###### By Joshua N. Winn, Princeton University

## The tapestry of galaxies that we see in all directions seems eternal. All those galaxies are rushing away from us at high speed, the further away the galaxy, the faster it’s receding. This observation—Hubble’s law—is one of the pillars of evidence supporting the Big Bang theory.

### Equation of Hubble’s Law

In equation form, Hubble’s law is *v* = *H*_{0}*d*, where *d* is the distance to the galaxy, *v* is the velocity with which it’s receding, and the proportionality constant, * H_{0}*, is the Hubble constant.

There might be something bugging you about Hubble’s law. It implies that *v* increases without bound, as we look further and further away. That can’t be right, can it? The galaxies can’t exceed the speed of light, can they? Well, yes and no. It’s a tricky issue.

### Doppler Shift

What we observe directly is not velocity, but rather, the Doppler shift of a galaxy’s spectrum. The spectrum of a nearby star might have an absorption line at a wavelength of 0.656 microns. That’s the Hα line: it comes from electrons jumping between the second and third energy levels of hydrogen. We call 0.656 microns the “rest” wavelength because the star isn’t moving very fast, relative to us. It’s the same wavelength we’d observe in a physics laboratory.

But when we observe the starlight from a galaxy far, far away, the pattern of lines is shifted to longer wavelengths. Now, the Hα line is at, say, 0.689 microns, 5% longer than usual.

### Redshift Theory

We define the redshift, *z*, as *Delta-lambda* over *lambda*, the observed wavelength minus the rest wavelength, all over the rest wavelength. In this case, *z* is 0.05. And if we interpret that shift as an ordinary Doppler shift, *z* = *v* / *c*.

An even more distant galaxy might show the Hα line at 1.3 microns, twice as large as usual! That would seem to imply that *v* is twice the speed of light, contradicting the fundamental principle that nothing can travel faster than light.

### Not an Inherent Property

The formal resolution of this apparent paradox is that the expanding universe needs to be described with general relativity, and in general relativity, the “velocity” of a distant galaxy is not a meaningful concept. There’s no unique way to define the relative velocity between two objects in widely separated locations, when space is changing with time. So, the velocity, *c* times *z*, that we would naively compute for a distant galaxy has no physical significance.

The redshift of a distant galaxy is not an inherent property of the galaxy; it doesn’t depend on how fast the galaxy is moving with respect to anything else. Instead, the redshift is something that happens to photons as they travel from that galaxy to our telescopes. It’s a consequence of expanding space.

This article comes directly from content in the video seriesIntroduction to Astrophysics. Watch it now, on Wondrium.

### Understanding the Hubble Constant

Consider a galaxy at a co-moving distance of *s*. At any time, *t*, the physical distance, *r*, is *a* of *t* times *s*. The recession velocity, *v*, is *dr/dt*, which is equal to *da/dt* times *s*.

To put that purely in terms of physical distance, we’ll replace *s* with *r /* *a*. Then we can rearrange the equation to say *v* equals one over *a* da*/dt* times *r*. That’s Hubble’s law—velocity is proportional to distance. The constant of proportionality—the Hubble constant—is equal to one over *a* *da/dt*.

The purists refer to one over *a* *da/dt* as the Hubble parameter, *H*, and reserve the name “Hubble constant” and the symbol *H*-naught, for the currently measured value of 70 kilometers per second per megaparsec.

### Interpreting Galaxy Redshift

Next, let’s use our new concept, the cosmological scale factor, to help us interpret galaxy redshifts. Suppose at some time *t* in the past, a distant galaxy emits photons, which travel for billions of years through an expanding universe, and end up inside our telescope at time *t*-naught, the present day. It will help to conceptually divide the journey up into lots of tiny steps, and pretend there are alien astronomers all along the way, who are observing the light from that same galaxy.

The first alien is at a physical distance of *dr *from the source, and crucially, we’ll let *dr* be such a short distance that the subtleties of relativity can’t possibly matter. Since Hubble’s law applies to everybody, the alien observes the galaxy to be receding with a small velocity *dv* equals *H* *dr*. Since the velocity is small, we can rely on the familiar, non-relativistic formula for the Doppler shift, *d-lambda* over *lambda* equals *dv* over *c*, which we can write as *H* *dr* over *c*.

### Solving the Equation

We learned a minute ago that *H* equals one over *a* *da/dt*, so let’s make that replacement. We can also replace *dr* over *c* by *dt*, the time that it takes for the light to travel a physical distance *dr*. Those replacements lead to a simple equation: *d-lambda* over *lambda* equals *da* over *a*. The fractional change in wavelength equals the fractional change in the scale factor during the time interval *dt*.

The result is *lambda*-observed over *lambda*-rest equals one over *a* of *t*. The wavelength gets stretched by the same factor the universe has expanded throughout the photon’s journey. That implies the redshift, *z*, as we defined it earlier, is equal to one over *a* minus one.

Or, equivalently, one over *a* equals one plus *z*. That’s the interpretation of the redshift that we’ve been seeking. When we observe a galaxy to have a redshift of 2, instead of saying the galaxy is rushing away at twice the speed of light, it makes more sense to say the universe has expanded by a factor of 3 since the light was emitted.

### Common Questions about How Hubble’s Law Supports the Big Bang Theory

**Q: What is the equation of Hubble’s Law?**

In equation form, Hubble’s law is *v* = *H*_{0}*d*, where *d* is the distance to the galaxy, *v* is the velocity with which it’s receding, and the proportionality constant, * H_{0}*, is the Hubble constant.

**Q: How does Hubble’s Law support the Big Bang theory?**

The tapestry of galaxies that we see in all directions seems eternal and they are rushing away from us at high speed, the further away the galaxy, the faster it’s receding. This observation—Hubble’s law—supports the Big Bang theory.

**Q: Is Redshift one of the properties of a galaxy?**

The redshift of a distant galaxy is not an inherent property of the galaxy; it is something that happens to photons as they travel from that galaxy to our telescopes. It’s a consequence of expanding space.