerprint appears: the sharp lines of hydrogen, calcium, sodium and iron, at wavelengths a laboratory can predict to many decimal places. Almost always they are not quite where the laboratory puts them. The entire pattern is slid bodily toward the red or the blue, and the fractional size of that slide is a single dimensionless number, the redshift $z\equiv(\lambda_{\rm obs}-\lambda_{\rm emit})/\lambda_{\rm emit}$ . Because it is read off the positions of identified lines rather than off a brightness, $z$ is among the most precisely measured quantities in astronomy — six significant figures for a bright quasar. That one number hides three utterly different pieces of physics, and conflating them is the classic error of the subject. A star or galaxy moving through space carries its lines with it by the Doppler effect. A photon crawling out of a deep gravitational potential arrives reddened because the emitting and receiving clocks disagree — the gravitational shift demanded by general relativity. And a photon crossing billions of light-years of expanding space is stretched by the expansion itself, the cosmological redshift, which is not a velocity through space at all. The exact formulae differ and the mechanisms compound multiplicatively in $(1+z)$ . This article assembles all three, checks each against the limits of the others, and shows why $z$ is the master key that turns a spectrum into a position in the three-dimensional universe.

Key takeaways

Redshift is a pure ratio, $z=(\lambda_{\rm obs}-\lambda_{\rm emit})/\lambda_{\rm emit}$ : positive for a shift to longer wavelengths, negative for a blueshift, read directly off identified spectral lines.
Three distinct mechanisms produce it: the Doppler effect from relative motion, the gravitational shift from light climbing a potential well, and the cosmological stretch from the expansion of space.
For slow motion all reduce to $z\approx v_r/c$ ; the exact radial law is $1+z=\sqrt{(1+\beta)/(1-\beta)}$ , and even transverse motion reddens light by the factor $\gamma$ .
The cosmological redshift is not a Doppler shift — wavelengths ride the scale factor, $1+z=a(t_{\rm obs})/a(t_{\rm emit})$ — which is why cosmic recession can exceed $c$ without breaking relativity.
Redshift is the workhorse of extragalactic astronomy: radial velocities, gravitational potentials of white dwarfs and clusters, and cosmological distances through Hubble's law $cz\approx H_0 d$ .
Key idea: one dimensionless number $z$ records the wavelength shift, but its meaning — velocity, potential, or expansion — is set entirely by the mechanism, and the exact relations must never be interchanged.

The core insight

A single dimensionless number $z$ encodes the wavelength shift, but its physical interpretation — velocity, gravitational potential, or cosmic expansion — depends entirely on the mechanism; the exact relations differ and must not be interchanged.

Where this lives today

Redshift is the single most productive measurement in modern cosmology. Massive spectroscopic surveys — the Sloan Digital Sky Survey, and now DESI, measuring tens of millions of galaxy and quasar redshifts — turn spectra into three-dimensional maps, reading the baryon acoustic oscillation scale off the clustering to pin down the expansion history and dark energy. The same physics drives the Hubble tension: the local distance ladder returns $H_0\approx73\,\mathrm{km\,s^{-1}\,Mpc^{-1}}$ while the cosmic microwave background, whose $z\approx1100$ is the largest redshift we observe, demands $\approx67$ — a discrepancy past five sigma that may signal missing physics. Gravitational redshift has moved from null test to working probe: the GRAVITY collaboration detected the combined gravitational and transverse-Doppler shift of the star S2 at pericenter around the four-million-solar-mass black hole at the Galactic Centre in 2018, and drifting Galileo satellites in eccentric orbits have confirmed the effect to a few parts in $10^5$ . At the extreme, JWST spectroscopy is confirming galaxies at $z>13$ — light emitted when the universe was under $350$ million years old — while a new observable, the slow drift of a redshift over decades ( $\dot z$ , the Sandage–Loeb effect), is a design goal of the Extremely Large Telescope.

The physical picture

Start with the most intuitive origin. A source emits wave crests at a fixed rate, one every period. If it is receding, each successive crest is launched from farther away and has farther to travel, so the crests arrive spaced out in time: the period lengthens, the wavelength stretches, and the light reddens. Approaching motion packs the crests into a blueshift. This is the Doppler effect, and in the slow limit it gives the workhorse relation $z\approx v_r/c$ directly, no relativity required. The gravitational shift follows from one sentence of energy accounting: a photon has energy $E=h\nu$ , and climbing out of a well it must do work against gravity. It cannot slow down — its speed is locked at $c$ — so the only way to pay the toll is to lower its frequency. Light leaving a compact star arrives reddened by the depth of the potential, $z\approx GM/Rc^2$ . Equivalently, the clock deep in the well runs slow, and a slow clock ticks out a longer period. The cosmological redshift is different in kind. The source is not chasing the light; instead the ruler of space stretches while the wave is in flight. A crest launched when the universe was half its present size arrives with twice the wavelength — the photon rides the expansion, and the shift measures the ratio of scale factors, not a velocity. The first figure sets the three side by side; the second plots the exact Doppler law and shows why the naive $z=\beta$ fails near the speed of light.

Figure

Three physics, one fingerprint: a Doppler shift from motion, a gravitational shift from a photon climbing out of a potential well, and a cosmological stretch from expanding space — for the Hα line at 6563 Å seen at 6600 Å the redshift is z = 0.0056

The Doppler picture is exact only in the slow limit; relativity modifies it in two ways. The source clock is time-dilated, so the classical crest-counting must be multiplied by $\gamma$ , and a source moving purely across the line of sight is still reddened by that dilation alone. The exact radial law $1+z=\sqrt{(1+\beta)/(1-\beta)}$ agrees with $z\approx\beta$ only when $\beta\ll1$ ; as $\beta\to1$ the true redshift diverges while the naive estimate creeps toward $z=1$ . The second figure is the quantity itself, and the gap between the curves is the correction relativity supplies.

Figure

The exact special-relativistic redshift 1 + z = √((1+β)/(1−β)) plotted against recession speed β = v/c, contrasted with the naive linear z = β: they agree for β ≪ 1 but diverge sharply near the speed of light — at β = 0.5 the true z = 0.732 not 0.5, at β = 0.9 the true z = 3.36 not 0.9, and z → ∞ as β → 1

The core equations

Everything begins with the operational definition: the fractional change in the wavelength of an identified spectral feature, positive when the light reddens.

z\equiv\frac{\lambda_{\rm obs}-\lambda_{\rm emit}}{\lambda_{\rm emit}}=\frac{\lambda_{\rm obs}}{\lambda_{\rm emit}}-1 \qquad (1)

where: $z$ — dimensionless redshift, negative for a blueshift; $\lambda_{\rm obs}$ — observed wavelength; $\lambda_{\rm emit}$ — rest-frame wavelength of the same transition, known from the laboratory. Reading: measure where a known line lands, divide the shift by the rest wavelength — you have $z$ before committing to any cause. For a source receding slowly, counting wave crests gives the first-order Doppler relation — the everyday tool of stellar spectroscopy, accurate whenever $v_r\ll c$ .

z\approx\frac{\Delta\lambda}{\lambda}=\frac{v_r}{c}\qquad (v_r\ll c) \qquad (2)

where: $v_r$ — radial (line-of-sight) velocity, positive for recession; $c=2.998\times10^{5}\,\mathrm{km\,s^{-1}}$ ; $\Delta\lambda=\lambda_{\rm obs}-\lambda_{\rm emit}$ . Reading: multiply $z$ by $c$ for the radial velocity — what "receding at 1700 km/s" means operationally. When the speed is not small the classical formula breaks down and the exact special-relativistic law takes over, folding in time dilation.

1+z=\sqrt{\frac{1+\beta}{1-\beta}}=\gamma\,(1+\beta),\qquad \gamma=\frac{1}{\sqrt{1-\beta^2}} \qquad (3)

where: $\beta=v/c$ — radial speed fraction, positive for recession; $\gamma$ — Lorentz factor. Reading: for $\beta\ll1$ this collapses to $z\approx\beta$ , but as $\beta\to1$ the redshift diverges — $z$ is unbounded for a source approaching the speed of light. Motion is rarely purely radial, so the general Doppler formula carries the angle between velocity and line of sight, giving a shift even at zero radial velocity.

1+z=\gamma\left(1+\frac{v}{c}\cos\theta\right) \qquad (4)

where: $\theta$ — angle between source velocity and line of sight; $v$ — source speed. Reading: $\theta=90^\circ$ leaves $1+z=\gamma$ , the transverse Doppler redshift from time dilation alone; $\theta=0$ recovers equation (3). The gravitational shift comes from the Schwarzschild metric: a photon emitted at radius $R$ from a mass $M$ and received far away is reddened by the depth of the potential it climbed.

1+z=\left(1-\frac{2GM}{Rc^2}\right)^{-1/2}\approx 1+\frac{GM}{Rc^2} \qquad (5)

where: $G=6.674\times10^{-11}\,\mathrm{m^3\,kg^{-1}\,s^{-2}}$ ; $M$ — mass of the body; $R$ — emission radius, observer at infinity. Reading: the exact form diverges as $R\to2GM/c^2$ (the horizon, infinite redshift), while $z\approx GM/Rc^2$ suffices for stars and white dwarfs. For light crossing the expanding universe the shift is not a velocity but the ratio of the scale factor now to its value at emission — the backbone of cosmology.

1+z=\frac{a(t_0)}{a(t_e)}\equiv\frac{a_0}{a_e} \qquad (6)

where: $a(t)$ — scale factor, with $a_0\equiv a(t_0)=1$ today by convention; $t_e,\,t_0$ — times of emission and observation. Reading: a galaxy at $z=1$ emitted its light when the universe was half its present size ( $a_e=1/2$ ); $z$ is a model-independent clock reading of cosmic epoch. At small redshift the cosmological stretch is linear in distance — Hubble's law — how redshift became a distance indicator and revealed the expansion.

cz\approx v_r=H_0\,d\qquad (z\ll1) \qquad (7)

where: $H_0\approx70\,\mathrm{km\,s^{-1}\,Mpc^{-1}}$ — Hubble constant; $d$ — proper distance. Reading: in the Hubble flow, redshift and distance are proportional, so a spectrum plus $H_0$ gives a distance with no standard candle. When several mechanisms act on one photon they do not add — they multiply in $(1+z)$ , because each stage rescales the wavelength the previous one produced.

1+z_{\rm tot}=(1+z_{\rm Dopp})(1+z_{\rm grav})(1+z_{\rm cosmo}) \qquad (8)

where: $z_{\rm tot}$ — total observed redshift; $z_i$ — contribution from each mechanism. Reading: a supernova in a distant cluster is Doppler-shifted by its peculiar velocity, gravitationally shifted leaving the cluster, and cosmologically stretched en route — only the product is observed.

Derivation AExact radial Doppler from Lorentz invariance of the phase

The goal is to derive the exact special-relativistic redshift $1+z=\sqrt{(1+\beta)/(1-\beta)}$ for a radially receding source, starting from a principle no observer can dispute — that all frames count the same number of wave crests between two events — combined with the time dilation of the moving source's clock. We want to see how the classical crest-counting factor $(1+\beta)$ and the relativistic factor $\gamma$ enter as two separate physical effects, and to confirm the law reduces to $z\approx\beta$ in the slow limit.

Step 1. Write the phase of a plane wave, $\phi=\omega t-\mathbf{k}\cdot\mathbf{x}$ , and note it is a Lorentz scalar: a crest is a crest in every frame, so crests emitted equal crests received. Why: this invariance guarantees frequency transforms as a physical quantity, not an artifact of coordinates.

Step 2. In the observer frame the source clock runs slow, so one emitted period $\Delta t'=1/\nu_{\rm emit}$ is stretched to $\Delta t=\gamma\,\Delta t'$ . Why: time dilation is the relativistic ingredient absent from the classical treatment, and it survives even for transverse motion.

Step 3. During that dilated period the source recedes $v\,\Delta t$ , so the next crest travels farther and the observed interval becomes $\Delta t_{\rm obs}=\gamma\,\Delta t'(1+\beta)$ . Why: this extra light-travel term is the classical Doppler factor; combining it with dilation is the whole content of the relativistic law.

Step 4. Convert to frequency via $\nu\propto1/\Delta t$ : $\nu_{\rm emit}/\nu_{\rm obs}=\gamma(1+\beta)=(1+\beta)/\sqrt{1-\beta^2}$ . Why: expressing the ratio in $\beta$ alone collapses the algebra to a compact form.

Step 5. Simplify $\sqrt{1-\beta^2}=\sqrt{(1-\beta)(1+\beta)}$ , so the ratio becomes $\sqrt{(1+\beta)/(1-\beta)}=\lambda_{\rm obs}/\lambda_{\rm emit}=1+z$ . Why: the symmetric form makes the limits clear — first order recovers $z\approx\beta$ , while $\beta\to1$ sends $z\to\infty$ . *Why the limits work: as $\beta\to0$ the square root collapses to $1+\beta\approx1+z$ , recovering $z\approx\beta$ ; as $\beta\to1$ the denominator vanishes and $z\to\infty$ , confirming the divergence seen in the figure.*

Go deeper · Derivation B · 1 minGravitational redshift from the equivalence principle

Derivation BGravitational redshift from the equivalence principle

The goal is to obtain the weak-field gravitational redshift $z\approx GM/Rc^2$ using nothing more than the equivalence principle, the photon energy relation $E=h\nu$ , and energy conservation — showing that a photon must redden as it climbs a potential well without invoking the full Schwarzschild metric. We treat the photon's energy as an effective gravitational mass $E/c^2$ , follow it up through a height, identify the fractional frequency loss with the potential change, then generalize from a uniform field to a point mass.

Step 1. Assign the photon an effective mass from its energy, $m=E/c^2=h_P\nu/c^2$ , with $h_P$ Planck's constant. Why: the equivalence principle insists all energy gravitates, so a photon feels the field though it is massless in the rest sense.

Step 2. Climbing a height $h$ against gravity $g$ costs work $\Delta E=-mgh=-(h_P\nu/c^2)gh$ . Why: it cannot pay by slowing down, its speed being fixed at $c$ , so the energy comes out of its frequency.

Step 3. With $E=h_P\nu$ the energy loss is a frequency loss, $\Delta\nu/\nu=-gh/c^2=-\Delta\Phi/c^2$ . Why: this is the master result — the fractional shift equals the potential difference in units of $c^2$ , valid for any weak field, not just a uniform one.

Step 4. For a spherical mass the potential difference between radius $R$ and infinity is $\Delta\Phi=GM/R$ , so the received light is redshifted by $z=-\Delta\nu/\nu\approx GM/Rc^2$ . Why: this converts the potential into the observable, exactly the leading term of the Schwarzschild factor in equation (5). *Why this is consistent: expanding $(1-2GM/Rc^2)^{-1/2}$ to first order in $GM/Rc^2$ returns exactly $z\approx GM/Rc^2$ , and the result is confirmed experimentally by the Pound–Rebka measurement and by observed white-dwarf and solar line shifts.*

Go deeper · Derivation C · 1 minCosmological redshift from null geodesics in an expanding universe

Derivation CCosmological redshift from null geodesics in an expanding universe

The goal is to derive the exact cosmological relation $1+z=a_0/a_e$ from the geometry of an expanding spacetime alone, showing it is not a Doppler shift of any kind but a consequence of light following null paths in a metric whose spatial part grows with time. We work in the flat Friedmann–Lemaître–Robertson–Walker metric, track two successive crests along the same comoving path, and show the ratio of received to emitted wavelength is exactly the ratio of scale factors — independent of the expansion history in between.

Step 1. Adopt the flat FLRW line element $\mathrm{d}s^2=-c^2\mathrm{d}t^2+a^2(t)\,\mathrm{d}\chi^2$ and impose $\mathrm{d}s=0$ for light, giving $c\,\mathrm{d}t=a(t)\,\mathrm{d}\chi$ along a radial ray. Why: the comoving coordinate $\chi$ is fixed to the galaxies, so all expansion is carried by $a(t)$ , isolating the geometry.

Step 2. A crest emitted at $t_e$ and one a period later at $t_e+\delta t_e$ each traverse the same comoving distance to the observer, so $\int_{t_e}^{t_0}c\,\mathrm{d}t/a=\int_{t_e+\delta t_e}^{t_0+\delta t_0}c\,\mathrm{d}t/a$ . Why: source and observer are both comoving, so the comoving separation is the same for both crests — the content of "the light rides the expansion".

Step 3. Subtracting the two integrals over infinitesimal periods leaves $\delta t_e/a(t_e)=\delta t_0/a(t_0)$ , so the observed period is stretched by the ratio of scale factors. Why: the integrands cancel except at the endpoints, turning a global path condition into a local statement about periods.

Step 4. Since wavelength is $c$ times the period, $\lambda_0/\lambda_e=\delta t_0/\delta t_e=a(t_0)/a(t_e)$ , so $1+z=a_0/a_e$ . Why: the result depends only on the endpoints $a_e$ and $a_0$ , never on the expansion rate in between — why $z$ is a clean clock reading of cosmic epoch. *Why the result is clean: the derivation depends only on the scale-factor values at the endpoints, never on the expansion rate $H(t)$ in between — so, for example, $a_e=1/2$ gives $1+z=2$ exactly, regardless of how the universe expanded along the way.*

Limiting cases

The general framework specializes into five regimes, each with its own law and physical meaning.

Case	Result	Meaning
Non-relativistic Doppler	$z\approx v_r/c,\ \ v_r\ll c$	radial velocity read directly from the line shift; sign positive for recession
Relativistic radial Doppler	$1+z=\sqrt{(1+\beta)/(1-\beta)}$	exact line-of-sight motion including time dilation; diverges as $\beta\to1$
Transverse Doppler	$1+z=\gamma=1/\sqrt{1-\beta^2}$	a source moving purely across the line of sight is still reddened, by dilation alone
Weak-field gravitational	$z\approx GM/Rc^2=\Delta\Phi/c^2$	photon climbing out of a potential well, observer at infinity; needs $GM/Rc^2\ll1$
Cosmological	$1+z=a(t_{\rm obs})/a(t_{\rm emit})$	exact in FLRW spacetime, independent of the expansion history in between

Common trapTreating the cosmological redshift as a Doppler velocity through space

The most common error in the subject is to read a cosmological $z$ as a special-relativistic recession speed. For a galaxy at $z=7$ the naive $v=cz$ gives $v=7c$ , seven times the speed of light. This is wrong, because the cosmological redshift is not motion through space — it is the stretching of space itself, governed exactly by $1+z=a_0/a_e$ , equation (6). The "recession velocity" $H_0 d$ is the rate of change of a proper distance, not a local speed past a local observer, and it genuinely exceeds $c$ beyond the Hubble radius without violating relativity, because no inertial observer ever measures another passing faster than light. The Doppler and cosmological formulae agree only in the low- $z$ limit where both reduce to $z\approx v/c$ ; beyond that they are different physics and must not be interchanged.

Worked examples

1 · Redshift and radial velocity of a galaxy (fully worked)

A galaxy's spectrum shows the Hα line, whose rest wavelength is 6562.8 Å, landing at 6600 Å. Find its redshift and its radial velocity in the low-z interpretation.

Step 1. Apply the definition (1): $z=(6600-6562.8)/6562.8=37.2/6562.8=\mathbf{5.67\times10^{-3}}$ — small, so the low- $z$ machinery applies.

Step 2. Because $z\ll1$ , equation (2) gives the velocity directly: $v_r\approx cz=(2.998\times10^{5})(5.67\times10^{-3})\approx\mathbf{1.70\times10^{3}\,km\,s^{-1}}$ , positive, so the galaxy is receding.

Step 3. Sanity check against (3): $\beta=5.67\times10^{-3}$ is so small that the relativistic $\beta^2$ correction is about $1.6\times10^{-5}$ — utterly negligible here. z = 0.00567, v_r ≈ 1700 km/s, recession, and relativity is irrelevant at this speed.

2 · A quasar at z = 2: what the redshift does and does not mean

A quasar has a cosmological redshift $z=2.00$ . Find the scale factor at emission, and contrast the meaningless "naive velocity" with the speed you would infer if you wrongly treated the shift as special-relativistic recession.

Step 1. Cosmic epoch from (6): $a_e=a_0/(1+z)=1/3$ . The universe was one third of its present linear size when this light set out, so every volume was $27$ times denser.

Step 2. The naive $v=cz$ gives $v=2c$ — nonsense, confirming the warning: at $z=2$ the linear formula is not merely inaccurate but unphysical, since the shift is cosmological.

Step 3. If instead one misread the same $z=2$ as special-relativistic recession, invert (3): $(1+z)^2=(1+\beta)/(1-\beta)=9$ , so $\beta=8/10=\mathbf{0.80}$ , i.e. $v=0.8c$ . This is legitimate for a genuinely moving source but the wrong interpretation for a cosmological redshift — quoted only to show how different the two frameworks are at the same $z$ . a_e = 1/3; the honest statement is the scale-factor ratio, not any velocity.

3 · Gravitational redshift of a white dwarf (fully worked)

Estimate the gravitational redshift from the surface of Sirius B, a white dwarf with $M\approx1.0\,M_\odot$ and $R\approx5900$ km, and compare with the HST-measured value.

Step 1. Use the weak-field form of (5), $z\approx GM/Rc^2$ . With the constant $GM_\odot/c^2=1.477$ km, $z\approx(1.477\,\mathrm{km})/(5900\,\mathrm{km})=\mathbf{2.50\times10^{-4}}$ .

Step 2. As an equivalent velocity, $cz=(2.998\times10^{5})(2.50\times10^{-4})\approx\mathbf{75\,km\,s^{-1}}$ — far larger than the $\sim10^{-6}$ shift at the Sun's surface, because a white dwarf packs a solar mass into an Earth-sized radius.

Step 3. The measured gravitational redshift of Sirius B is $80.6\pm0.8\,\mathrm{km\,s^{-1}}$ (Barstow et al., HST 2005). Our estimate is within $\sim7\%$ , the gap reflecting a slightly larger mass and the exact-versus-weak-field distinction. z ≈ 2.5×10⁻⁴ ≈ 75 km/s, matching the measured 80 km/s and confirming general relativity in a stellar field.

Physics of the portrait

Redshift is usually remembered as Edwin Hubble's discovery, but the velocities Hubble plotted were not his own. They were measured a decade earlier, one painstaking spectrum at a time, by a quiet observer in Arizona who found the faint spiral nebulae almost all rushing away — the first hint that the universe was not still.

Origins · 1 minVesto Slipher and how this was discovered

Vesto Slipher

1875–1969 · the first radial velocities of spiral nebulae, the empirical seed of the expanding universe

Vesto Melvin Slipher spent his entire career at the Lowell Observatory in Flagstaff, hired in 1901 and eventually its director, working with a spectrograph on a 24-inch refractor never meant for the task he set it. Photographing the spectrum of a faint spiral nebula meant guiding by hand for many nights on a single plate. In 1912 he obtained the spectrum of the Andromeda nebula and found its lines blueshifted, implying approach at about 300 km/s — a startlingly large speed. Over the next five years he measured some forty more, and the pattern was overwhelming: with a few nearby exceptions like Andromeda, the spirals were redshifted, receding at hundreds and then thousands of kilometres per second, faster than anything known in the Galaxy. Slipher could not know their distances, so he could not draw the velocity–distance line himself; that waited for Hubble, who in 1929 combined Slipher's velocities with his own Cepheid distances to reveal the linear expansion law. It is a case study in how the decisive observation often precedes, by years, the theory that makes it famous — the redshifts were on his plates long before anyone knew they signalled an expanding cosmos.

History

1842

— Christian Doppler predicts that motion shifts the frequency of waves, light included

1868

— William Huggins measures a stellar radial velocity from the shifted lines of Sirius

1912

— Vesto Slipher records the first spiral-nebula velocities, most of them large redshifts

1929

— Edwin Hubble combines redshifts with distances to reveal the expanding universe

1960

— Pound and Rebka detect the gravitational shift of gamma rays in a 22.5 m tower

Going further

Texts. Carroll and Ostlie, An Introduction to Modern Astrophysics, 2nd ed., §4.3 (Doppler shift) and Ch. 29 (cosmological redshift) · Ryden, Introduction to Cosmology, 2nd ed., Ch. 2 and Ch. 5 · Dodelson, Modern Cosmology, 2nd ed., Ch. 2 · Mo, van den Bosch and White, Galaxy Formation and Evolution, §3.1 · Binney and Tremaine, Galactic Dynamics, 2nd ed., §1.1. The key numerical anchors used throughout this article are: the Hα check gives $z=5.67\times10^{-3}$ and $v_r\approx1700\,\mathrm{km\,s^{-1}}$ ; the quasar at $z=2$ corresponds to $a_e=1/3$ and, under a (incorrect) special-relativistic reading, $\beta=0.8$ ; the Sirius B estimate yields $z\approx2.5\times10^{-4}\approx75\,\mathrm{km\,s^{-1}}$ against the measured $80.6\,\mathrm{km\,s^{-1}}$ ; and the exact Doppler curve diverges as $\beta\to1$ . Constants used: $c=2.998\times10^{5}\,\mathrm{km\,s^{-1}}$ , $G=6.674\times10^{-11}$ SI, $GM_\odot/c^2=1.477$ km, $H_0\approx70\,\mathrm{km\,s^{-1}\,Mpc^{-1}}$ .

Prerequisites. Luminosity and Radiant Flux · The Magnitude System. Leads to. The Cosmic Distance Ladder · Standard Candles.

Ask the tutor: Ask me to walk through disentangling the Doppler, gravitational, and cosmological contributions to a single observed redshift for a specific object — say a supernova in a distant galaxy cluster.

Is redshift the same thing as a Doppler shift?

Not in general. A Doppler shift is one of three ways to produce a redshift, applying specifically to motion through space. The gravitational redshift comes from light climbing a potential well, and the cosmological redshift from the expansion of space itself — neither is motion, and each has its own exact formula. For nearby stars the Doppler interpretation dominates and the two terms are used almost interchangeably, but for distant galaxies the cosmological term takes over and the identification breaks down. The safe habit is to treat "redshift" as the observed $z$ and name the mechanism separately.

Why does a transverse-moving source redshift at all, with no radial velocity?

Because of time dilation, a purely relativistic effect with no classical counterpart. Even when a source has zero velocity along your line of sight — passing broadside — its internal clock runs slow in your frame by $\gamma$ . A slower clock emits crests less often as you receive them, so the light is reddened by exactly $1+z=\gamma$ . Classically there is no first-order transverse shift at all, which is why the transverse Doppler effect was an early, clean confirmation of special relativity. It is small — second order in $\beta$ — but real, and contributes to the shift of fast sources like the star S2 near the Galactic Centre black hole.

How can recession velocities be larger than the speed of light?

They can because they are not velocities in the special-relativistic sense. The cosmological "recession velocity" $v=H_0 d$ is the rate at which the proper distance between two comoving points grows as space expands — not the speed of anything moving past a local observer. Special relativity forbids overtaking a light signal locally, and that is never broken: no galaxy passes another faster than light on the spot. But the sum of many small local expansions across a huge distance can make the total proper distance grow faster than $c$ , and galaxies beyond the Hubble radius do recede superluminally. We still see some of them because the Hubble radius itself grows with time.

Why is the frequency of the emitted line the thing that changes, not the emitting atom?

The atom emits at its fixed rest-frame frequency every time — set by unchanging atomic physics, which is why spectral lines are useful. What changes is the relationship between emission and reception: the source may be moving, sitting deep in a potential, or separated from you by expanding space. Each alters how the emitted crests are spaced by the time they reach your detector, which you measure as a shifted wavelength. So the redshift lives entirely in the geometry between emitter and observer, never in the atom — which is what makes $z$ a clean probe of that geometry.

What sets the sign convention, and can z be negative?

By convention $z$ is positive for reddening, and yes, $z$ can be negative: that is a blueshift. Sources approaching you Doppler-blueshift ( $z<0$ ), the classic example being the Andromeda galaxy, falling toward the Milky Way at $z\approx-0.001$ . Light falling into a potential well is gravitationally blueshifted, as in the downward arm of the Pound–Rebka experiment. What has no negative counterpart is the cosmological redshift of a distant source: the universe only expands on large scales, so cosmological $z$ is always positive and grows monotonically with look-back time.

How is redshift actually measured to six decimal places?

By cross-correlating an entire spectrum against a rest-frame template, not by eyeballing one line. A galaxy spectrum carries dozens of features whose rest wavelengths are known precisely; sliding the template in $\log\lambda$ until the whole forest lines up yields a single best-fit $1+z$ far more precisely than any one line allows, because the information adds in quadrature. Calibration is tied to arc lamps or sky lines of known wavelength, and modern instruments reach metres per second for exoplanet work. This is why redshift, despite its threefold ambiguity of cause, remains among the most exact numbers in observational astronomy.