# What is Proper distance: Definition and 26 Discussions

Proper length or rest length is the length of an object in the object's rest frame.
The measurement of lengths is more complicated in the theory of relativity than in classical mechanics. In classical mechanics, lengths are measured based on the assumption that the locations of all points involved are measured simultaneously. But in the theory of relativity, the notion of simultaneity is dependent on the observer.
A different term, proper distance, provides an invariant measure whose value is the same for all observers.
Proper distance is analogous to proper time. The difference is that the proper distance is defined between two spacelike-separated events (or along a spacelike path), while the proper time is defined between two timelike-separated events (or along a timelike path).

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1. ### A Schwarzschild Geometry: Evaluating Proper Distance

Schwarzschild Geometry-proper distance. From what I have studied when the Schwarzschild line element is evaluated at constant time and at a constant radius , proper distance becomes a Euclidean distance on the surface of a sphere. What I don't understand is how to evaluate the integral...
2. ### B Proper distance in cosmology

While studying the proper distance in cosmology I came across the thing as The FRW metric ds2=c2dt2-a(t)2[dr2+Sk(r)2dω2] And Sk(r)=Rsin(r/R) Where a(t) is the scale factor and dω2= dθ2+sin2θdΦ2, While calculating the proper distance at the time of emmission of light the term ds and dω are...
3. ### Proper distance in Schwarzschild metric

Homework Statement Let the line element be defined as ##ds^2 = -(1-\frac{2m}{r})dt^2+\frac{dr^2}{1-\frac{2m}{r}}+r^2 d\theta^2 + r^2 \sin^2{\theta} d\phi^2## a) Find a formula for proper distance between nearby spherical shells, assuming only the radius changes, and ## r > 2m ## b) Now look...
4. A

### I Understanding the Role of a(t)*dr in Calculating Actual Distance

actually I want to clarify that is the proper distance a(t)*dr is the elementary length along the radial direction which is changing with respect to time and if integrating a(t)*dr and dr ,which may be considered the actual distance
5. ### I Radial Proper Dist. Objects Free Fall: Calculating Δs

If we consider 2 shells in Schwarzschild spacetime their radial proper distance ##d s^2 = \frac{1}{1-\frac{r_s}{r}}d r^2## follows from the metric with ##dt=0## and ##d\Phi=0##. Integrating yields ##\Delta s## as a function of the r-coordinates of the shells. A reference with the formula for...
6. ### I Geodesics on S2 Surface: Arc-Length Parametrisation

Consider the metric of ##S^{2}##: $$ds^{2}=d\theta^{2}+\sin^{2}(\theta)d\phi^{2}$$ Then in order to determine the geodesics on this surface one can minimise the integral $$s=\int_{l_{1}}^{l_{2}}\sqrt{\left(\frac{d\theta}{dl}\right)^{2}+\sin^{2}(\theta)\left(\frac{d\phi}{dl}\right)^{2}}dl$$ where...
7. ### I Proper Distance Between Surfaces of Constant R in Curved Spacetime

How about proper length in curved spacetime? Let's consider the radial distance between two spherical shells in Schwarzschild spacetime. The proper distance between them follows from the spacelike form of the metric with ##dt=0## for simultaneity. So I think having the r-values of the shells the...
8. ### I Definition and measurement of proper length

As I understand it, the proper length, ##L## of an object is equal to the length of the space-like interval between the two space-time points labelling its endpoints, i.e. (in terms of the corresponding differentials) $$dL=\sqrt{ds^{2}}$$ (using the "mostly plus" signature). Furthermore, this is...
9. ### Angular diameter distance to surface of last scattering

Homework Statement 1) Calculate the angular diameter distance to the last scattering surface in the following cosmological models: i) Open universe, ΩΛ= 0.65, Ωm = 0.30 ii) Closed universe, ΩΛ = 0.75, Ωm = 0.30 ii) Flat universe, ΩΛ = 0.75, Ωm = 0.25 Describe how the CMB power spectrum...
10. ### Proper distance for co-moving bodies

I was told in my cosmology class that there are two planets separated by a distance L at time t = 0. L is known as the co-moving distance between the planets. I was told the equation for proper distance, ##d_p##, is given by: ## d_p = a(t) L ## where ##a(t)## is the spatial expansion factor of...
11. ### Proper distance integral limits seem wrong

I've seen in some lecture notes that the proper distance dp(t) can be written as ##\int_{t_e}^{t_0} c dt/a = \int_0^z c dz /H(z)## I can perform this integral ok using ##H =\dot a/a## and the fact that ##1 + z = 1/a(t_e)## but it requires associating the limits of the integration as te...
12. ### Proper distance problem/interpretation

Hi I'm trying to put some notes together but have run into an anomaly which I seem to have overlooked in the past but puzzles me now. I've included a jpg file of the page I've written up so far with the problem indicated right at the end. I'm using Barbara Ryden's book as my source, but it...
13. ### Proper distance from the FRW metric

Homework Statement Question: Homework Equations [/B] Dp=R(t)Dc where R(t) is the scale factor, Dc is the comoving distance The Attempt at a Solution [/B] I must have tried solving this starting 10 different ways now, starting with the fact that distance=integral over velocity dt...
14. ### Measuring Motion: What STAY Knows About TRAVEL

Dear PF Forum, I'm sorry if I ask the basic question here again. Just need confirmation. V = 0.6; Gamma = 1.25 TRAVEL travels at 0.6c. STAY stays. Pic 02 is Pic 01 boosted -V. 1. All STAY knows about TRAVEL is: Proper Time Speed Is this true? And mutually for TRAVEL 2. At B (and C)...
15. ### Space-intervals, proper time, and proper distance

Can I ask a question here? What is proper distance? V = 0.6c What is proper distance for Blue? What is proper distance for Green? Thanks.
16. ### Proper distance, Area and Volume given a Metric

Homework Statement [/B] (a) Find the proper distance (b) Find the proper area (c) Find the proper volume (d) Find the four-volume Homework EquationsThe Attempt at a Solution Part (a) Letting ##d\theta = dt = d\phi = 0##: \Delta s = \int_0^R \left( 1-Ar^2 \right) dr = R \left(1 -...
17. ### Proper Distance, Rest Length: What to Know

Hello, Could someone check my following statement: The proper distance between two spacelike separated events can be thought of as the rest length of a rod that connects both events in an inertial frame in which both events happen simultaneous. Thanks in advance!
18. ### Physical meaning of max. proper distance at time of emisson

I cannot picture the physical meaning of the maximum proper distance at time of emisson. In the benchmark comological model the plot of (H/c) * (Proper distance at time of emission) versus z (the redshift) shows a maximum at z=1.6 for a proper distance of emission = 1800 Mpc (mega parsecs). I...
19. ### Cosmic rest frame, comoving distance, and proper distance

Hi I am trying to understand more about these terms. I am currently studying a course about relativity and cosmology, but I am finding the textbook (Open University) difficult to follow. Can anyone help me untangle and make some simple sense of these different terms? Thanks (I accidentally...
20. ### Defining proper distance in GR.

At page 234 in Landau and Lifshitz' Classical Theory of Fields the proper time element is defined through the line element by ##ds^2 = c^2 d\tau^2##, then for a stationary observer, setting ##dx^i=0## for ##i=1,2,3##. One then obtains the relation $$c^2 d\tau^2 = g_{00}(dx^{0})^2.$$ He then...
21. ### Is the proper distance the shortest distance in FLRW universe?

In Appendix A of Davis & Lineweaver (2003) proper distance to a faraway galaxy is defined as the distance along a curve of constant time in the RW metric. I was wondering whether that line of constant time is a geodesic of spacetime. If not then there will be a shorter-distance path from here...
22. ### Cosmological Proper Distance

This is based on a side discussion in the balloon analogy thread, see #49 and #53. How is the definition of cosmological proper distance ("CPD" from now on) different from the usual definition of distance? Here, I want to discuss the respective defintions, what these definitions "really" mean...
23. ### Understanding proper distance in Schwarzschild solution.

I'm trying to understand the Schwarzschild solution concept of proper distance. Given the proper distance equation d\sigma=\frac{dr}{\left(1-\frac{R_{S}}{r}\right)^{1/2}} how would I calculate the coordinate distance. For example - assuming the distance from the Earth to the Sun is...
24. ### GR: "Proper Distance" Meaning and Usage

"Proper distance" in GR I am aware of two meanings of the term "proper distance" in GR. The first is when you have points in flat space-time, or space-time that's locally "flat enough", in which case it is defined as it is in SR, as the Lorentz interval between the two points. This usage of...
25. ### 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...
26. ### Exploring the Definition of "Proper Distance" in Cosmology

http://arxiv.org/PS_cache/astro-ph/pdf/0310/0310808.pdf [Broken] uses the term "proper distance", but doesn't define it. Presumably this must be a standard defintion. So far, though, I have not been able to track down a definitive defintion (I'm still looking). My guess is that this distance...