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Relativity and Cosmology

This is a Blog on relativity and cosmology for engineers and the like. My website "Relativity-4-Engineers" has more in-depth stuff.

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Relativistic Acceleration Part 6

Posted October 12, 2006 12:57 PM by Jorrie

In this final part of relativistic acceleration out of Relativity 4 Engineers, we take a look at perhaps one of the most baffling and least understood phenomena associated with relativistic acceleration. It is the opposing transverse (or tangential) acceleration that pops up in a strong gravitational field when both radial and transverse velocity components are present.

This situation arises in elliptical orbits, which have for the most part both radial and transverse velocity components. The exceptions are obviously at the peri-and apo-apses, where the movement is purely transverse. It can be shown that this is the dominant cause of the relativistic perihelion shift of planet Mercury.

Before describing the cause of the phenomenon, let us first give the equation for the opposing relativistic transverse acceleration:

It is called at(opp) because in essence, it is an acceleration that opposes the normal Newtonian transverse (or angular) acceleration in an elliptical orbit. We know that when a Newtonian orbit falls inwards towards peri-apsis, the transverse velocity increases, i.e., there is a positive angular acceleration. The orbit has to sweep out a constant area with a reduced radius. In such a case, the radial velocity vr is negative and at(opp) will be negative, thus opposing the positive Newtonian angular acceleration.

After the peri-apsis has been passed, the Newtonian transverse velocity decreases, i.e., there is a negative angular acceleration. Because the radial velocity vr is now positive, at(opp) will be positive, meaning the 'opposing' acceleration is now positive, again working against the normal Newtonian situation.

All very confusing, but this is what Einstein's theory of relativity tells us and this is what we observe in the planet Mercury and in many binary star systems observed so far. The opposing acceleration is mainly caused by the curvature of space.

Elliptical orbits mostly describe movement in directions that are neither with the space curvature (radial), nor normal to it (transverse). Therefore, elliptical orbits 'suffer' the effects of space curvature most of the way. When viewed in a distant observer's coordinate system, we observe this 'suffering' as spurious 'opposing' accelerations.

A much better treatment is given in Relativity 4 Engineers, with a plot of an orbit around a black hole, where the perihelion shift amounts to 360 degrees - see diagram below. The algorithm to plot such an orbit, using the equations described in this 'mini-series', is elaborated upon in Relativity 4 Engineers.

[edit] The above diagram and caption are much, much more readable in the PDF format of the eBook. There you can chose the 'zoom' value without degrading the quality. The symbols used in the caption are fully described in the eBook. [/edit]

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#1

Re: Relativistic Acceleration Part 6

10/12/2006 2:22 PM

Hey Jorrie, this is quite intriguing! Are you sure about this 'opposing acceleration'? Are there any references that confirm this?

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Guru
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#2
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Re: Relativistic Acceleration Part 6

10/13/2006 12:12 AM

Hi Guest, good question!

I must confess that there are no direct references from other sources - relativists simply do not work with the acceleration the way I work with them. However, I have taken their equations and numerically check that mine give the same answers as theirs.

I have also gone to a lot of bother to actually derive these equations from 'second principles', i.e., taking some solutions of Einstein's field equations at face value and derive my equations from them. This is documented in Appendix C of Relativity 4 Engineers. Take a look at the eBook's Table of Contents.

Jorrie

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Guru
Engineering Fields - Aerospace Engineering - Retired South Africa - Member - The Rainbow-nation Engineering Fields - Engineering Physics - Relativity & Cosmology Popular Science - Cosmology - The Big Picture!

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Re: Relativistic Acceleration Part 6

10/13/2006 3:15 AM

Quoting Jorrie: "I must confess that there are no direct references from other sources - relativists simply do not work with the accelerations the way I work with them".

To prevent misunderstanding, the eBook is full of references to works of specialists, e.g., Misner, Thorne and Wheeler. It is just that they do not give the accelerations in my 'engineering form'.

The eBook's bibliography can be downloaded from the website, if you are interested.

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