Dear All,
I would like to ask you to help me understand the use of "Isotropic
Coordintes" in regard to the Schwarzschild metric.
(1) What exactly is the purpose, or reason why one uses these
coordinates?
(2) Is it possible to derive the deflection of light expression in
these coordinates?

I have read the discussion of these coordinates in "Introduction to
General Relativity" by Adler, Bazin, and Schiffer, which was not
illuminating enough for me. The MTW treatment is not worth mentioning.
Is there a reference where these coordinates are discussed in detail?
Thank you for your help.

Regards,

Learner

Learner wrote:
Dear All,
I would like to ask you to help me understand the use of "Isotropic
Coordintes" in regard to the Schwarzschild metric.

(1) What exactly is the purpose, or reason why one uses these
coordinates?

Isotropic coordinates are spatially isotropic -- that is, for
an infintesimal coordinate change a, the physical distance between
(x,y,z) and (x+a,y,z) is the same as the physical distance between
(x,y,z) and (x,y+a,z), which is the same as the physical distance
between (x,y,z) and (x,y,z+a). This is not true for standard
Schwarzschild coordinates, in which a displacement a in the r
coordinate corresponds to a different physical distance than the
same displacement in a coordinate transverse to r. So using
isotropic coordinates may make some interpretations slightly
easier.

You can compute in any coordinate system you want to, although any
particular computation will be easier in some coordinates than in
others. (This is just like classical mechanics -- if you want to
find planetary orbits, you *can* use Cartesian coordinates, but
it's easier if you use spherical coordinates.) Off hand, I don't
know of anything that's easier to compute in isotropic coordinates
than in Schwarzschild coordinates, but there may be such a thing.

Isotropic coordinates *are* useful if you want to compare two
different models of gravity (say, general relativity and a scalar-
tensor theory). The requirement that the coordinates be isotropic
removes some ambiguity and makes it easier to be sure that you're
comparing the same things.

(2) Is it possible to derive the deflection of light expression in
these coordinates?

Yes. The deflection of light is a physical process; as long as you
are careful to compute the results of actual measurements, the
answer doesn't depend on what coordinate system you choose.

On the other hand, Schwarzschild and isotropic coordinates both have
a quantity called "r", but these have slightly different meanings.
So if you compute the deflection of light in terms of r, you will
get slightly different answers in the two coordinate systems. The
*physical* deflection is the same, but the symbols in the equation
don't mean the same thing.

The reference you want is Bodenner and Will, "Deflection of light
to second order: a tool for illustrating principles of general
relativity," Amer. J. Phys. 71 (2003) 770-3. Their abstract is:

We calculate the deflection of light by a spherically symmetric
body in general relativity, to second order in the quantity GM/dc^2,
where M is the mass of the body and d is a measure of the distance
of closest approach of the ray. Using three different coordinate
systems for the Schwarzschild metric we show that the answers for
the deflection, while the same at order GM/dc^2, differ at order
(GM/dc^2)^2. We demonstrate that all three expressions are really
the same by expressing them in terms of measurable, coordinate-
independent quantities. These results provide concrete illustrations
of the meaning of coordinates and coordinate invariance, which may
be useful in teaching general relativity.

Steve Carlip

"Learner" wrote in message
oups.com...
Dear All,
I would like to ask you to help me understand the use of "Isotropic
Coordintes" in regard to the Schwarzschild metric.
(1) What exactly is the purpose, or reason why one uses these
coordinates?
(2) Is it possible to derive the deflection of light expression in
these coordinates?

I have read the discussion of these coordinates in "Introduction to
General Relativity" by Adler, Bazin, and Schiffer, which was not
illuminating enough for me. The MTW treatment is not worth mentioning.
Is there a reference where these coordinates are discussed in detail?
Thank you for your help.

Regards,

Learner


Learner,

Foster and Nightingale in their text : 'A Short Course in General
Relativity: Second Edition' briefly discuss isotropic coordinates in Section
4.9. They then briefly use it in problem 5.3 on calculating gravitation
radiation in the far zone.

I think that one of the advantages of isotropic coordinates is that you can
fairly easily switch between spherical coordinates rho, theta, phi and
Cartesian coordinates x,y,z for the spatial part of the metric.

David Park

http://home.earthlink.net/~djmp/

Learner wrote:
Dear All,
I would like to ask you to help me understand the use of "Isotropic
Coordintes" in regard to the Schwarzschild metric.
(1) What exactly is the purpose, or reason why one uses these
coordinates?

This is an adjunct to the good replies posted...
One really only needs HS physics, an appreciation
of the conservation of energy and a bit of common
sense, to get an order of accuracy for GR effects
in our solar system using K. Schwarzschild (KS).

Begin with a massless lightbulb, and measure
the intensity of the bulb at various distances
using a 1 meter^2 receiver, basically measuring
light energy flux.
Well we know the photon intensity on the meter^2
is proportional to 1/r^2 in Newtonian physics.

In the early 1900's Planck established E=hf
(E=energy, h= constant, f=frequency).

Because of conservation of energy, light rays
(photons) going upward in a g-field, must loose
frequency, aka the Einstein shift. This gets
the metric g_00.

So now if we place a large mass M where the bulb
is, the intensity will reduce where the receiver
is, compared to when M=0 because of the *red shift*
of photons moving upward due to mass M0.

To get the same photon intensity as when M=0,
the reciever will need to be set closer to the
M0 bulb. That is why radial *length* shrinks
in a g-field, (g_11).

The Newtonian coordinates to retain the
conservation of energy are replaced by the KS,
coordinates accounting for E=hf in a g-field.

For ref see Weinberg's "Grav & Cosmo" pg. 84,
beginning with "Incidentally...", it's
uncredited, but I think reliable.

(2) Is it possible to derive the deflection of light expression in
these coordinates?

Yes, as Dr. Carlip confirms.

I have read the discussion of these coordinates in "Introduction to
General Relativity" by Adler, Bazin, and Schiffer, which was not
illuminating enough for me. The MTW treatment is not worth
mentioning.
Is there a reference where these coordinates are discussed in detail?

Well, Weinberg's "Grav & Cosmo" chpt. 8, is
fairly detailed, are you a mathematical masochist?

Thank you for your help.
Regards,
Learner

It's a good topic, GR shouldn't be intimidating.
As Dr. Carlip's post shows, one need to careful
well coordinates are concerned at a more advanced
level.

Regards
Ken S. Tucker
kxsxt