# Proper distance at time of emission

We have that the proper distance to an object is given by

$$d_p (t_0 ) = c\int_{t_e }^{t_0 } {\frac{{{\rm{d}}t}}{{a(t)}}}$$

and this goes for all possible universes described by the Robertson-Walker metric. Since we know that

$$1 + z = \frac{1}{a(t_e)}$$

does this mean that the proper distance at the time of emission is always smaller than the proper distance at the time of observation by a factor of 1+z? And in this true for multi-component universes (e.g. matter + radiation) as well as single-component (e.g. only radiation, only matter ...) ?

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I am sure not everybody agrees with it but I find the term "proper distance" a misnomer and yet another cosmological term that is doomed to confuse more than it explains. There is obviously nothing wrong with the concept itself but I do not think there is anything proper about it. Distance is not very well defined in GR especially in non-stationary spacetimes.

marcus
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We have that the proper distance to an object is given by

$$d_p (t_0 ) = c\int_{t_e }^{t_0 } {\frac{{{\rm{d}}t}}{{a(t)}}}$$

and this goes for all possible universes described by the Robertson-Walker metric. Since we know that

$$1 + z = \frac{1}{a(t_e)}$$

does this mean that the proper distance at the time of emission ...

How would you define the proper distance to the object at the time of emission? Could you write down the integral for us, in that case?

Just trying to get the concept clearly in mind. Jennifer may have a point.

I wouldn't define it, I would just think that d_p(t_0) is the distance to the light source when the light is observed. At the time of emission, the proper distance was smaller by a factor a(t_e) / a(t_0) = 1/(1+z).

But I'm not sure this is consistent?

Wallace
I'd like to clarify the issue of proper distance. The term 'proper' has a very specific meaning like all words in physics and of course doesn't embody the everyday meaning, which could be paraphrased as the 'correct' distance. No, the way proper distance is defined is that it is the length of the interval between two points integrated over a surface of constant cosmic time. It is one of many ways to define distance in cosmology.

The statement 'Distance is not very well defined in GR especially in non-stationary spacetimes' is clearly erroneous, since proper distance is perfectly well defined, as are a number of other distance measures. What is certainly true however is that proper distance cannot be measured since to do so would violate simultaneity. What this means is to actual measure proper distance you would have to someone freeze time, then lay out a ruler between you and the object you wish to measure the distance to. This clearly cannot be done.

To answer to question in the OP, proper distance to co-moving objects can be written as

$$r_p = a(t) \chi$$

where $$\chi$$ is the co-moving co-ordinate. This shows that as long as the Universe is expanding for between the time of emission and the time of observation then the proper distance will be smaller at the time of emission than it will be at the time of observation. It doesn't matter what components there are in the Universe, only that it is expanding the whole time.

The statement 'Distance is not very well defined in GR especially in non-stationary spacetimes' is clearly erroneous, since proper distance is perfectly well defined, as are a number of other distance measures.
Just because we define something and label it some kind of a 'distance' does not imply it is anything like distance as we intuitively know it. Distance as we know it in the Newtonian and special relativity world simply does not exist under general relativity.

Wallace
Just because we define something and label it some kind of a 'distance' does not imply it is anything like distance as we intuitively know it.
Right, but we are not talking about distance 'as we intuitively know it'. Newtonian Momentum is not 'momentum as we intuitively know it' either, in terms of the everyday use of the word. Don't judge the merit of a scientific concept by the everyday use of a word!

Distance as we know it in the Newtonian and special relativity world simply does not exist under general relativity.

Exactly, which is why we need to define ways of talking about distance that are relevant to GR. For a good introduction to distance measures in cosmology, see http://arxiv.org/abs/astro-ph/9905116" [Broken] classic primer by David Hogg.

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Exactly, which is why we need to define ways of talking about distance that are relevant to GR.
Indeed and my point is that it is confusing to then label something as 'proper distance'. For instance the term 'comoving distance' is in that respect a bit less confusing.

Wallace