# Proper distance at time of emission

• Niles
In summary: So again, I don't see the point of getting hung up on the terminology, it is what it is and if you want to learn cosmology then that is the way it is described. Not much more to be said about it.In summary, proper distance in cosmology is a specific term that refers to 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 and is perfectly well defined in GR, despite not being able to be measured in practice. The term 'proper' does not imply that it is the 'correct' distance, but rather a specific distance measure relevant to GR.
Niles
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.

Niles said:
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?

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.

Wallace said:
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.

MeJennifer said:
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!

MeJennifer said:
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" classic primer by David Hogg.

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Wallace said:
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.

But as I've said, like so many words used in physics 'proper distance' has a specific meaning and is only confusing if you want to ascribe the everyday meaning to the word 'proper'. You could make this argument about so much of physics terminology, it is nothing unique to cosmology or GR. The first thing students of physics learn is that they need to understand the very specific uses of language in the terminology of physics.

This is why you don't see terms like 'proper distance' in pop sci. In that context it clearly would be confusing, but not in the context that it is used within cosmology.

## 1. What is proper distance at time of emission?

Proper distance at time of emission is a concept in cosmology that refers to the distance between two objects at the time that light is emitted from one object and received by the other. It takes into account the expansion of the universe over time.

## 2. How is proper distance at time of emission different from comoving distance?

Comoving distance is a measure of the current distance between two objects in the universe, while proper distance at time of emission takes into account the expansion of the universe and calculates the distance at the time the light was emitted.

## 3. Why is proper distance at time of emission important in cosmology?

Proper distance at time of emission is important because it allows us to accurately measure the distances between objects in the universe, taking into account the expansion of the universe over time. This is crucial for understanding the evolution and structure of the universe.

## 4. How is proper distance at time of emission calculated?

Proper distance at time of emission is calculated using the formula d(t) = a(t)*dc, where d(t) is the proper distance at time t, a(t) is the scale factor of the universe at time t, and dc is the comoving distance between the two objects.

## 5. Can proper distance at time of emission be measured?

Yes, proper distance at time of emission can be measured using astronomical observations and data. However, due to the vast distances involved, it is often measured in terms of redshift, which is the amount that light from an object has been stretched due to the expansion of the universe.

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