# How do we measure the range of distant objects despite relativistic effects?

When we observe astronomical objects like distant galaxies there are several complicating factors for estimating the distance:

• Relativistic speed result in length contraction
• Relativistic speed results in Doppler-shifts in the frequency of light
• Simultaneity is skewed due to our differing velocities

With these effects, how are we able to state with decent accuracy how far away all of the distant galaxies are from us? Do the above effects have an impact on the measurements?

I know that determining far distances is mainly done by measuring the red-shift of light from those sources. A higher z number normally means it is farther away (due to the accelerating expansion of the Universe). My question is really about how relativistic effects are accounted for and whether or not they play a big role in the uncertainty of our calculations.

I'm looking for an answer geared towards the layperson, but formulas are always a welcome challenge.

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Astronomers know about this trouble, and stick with what they can measure. For distant galaxies, quasars, cosmic background radiation, etc. they use only the redshift, the "z" value. This is defined by the measured wavelength and the known laboratory value - assuming any spectral emission and absorption lines are correctly identified.

To say anything about distance (as in kilometers, parsecs, light years etc) beyond a couple billion lightyears requires defining a frame of reference, careful thought about what "distance" means (never mind cosmology, just what does it mean with moving objects?) and this takes cosmological models such as the Friedmann-Lemaître which provide the mathematical ground for calculating null geodesics, spacelike hyperplanes and all that cool stuff that laymen go to physics school to become physicists for.

Less than a couple billion light years, yes, sure there's a problem of seeing a galaxy where it was, but wanting to speak of how far away it is "now", which we can estimate from what we can know of the galaxy's motion. But the errors due to not using GTR and the right cosmological model amount to no more than the uncertainties in our measurements or desired accuracy of results.

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+1 good answer. But it needs more. Could you by any chance expand on the first two paragraphs? That would make it a great accepted answer – Jack Dozer May 7 '13 at 14:33

The difficulties are in some ways worse than you imagined and in other ways not as bad.

The difficulties are worse in the sense that when we're dealing with distant galaxies, we need general relativity, not just special relativity. General relativity does not even have a well-defined notion of a global frame of reference, so it doesn't offer a uniquely well-defined way of even defining cosmological distances or the velocities of distant objects.

The difficulties are not as bad in the sense that the structure of spacetime on cosmological scales is relatively well understood these days, so we can calculate all the relativistic effects with pretty decent precision.

Relativity doesn't have a preferred frame. However, in cosmology there is a locally defined frame that is special, which is the frame of the cosmic microwave background photons. When we say that the distance from our galaxy A to a cosmologically distant galaxy B equals L, what we mean is this. We mean that L is the result we would get from a chain of rulers stretching from A to B, each ruler being at rest relative to the CMB in its own neighborhood.

There is also a standard notion of cosmological time, which is the time measured on a clock that is at rest relative to the CMB. For example, when we say that the universe is 14 billion years old, we mean in terms of this time.

General relativity doesn't require that we use these specific, standard measures of cosmological time and distance. In fact, GR is completely agnostic about coordinates. But these measures are physically well motived in terms of their relation to the physical characteristics of the universe.

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But isn't it practically impossible to talk about the frame of reference of a photon? Based on our understanding, in the frame of a photon, wouldn't all time and length differences go to zero? After all, a clock at rest relative to the CMB would measure zero time having passed since it was emitted. I was under the impression that the CMB timing and distance was measured with references to our own inertial frame. – Jack Dozer May 7 '13 at 14:31
You can't define the frame of an individual photon, but the frame of the CMB is the frame in which the average motion of the CMB cancels out. – Ben Crowell May 9 '13 at 3:10
.... I understood the word frame. Okay, since the CMB is isotropic, wouldn't we observe its average motion cancel out in our inertial frame and others in their frame, etc? – Jack Dozer May 9 '13 at 14:15

The first thing to note is that most accurate distance measurements are at about the 5% level (except for a few exceptions), but 50% is more common - and often satisfactory.

• Length contraction is not an issue for two reasons. First, relative velocities between us and entire galaxies are generally small1 (especially due to 'peculiar velocities'). Second, length contraction effects the observed length of the moving object, not the distance that we measure to it.

• Relativistic doppler shifting is an important effect in when measuring things like high energy astrophysical jets; but when measuring distances to galaxies (for example), you're really looking at the light from millions to billions of stars averaged together---where the typical velocity is small.

• Simultaneity is mostly irrelevant, unless I'm misunderstanding your question, as distance measures don't depend on absolute timing.

1: Even at a distance of about 2000 Mpc ($z \approx 0.5$), the apparent velocity is less than 20% the speed of light --- which is negligible.

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It's not true that relative velocities are usually small compared to c. This is only true for nearby galaxies. The question is specifically about distant galaxies. It's also not true that length contraction only affects the size of objects rather than their positions. – Ben Crowell May 2 '13 at 22:41
That is very paradoxical. It is impossible that objects are shrinked but stay at the same position. This blows my mind. – Val May 5 '13 at 9:06