How Do You Derive Transverse Comoving Distance?

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Discussion Overview

The discussion revolves around deriving the transverse comoving distance in cosmology, specifically referencing a paper on the Friedmann-Robertson-Walker metric. Participants seek clarification on the derivation process and relevant concepts.

Discussion Character

  • Homework-related, Technical explanation, Conceptual clarification

Main Points Raised

  • One participant requests assistance in deriving the transverse comoving distance, indicating difficulty in starting the process.
  • Another participant asks for more details to better understand the request, suggesting that the original paper may not be sufficient for all readers.
  • A third participant mentions the importance of the Friedmann-Robertson-Walker metric and its relation to the derivation, noting that trigonometric terms arise from coordinate transformations.
  • This participant also states that transverse comoving distance is connected to angular diameter distance through the scale factor, which is equal to one at the current time.
  • There is an acknowledgment of the challenge in explaining the derivation without going through the entire process, particularly for those familiar with Einstein notation.

Areas of Agreement / Disagreement

Participants express varying levels of understanding and familiarity with the concepts involved, and there is no consensus on how to approach the derivation or what specific information is missing.

Contextual Notes

Some participants note that the original paper may not provide all necessary details for deriving the transverse comoving distance, and there is an implication that familiarity with certain mathematical concepts is required.

S0uris
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Hi, i need derive the comoving distance (transverse)

im working with https://arxiv.org/pdf/astro-ph/9905116.pdf

I've had trouble starting, please help :)

Captura de pantalla de 2020-06-03 19-12-29.png
 
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Could you go into more details? You cannot expect the readers read the entire paper.
 
ok :) i just need this form, i don't know how to start.

Captura de pantalla de 2020-06-03 19-12-29.png
 
S0uris said:
i need derive the comoving distance (transverse)

What do you mean by "derive" and what information that you need to do so isn't in the paper you linked to?
 
S0uris said:
Hi, i need derive the comoving distance (transverse)

im working with https://arxiv.org/pdf/astro-ph/9905116.pdf

I've had trouble starting, please help :)

View attachment 264089

I think you need to read up on the Friedmann-Robertson-Walker metric. If you do you will know those trigonometric terms come from the coordinate transformation to hyperspherical coordinates. This is done because the FRW metric (which is really the metric of a 3-sphere in spacetime) has a coordinate singularity when the curvature of the universe is non-zero.

The second point is that transverse comoving distance is related to angular diameter distance via the scale factor and in fact in the current time the scale factor is equal to one.

This explanation is terrible I know, but It's difficult to explain without actually deriving everything. If you are familiar with Einstein notation, then pages 3 to 6 inclusive of this document might help.
 
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