Suppose that I want to study the trajectory of a test particle
that is acted upon by a massive spherical object, say a star.
Suppose the test particle comes in from infinity at an initial
impact parameter b and is deflected by the gravitational
attraction of the star. Assuming that no other forces act on
the particle,the equation for the trajectory can be obtained
from the geodesic equation sraightforwardly. My question is,
which coordinates for the Schwarzschild metric should one use,
the Schwarzschild coordinates or the Isotropic coordinates?
This is a relavent question because the equation, the initial
conditions, etc. for the trajectory are different in different
coordinates. Hence, the distance from the center of the star
to the particle at any moment will be different in different
coordinates. If it were possible to actually measure this
distance, which coordinates' predictions would agree with
experiment?
I am inclined towards the Isotropic coordinates but would like
to hear your opinions too.

Regards,

Murat Ozer

On Tue, 27 Sep 2005 18:39:12 +0000 (UTC), Murat Ozer
wrote:

Suppose that I want to study the trajectory of a test particle
that is acted upon by a massive spherical object, say a star.
Suppose the test particle comes in from infinity at an initial
impact parameter b and is deflected by the gravitational
attraction of the star. Assuming that no other forces act on
the particle,the equation for the trajectory can be obtained
from the geodesic equation sraightforwardly. My question is,
which coordinates for the Schwarzschild metric should one use,
the Schwarzschild coordinates or the Isotropic coordinates?
This is a relavent question because the equation, the initial
conditions, etc. for the trajectory are different in different
coordinates. Hence, the distance from the center of the star
to the particle at any moment will be different in different
coordinates. If it were possible to actually measure this
distance, which coordinates' predictions would agree with
experiment?
I am inclined towards the Isotropic coordinates but would like
to hear your opinions too.

Regards,

Murat Ozer

It will not make any difference as to the physics of the result. Use
whichever coordinates system you feel most comfortable working with
mathematically.

the softrat
Unless Barad-dur is rebuilt, twice as evil as before, Frodo has triumphed!

--
"So tell me, just how long have you had this feeling that no one
is watching you?" (Christopher Locke: Entropy Gradient Reversals)

On Tue, 27 Sep 2005, Murat Ozer wrote:

Suppose that I want to study the trajectory of a test particle that is
acted upon by a massive spherical object, say a star. Suppose the test
particle comes in from infinity at an initial impact parameter b and is
deflected by the gravitational attraction of the star. Assuming that no
other forces act on the particle,the equation for the trajectory can be
obtained from the geodesic equation sraightforwardly. My question is,
which coordinates for the Schwarzschild metric should one use, the
Schwarzschild coordinates

ds^2 = -(1-2m/r) dt^2 + dr^2/(1-2m/r) + r^2 (du^2 + sin(u)^2 dv^2),

-infty t infty, 2m r infty, 0 u pi, -pi v pi

or the Isotropic coordinates?

ds^2 = -(1-M/2/R)^2/(1+M/2/R)^2 dt^2

+ (1+M/2/R)^4 [dR^2 + R^2 (du^2 + sin(u)^2 dv^2)],

-infty t infty, M/2 R infty, 0 u pi, -pi v pi

(where the coordinate ranges indicate the exterior region with
approximately a half plane removed, as in a local coordinate patch for
ordinary spherical coordinates).

The answer to your question depends on what you are trying to compute.
If you do everything correctly you will of course always get the -same-
answer for any physically meaningful quantity (compared at the same
event!), but one or the other may be more -convenient- for a given
purpose.

For example, the Schwarzschild radial coordinate r has a simple geometric
meaning: 4 pi r0^2 is the surface area of the surface t=t0, r=r0, happens
to be a geometric sphere with Gaussian curvature 1/r0^2. This is often
very useful to know.

OTH, the polar spherical isotropic coordinates may be more convenient for
studying "strong lensing" since the angles of null geodesics coming into
an event are correctly represented, while in the Schwarzschild chart, the
light cones do not appear to be "circular" cones, but rather are
flattened. This is because in the Schwarzschild chart, "transverse" but
not radial distances are not faithfully represented.

(Actually, for strong lensing the Costa chart is even better, but the
isotropic chart has still other uses.)

Another possible consideration in studying geodesics is that the geodesic
equations in one or the other chart may be -integrable in closed form- for
particular "special" values of the parameters. This is in fact the case
for tracks of null geodesics in the Schwarzschild chart.

This is a relavent question because the equation, the initial
conditions, etc. for the trajectory are different in different
coordinates. Hence, the distance from the center of the star to the
particle at any moment will be different in different coordinates.

Ah. We've discussed this issue before in this group, but this is a good
question since (rather amazingly) I haven't seen many problems in gtr
textbooks which ask the student to compute two quantities (in this case
lengths) in two charts and checking they give the same answer. This is
unfortunate. In particular, in this example, the computation you are
asking about raises at least two interesting conceptual issues.

But first, let's set the stage. For concreteness, I will assume you are
now talking about a stellar model in which a Schwarzschild fluid (or some
other static spherically symmetric perfect fluid model) is matched across
a surface of zero pressure (but positive matter density) to an exterior
Schwarzschild vacuum region. In this case, "distance to the center" would
make sense if understood as the length of a spacelike geodesic arc
everywhere orthogonal to the timelike Killing vector @/@t (which is
vorcity free, since the solution is static, hence hypersurface orthogonal,
and the geodesic is just a "radius" in some spatial hyperslice t=t0).
However, our discussion would also work for distance between any two
static observers outside the event horizon of a Schwarzschild hole.

In fact, it is probably easier to consider a closely related problem,
determining the distance along a "radial arc" (in a slice orthogonal to
static observers) from the surface of the star to the world line of one of
the static observers, since if you really were interested in the stellar
model you should find it easy to find the radial distance between the
(very hot) static observer at the center of the star and a static observer
on the surface.

Consider first the situation represented in the Schwarzchild chart. Say
the surface of the star (zero pressure sphere) occurs at r = r1, and that
our static observer's world line is
r = r2,u=Pi/2, v =0

where of course r2 r1 2m. (Actually, the surface of any compact
object would have to lie somewhat outside r = 2m by Buchdahl's theorem,
but never mind that.)

To find the length of the spacelike arc (which is in fact a geodesic arc,
an integral curve of the spacelike geodesic vector field sqrt(1-2m/r)
@/@r), we need to integrate

ds = dr/sqrt(1-2m/r)

from r1 to r2 r1. Note that to first order in m, this is

ds = 1+m/r

Plugging in some value, say m=1, into the exact integral, we have some
length, which we can convert to meters by plugging in the appropriate
factors to convert from geometric units (in which G = c = 1) to kms units.

To compare with the isotropic chart, we proceed similarly, but there are
two issues we need to address:

1. Which value of the isotropic radial coordinates R1 R2 correspond to
given values of the Schwarzschild radial coordinates r1 r2?

2. Which value of the mass parameter M corresponds to a given value of the
mass parameter m?

Of course you can anser the first question using the transformation
between the two charts (or directly, by considering surface areas of
spheres). Then we integrate

ds = (1+M/2/R)^2

from R1 to R2. Note that to first order in M, this is

ds = 1+M/R

which agrees with our previous expression when M = m, and suggests that
for weak fields, far from the star the distinction between the two charts
is moot (see below).

Now we have expressions in f(r,m) and g(R,M) which should give the same
numerical value when we plug in corresponding values for m, M. But we
already saw a hint that we should simply identify m,M. Indeed, expanding
the Schwarzschild line element to O(1/r) gives

ds^2 = -(1-2m/r) dt^2 + (1+2m/r) dr^2 + r^2 (du^2 + sin(u)^2 dv^2)

and expanding the isotropic line element to O(1/R) gives

ds^2 = -(1-2m/r) dt^2 +

(1+2m/R) [ dR^2 + R^2 (du^2 + sin(u)^2 dv^2) ]

In particular, we see that to O(1/r) and O(1/R)

(1+ m/r) r du ~ (1+M/R) R du

This means that sufficiently far out, R,r essentially coincide, with m,M
playing analogous roles. Thus the mass parameters m,M can be identified.

(Even better, you can look up Misner-Sharpe mass in a textbook like
Carroll.)

If it were possible to actually measure this distance, which
coordinates' predictions would agree with experiment?

Both will, of course, according to gtr.

Here is an explicit example. Say in the Schwarzschild chart we have r1 =
4 and r2 = 8. These correspond to spheres of areas 64 pi and 256 pi
respectively. Setting m = 1 and integrating we have

delta s ~ 4.971

Even without working out the transformation to isotropic chart, from A1,
A2 we see that r1, r2 correspond to R1 ~ 2.914 and R2 ~ 6.964. Setting M
= m = 1 and integrating the expression for radial ds in isotropic chart,
we find again

delta s ~ 4.971

That's the beauty of "local diffeomorphism covariance"!

As an exercise, I leave you to consider a model in which two static
observers bounce a radar signal off each other and divide the net travel
time by two to compute a "light travel time distance". Do you predict
that this will give the same result as we just found by laying rulers (in
imagination) along a radial spacelike geodesic orthogonal to @/@t? (Call
this the "static rulers distance"; unlike the light travel time existence,
it clearly only makes sense for hypersurface orthogonal observers). The
same result for the inner observer bounding a signal off the outer
observer, as for the outer observer bounding a signal off the inner
observer? What about two Hagihara observers in stable circular orbits of
different radii, both lying in the equatorial plane?

Ambitious students can also think about defining an "optical distance" by
measuring through your telescope the visual height of an astronaut
(oriented orthogonally to a radial arc) whom you know is exactly 2 metres
tall. Does this notion of distance agree with either of the other two?

"T. Essel" (hiding somewhere in cyberspace)

Murat Ozer wrote:
Suppose that I want to study the trajectory of a test particle
that is acted upon by a massive spherical object, say a star.
Suppose the test particle comes in from infinity at an initial
impact parameter b and is deflected by the gravitational
attraction of the star. Assuming that no other forces act on
the particle,the equation for the trajectory can be obtained
from the geodesic equation sraightforwardly. My question is,
which coordinates for the Schwarzschild metric should one use,
the Schwarzschild coordinates or the Isotropic coordinates?

You can use either. But you have to be careful that the questions
you are asking are questions about genuine observables -- that is,
coordinate-independent quantities. For example: what, exactly, do
you mean by "impact parameter b"? Do you mean minimum *proper*
distance from the surface? That's fine. But if you mean "minimum
value of some radial coordinate," you have to specify which radial
coordinate you mean.

This is a relavent question because the equation, the initial
conditions, etc. for the trajectory are different in different
coordinates.

The equations are. The initial conditions are, too, but only if
you are careless enough to describe them in terms of a particular
set of coordinates (in which case, you'd better use those same
coordinates for the calculation).

Hence, the distance from the center of the star
to the particle at any moment will be different in different
coordinates.

"Distance" as measured how? "Any moment" as determined by what
clock? A particle follows a unique trajectory, which has different
*descriptions* in different coordinate systems.

(If I want to plot a course from Los Angeles to Shanghai, should I
use a map based on a Mercator projection or a polar projection? The
answer is that I can use either one, but that in either case I have
to recognize that they are maps, with distortions that need to be
taken into account.)

You might want to look at Boddener and Will, Am. J. Phys. 71 (2003)
770, "Deflection of light to second order: A tool for illustrating
principles of general relativity," which discusses almost exactly
your question, with detailed computations in Schwarzschild and
isotropic (and also harmonic) coordinates and a careful explanation
of their relation and physical equivalence.

Steve Carlip

wrote:
Murat Ozer wrote:
Suppose that I want to study the trajectory of a test particle
that is acted upon by a massive spherical object, say a star.
Suppose the test particle comes in from infinity at an initial
impact parameter b and is deflected by the gravitational
attraction of the star. Assuming that no other forces act on
the particle,the equation for the trajectory can be obtained
from the geodesic equation sraightforwardly. My question is,
which coordinates for the Schwarzschild metric should one use,
the Schwarzschild coordinates or the Isotropic coordinates?

You can use either. But you have to be careful that the questions
you are asking are questions about genuine observables -- that is,
coordinate-independent quantities. For example: what, exactly, do
you mean by "impact parameter b"? Do you mean minimum *proper*
distance from the surface? That's fine. But if you mean "minimum
value of some radial coordinate," you have to specify which radial
coordinate you mean.

This is a relavent question because the equation, the initial
conditions, etc. for the trajectory are different in different
coordinates.

The equations are. The initial conditions are, too, but only if
you are careless enough to describe them in terms of a particular
set of coordinates (in which case, you'd better use those same
coordinates for the calculation).

Hence, the distance from the center of the star
to the particle at any moment will be different in different
coordinates.

"Distance" as measured how? "Any moment" as determined by what
clock? A particle follows a unique trajectory, which has different
*descriptions* in different coordinate systems.

(If I want to plot a course from Los Angeles to Shanghai, should I
use a map based on a Mercator projection or a polar projection? The
answer is that I can use either one, but that in either case I have
to recognize that they are maps, with distortions that need to be
taken into account.)

You might want to look at Boddener and Will, Am. J. Phys. 71 (2003)
770, "Deflection of light to second order: A tool for illustrating
principles of general relativity," which discusses almost exactly
your question, with detailed computations in Schwarzschild and
isotropic (and also harmonic) coordinates and a careful explanation
of their relation and physical equivalence.

Steve Carlip

As everybody will, hopefully, agree with me, it's best to argue on a
realistic case. Suppose a spaceship is approaching the Earth from the
North towards the South. The astronauts determine, at a time of their
choice, the initial impact parameter b (the vertical distance from the
ship to the North axis), and the distance d (the vertical distance from
the ship to the West axis) by sending radar signals to space stations
whose positions are known (Note that the center of the East, North,
West, South axes coincide with the center of the earth.) This way the
initial position of the ship from the center of the earth and the
initial angle Phi can be determined (Phi is 0, and Pi/2 on the East
and North axes, respectively):

position_i = Sqrt(b^2 + d^2); Phi_i = Pi/2 + ArcTan(b/d)

Assume, also, that the astronauts determine the ship's velocity
relative to the earth at the same time, call it v_i.
I will try to determine the position of this spaceship when it just
reaches the South axis at which time the angle Phi = 3Pi/2.
The equations of motion in the Schwarzschild, the Isotropic, and the
Special Relativistic coordinates, respectively, are

u'' + u - m_g/h^2 -3m_g u^2 =0 (Schwarzschild.),

u'' + u - m_g/2*(1+m_g/2*u/h^2*[3(1+m_g/2*u)^2*(1-m_g/2*u)^(-2)
+(1+m_g/2*u)^3*(1-m_g/2*u)^(-3) -2]=0 (Isotropic.),

u'' + u - m_g/h^2 - m_g/h^2*u =0 (Special Relativistic).

Here u = 1/r, m_g = GM/(c^2r) with G and M being the gravitational
constant and the mass of the earth, u'=du/dPhi, and h = r^2 dPhi/ds
is a constant of the motion and can be expressed in terms of the
ordinary angular momentum L as follows:

h = L/mc*( 1-2m_g *u)^(-1) (Schwarzschild)

h = L/mc*(1+m_g/2*u)^6*(1-m_g/2*u)^(-2) (Isotropic)

h = L/mc*(1+m_g*u) (Special Relativistic)
, where m is the mass of the spaceship and cancels out in the
equations.

In the numerical examples that I will present below, the ratio m_g/R_E,
where R_E is the radius of the earth, is much less than 1 so that the
spacetime is very close to being flat in the general relativistic
cases. This enables one to derive the expression for L:

L^2 = m^2v^2/(u^2 + u'^2).

Next, one can write a, say Mathematica , program to numerically solve
for r_{final} = 1/u_{final} when Phi_{final} = 3Pi/2. (Note that I have
been using the same symbols u and r for all the coordinate systems.
Since I treat them separately there is no confusion. ) Here are some
examples:

R |b/R |d/R | v_i(m/s) |Schw./R |Iso./R | Spec.Rel./R |
__________________________________________________ _______________
R_E | 5 |20 | 10^4 | 20.3335 | 20.3335 | 20.3335 |
| 5 |20 | 10^5 | 2034.29 | 2034.29 | 2034.29 |
| 5 |20 | 10^6 | 213272 | 213277 | 213277 |
__________________________________________________ _______________
0.5m | 5 |20 | 10^7 | 1.58306 | 1.58049 | 1.5911 |
| 5 |20 | 5x10^7 | 37.5351 | 39.775 | 39.8371 |
| 5 |20 | 10^8 | 129.444 | 159.146 | 159.375 |

In the second set of values, I consider a very compact object, but not
a blackhole, with radius half a meter and mass that of the earth. The
last three columns are the coordinates of the spaceship in the relevant
coordinates in terms of R, the radius of the massive object. These
numbers show very clearly that there is a very strong speed dependence
and for a small but very heavy object the predictions of the
Schwarzschild and Isotropic coordinates may be very different. In all
these cases though, the spacetime is very close to being flat, and
therefore I believe the isotropic coordinates' predictions are the
actual values that the astronauts may measure at Phi_{final} = 3Pi/2
because they are very nearly the same as the (flat) Special
Realativistic predictions.

Murat Ozer

Murat Ozer wrote:

[...]
Suppose a spaceship is approaching the Earth from the
North towards the South.

This is harder than you think. For example, what do you mean
by "from the North towards the South"? I understand what that
means at the location of the Earth, but if the spaceship is far
from Earth, how do you transfer your definition of a direction?
You can't just say "in a direction parallel to the direction
from North to South on the Earth" -- the spacetime geometry
isn't Euclidean, so there's no "distant parallelism." You can
say, "parallel transport a North-to-South-pointing vector to
the spaceship," but the result is not unique unless you tell
me what path to use for that parallel transport.

In general, you can compare "North-to-South" here and "North-to-
South" there only if you have at least a piece of a coordinate
system already chosen.

The astronauts determine, at a time of their
choice, the initial impact parameter b (the vertical distance from the
ship to the North axis), and the distance d (the vertical distance from
the ship to the West axis) by sending radar signals to space stations
whose positions are known

What, exactly, does it mean to say that the positions of the space
stations are known? How are those positions described? In some
coordinate-independent way? (If so, what?) Or in a way that already
implies a choice of coordinates?

[...]
This way the
initial position of the ship from the center of the earth and the
initial angle Phi can be determined (Phi is 0, and Pi/2 on the East
and North axes, respectively):

position_i = Sqrt(b^2 + d^2); Phi_i = Pi/2 + ArcTan(b/d)

Here's a big problem. You are assuming Pythagoras's theorem -- that
is, you are assuming that the spatial geometry is flat. This is *not
true* for most coordinate systems. If this is the way you want to
define position and angle, and you want your results to coincide with
the distance and angle computed in some coordinate system, then you
must use a coordinate system in which space at a fixed time is flat.
For the Schwarzschild metric, though not, for example, the Kerr metric,
such coordinate systems exist; they are Painleve-Gullstrand coordinates
(see gr-qc/0001069 for a nice introduction). But in Schwarzschild, or
isotropic, coordinates, these expressions are simply not correct.

Assume, also, that the astronauts determine the ship's velocity
relative to the earth at the same time, call it v_i.

Again, you have to tell me what definition of velocity you're using.
Velocity is the rate of change of position with respect to time.
You've said, roughly, how you're defining position. But what time
coordinate are you using to define velocity? Proper time measured
by a clock on the spaceship? Time as measured at infinity (e.g.,
the Schwarzschild time coordinate)? If the latter, how does the
spaceship determine this time? The answer will, in general, depend
on your coordinate choices.

The basic problem is that you've made a bunch of hidden assumptions
about coordinate systems in setting up your problem. The biggest, I
suspect, is the assumption of Euclidean geometry in determining the
position. But the choice of a time to define your velocity may be as
important -- dx/dt is a physically different quantity depending on
whether t is time in Schwarzschild coordinates or isotropic coordinates.
To do this right, you will need to go back and give careful, coordinate-
independent descriptions of your parameters, or else make the coordinate
dependence explicit and keep track of it.

Steve Carlip

Kindly pardon my top post.

IMO Mr. Ozer is asking a very interesting question
that involves the Pound-Rebka experiment, Shapiro
and relating that with "radar" that involves the
gravitational (Einstein) "red-shift".

Mr. Ozer may be on the top of a tower shining a
laser to a reflector at the base of the tower and
will find, by virtue of energy conservation, the
reflected energy he receives from the reflector
to be precisely equal to his emitted energy.

To be more precise, let Mr. Ozer set his frequency
"f" so that there are "n" wavelengths between him
and the base of the tower.

I think that "n" is a scalar and an invariant.

Is that a reasonable approach?
Ken S. Tucker

Murat Ozer wrote:
wrote:
Murat Ozer wrote:
Suppose that I want to study the trajectory of a test particle
that is acted upon by a massive spherical object, say a star.
Suppose the test particle comes in from infinity at an initial
impact parameter b and is deflected by the gravitational
attraction of the star. Assuming that no other forces act on
the particle,the equation for the trajectory can be obtained
from the geodesic equation sraightforwardly. My question is,
which coordinates for the Schwarzschild metric should one use,
the Schwarzschild coordinates or the Isotropic coordinates?

You can use either. But you have to be careful that the questions
you are asking are questions about genuine observables -- that is,
coordinate-independent quantities. For example: what, exactly, do
you mean by "impact parameter b"? Do you mean minimum *proper*
distance from the surface? That's fine. But if you mean "minimum
value of some radial coordinate," you have to specify which radial
coordinate you mean.

This is a relavent question because the equation, the initial
conditions, etc. for the trajectory are different in different
coordinates.

The equations are. The initial conditions are, too, but only if
you are careless enough to describe them in terms of a particular
set of coordinates (in which case, you'd better use those same
coordinates for the calculation).

Hence, the distance from the center of the star
to the particle at any moment will be different in different
coordinates.

"Distance" as measured how? "Any moment" as determined by what
clock? A particle follows a unique trajectory, which has different
*descriptions* in different coordinate systems.

(If I want to plot a course from Los Angeles to Shanghai, should I
use a map based on a Mercator projection or a polar projection? The
answer is that I can use either one, but that in either case I have
to recognize that they are maps, with distortions that need to be
taken into account.)

You might want to look at Boddener and Will, Am. J. Phys. 71 (2003)
770, "Deflection of light to second order: A tool for illustrating
principles of general relativity," which discusses almost exactly
your question, with detailed computations in Schwarzschild and
isotropic (and also harmonic) coordinates and a careful explanation
of their relation and physical equivalence.

Steve Carlip

As everybody will, hopefully, agree with me, it's best to argue on a
realistic case. Suppose a spaceship is approaching the Earth from the
North towards the South. The astronauts determine, at a time of their
choice, the initial impact parameter b (the vertical distance from the
ship to the North axis), and the distance d (the vertical distance from
the ship to the West axis) by sending radar signals to space stations
whose positions are known (Note that the center of the East, North,
West, South axes coincide with the center of the earth.) This way the
initial position of the ship from the center of the earth and the
initial angle Phi can be determined (Phi is 0, and Pi/2 on the East
and North axes, respectively):

position_i = Sqrt(b^2 + d^2); Phi_i = Pi/2 + ArcTan(b/d)

Assume, also, that the astronauts determine the ship's velocity
relative to the earth at the same time, call it v_i.
I will try to determine the position of this spaceship when it just
reaches the South axis at which time the angle Phi = 3Pi/2.
The equations of motion in the Schwarzschild, the Isotropic, and the
Special Relativistic coordinates, respectively, are

u'' + u - m_g/h^2 -3m_g u^2 =0 (Schwarzschild.),

u'' + u - m_g/2*(1+m_g/2*u/h^2*[3(1+m_g/2*u)^2*(1-m_g/2*u)^(-2)
+(1+m_g/2*u)^3*(1-m_g/2*u)^(-3) -2]=0 (Isotropic.),

u'' + u - m_g/h^2 - m_g/h^2*u =0 (Special Relativistic).

Here u = 1/r, m_g = GM/(c^2r) with G and M being the gravitational
constant and the mass of the earth, u'=du/dPhi, and h = r^2 dPhi/ds
is a constant of the motion and can be expressed in terms of the
ordinary angular momentum L as follows:

h = L/mc*( 1-2m_g *u)^(-1) (Schwarzschild)

h = L/mc*(1+m_g/2*u)^6*(1-m_g/2*u)^(-2) (Isotropic)

h = L/mc*(1+m_g*u) (Special Relativistic)
, where m is the mass of the spaceship and cancels out in the
equations.

In the numerical examples that I will present below, the ratio m_g/R_E,
where R_E is the radius of the earth, is much less than 1 so that the
spacetime is very close to being flat in the general relativistic
cases. This enables one to derive the expression for L:

L^2 = m^2v^2/(u^2 + u'^2).

Next, one can write a, say Mathematica , program to numerically solve
for r_{final} = 1/u_{final} when Phi_{final} = 3Pi/2. (Note that I have
been using the same symbols u and r for all the coordinate systems.
Since I treat them separately there is no confusion. ) Here are some
examples:

R |b/R |d/R | v_i(m/s) |Schw./R |Iso./R | Spec.Rel./R |
__________________________________________________ _______________
R_E | 5 |20 | 10^4 | 20.3335 | 20.3335 | 20.3335 |
| 5 |20 | 10^5 | 2034.29 | 2034.29 | 2034.29 |
| 5 |20 | 10^6 | 213272 | 213277 | 213277 |
__________________________________________________ _______________
0.5m | 5 |20 | 10^7 | 1.58306 | 1.58049 | 1.5911 |
| 5 |20 | 5x10^7 | 37.5351 | 39.775 | 39.8371 |
| 5 |20 | 10^8 | 129.444 | 159.146 | 159.375 |

In the second set of values, I consider a very compact object, but not
a blackhole, with radius half a meter and mass that of the earth. The
last three columns are the coordinates of the spaceship in the relevant
coordinates in terms of R, the radius of the massive object. These
numbers show very clearly that there is a very strong speed dependence
and for a small but very heavy object the predictions of the
Schwarzschild and Isotropic coordinates may be very different. In all
these cases though, the spacetime is very close to being flat, and
therefore I believe the isotropic coordinates' predictions are the
actual values that the astronauts may measure at Phi_{final} = 3Pi/2
because they are very nearly the same as the (flat) Special
Realativistic predictions.

Murat Ozer

wrote:
Murat Ozer wrote:
This way the
initial position of the ship from the center of the earth and the
initial angle Phi can be determined (Phi is 0, and Pi/2 on the East
and North axes, respectively):

position_i = Sqrt(b^2 + d^2); Phi_i = Pi/2 + ArcTan(b/d)


Steve Carlip commented
[[many very cogent remarks snipped]]
The basic problem is that you've made a bunch of hidden assumptions
about coordinate systems in setting up your problem. The biggest, I
suspect, is the assumption of Euclidean geometry in determining the
position. But the choice of a time to define your velocity may be as
important -- dx/dt is a physically different quantity depending on
whether t is time in Schwarzschild coordinates or isotropic coordinates.
To do this right, you will need to go back and give careful, coordinate-
independent descriptions of your parameters, or else make the coordinate
dependence explicit and keep track of it.

As a further example of what Steve Carlip was talking about, let's
take a really simple problem: finding the distance between two specified
points (say one of them on the Earth's surface, the other one on the
moon's surface) at some specified time.

[As Steve noted, defining "at some specified time"
is a major problem in itself, but for what I'm going
to say that doesn't matter... so let's just assume
that the two points have already-synchronized-somehow
atomic clocks.]


One obvious way (let's call it #1) to measure the distance is to fire
a short laser pulse from the Earth station to the moon station, have
a mirror at the moon reflect (some of) that light back, and time its
receipt back at the original Earth station.

Another way (let's call it #2) would be to (logically, this is a
gedanken experiment, not a real one) stretch a long cable between
the Earth and the moon, make sure it's straight (by sighting along it),
and measure the length of the cable with meter sticks (i.e. count
how standard-1-meter-bars it takes to span the full length).

Another way (let's call it #3) would be to use the same long cable
as #2, but now don't keep it straight by sighting along it. Rather,
let's just (gedanken) take the *minimum* length (measured the same
way as in #2) over all possible shapes of the cable.

Finally, we might (call this #4) put 1-meter rods at the Earth and
moon stations, put a spacecraft S at (say) the Earth-moon system's
L4 or L4 point, and have S measure the angles E-S-M and the angles
subtended by the 1-meter rods at E and M, and do the appropriate
trigonometry.


Ok now the fun part: In general #1, #2, #3, and #4 will all give
*different* answers!
[For the Earth-Moon system I would expect
differences on the order of millimeters to
centimeters, since GM_earth/c^2 is about 1cm.]
So just saying "distance" isn't good enough: we have to say *how*
we're going to measure (define!) it, and we really ought to say why
we're choosing that way rather than one of the many other plausible
ways.

In other words, [switching back to relativity terminology] we have
to define (give an operational definition of) a coordinate system.


ciao,

--
-- "Jonathan Thornburg -- remove -animal to reply"
Max-Planck-Institut fuer Gravitationsphysik (Albert-Einstein-Institut),
Golm, Germany, "Old Europe" http://www.aei.mpg.de/~jthorn/home.html
"Washing one's hands of the conflict between the powerful and the
powerless means to side with the powerful, not to be neutral."
-- quote by Freire / poster by Oxfam

wrote:
Murat Ozer wrote:

[...]
Suppose a spaceship is approaching the Earth from the
North towards the South.

This is harder than you think. For example, what do you mean
by "from the North towards the South"? I understand what that
means at the location of the Earth, but if the spaceship is far
from Earth, how do you transfer your definition of a direction?
You can't just say "in a direction parallel to the direction
from North to South on the Earth" -- the spacetime geometry
isn't Euclidean, so there's no "distant parallelism." You can
say, "parallel transport a North-to-South-pointing vector to
the spaceship," but the result is not unique unless you tell
me what path to use for that parallel transport.


I was only trying to find a way to locate the spaceship with respect
to the center of the Earth. My suggestion may be at fault.No problem.
I'm sure there is a practical way of keeping track of a spaceship
all the time. Otherwise NASA's space missions would not have
materialized.


In general, you can compare "North-to-South" here and "North-to-
South" there only if you have at least a piece of a coordinate
system already chosen.

The astronauts determine, at a time of their
choice, the initial impact parameter b (the vertical distance from the
ship to the North axis), and the distance d (the vertical distance from
the ship to the West axis) by sending radar signals to space stations
whose positions are known

What, exactly, does it mean to say that the positions of the space
stations are known? How are those positions described? In some
coordinate-independent way? (If so, what?) Or in a way that already
implies a choice of coordinates?


Again, this really does not matter. I'm sure
astronomers/astrophysicists
know how to keep track of a spaceship.


[...]
This way the
initial position of the ship from the center of the earth and the
initial angle Phi can be determined (Phi is 0, and Pi/2 on the East
and North axes, respectively):

position_i = Sqrt(b^2 + d^2); Phi_i = Pi/2 + ArcTan(b/d)

Here's a big problem. You are assuming Pythagoras's theorem -- that
is, you are assuming that the spatial geometry is flat. This is *not
true* for most coordinate systems.

In the examples I gave the ratio M_G/R is much smaller than 1. To be
specific, it's 6.94x10^(-10) and 0.0088 for R=R_E (radius of the Earth)
and R=0.5m, respectively. Therefore, Pythogoras's theorem should be
a very good approximation.



If this is the way you want to
define position and angle, and you want your results to coincide with
the distance and angle computed in some coordinate system, then you
must use a coordinate system in which space at a fixed time is flat.
For the Schwarzschild metric, though not, for example, the Kerr metric,
such coordinate systems exist; they are Painleve-Gullstrand coordinates
(see gr-qc/0001069 for a nice introduction). But in Schwarzschild, or
isotropic, coordinates, these expressions are simply not correct.

Assume, also, that the astronauts determine the ship's velocity
relative to the earth at the same time, call it v_i.

Again, you have to tell me what definition of velocity you're using.
Velocity is the rate of change of position with respect to time.
You've said, roughly, how you're defining position. But what time
coordinate are you using to define velocity? Proper time measured
by a clock on the spaceship? Time as measured at infinity (e.g.,
the Schwarzschild time coordinate)? If the latter, how does the
spaceship determine this time? The answer will, in general, depend
on your coordinate choices.


I don't think such details are necessary when spacetime is almost
flat. What I had on my mind was the rate of change of the position
of the ship with respect to the Earth time. I expect that all the
different definitions would agree for nearly flat spacetimes. Again,
how do astronomers/astrophysicists determine the velocity of
spaceships?


The basic problem is that you've made a bunch of hidden assumptions
about coordinate systems in setting up your problem. The biggest, I
suspect, is the assumption of Euclidean geometry in determining the
position. But the choice of a time to define your velocity may be as
important -- dx/dt is a physically different quantity depending on
whether t is time in Schwarzschild coordinates or isotropic coordinates.
To do this right, you will need to go back and give careful, coordinate-
independent descriptions of your parameters, or else make the coordinate
dependence explicit and keep track of it.


I agree with everything you say. However, all these details, I believe,
are not necessary for nearly flat spacetimes. Calculating the position
of a spacehip in different coordinates without making approximations
might be, my guess is, impossible. For nearly flat spacetimes, as in
the examples I gave, I do not see why my procedure should not work.
Therefore, the calculations I presented indicate that Schwarzschild and
Isotropic coordinates do indeed predict different positions especially
when the velocity of the ship is large. I know this is against one's
immediate expectations.
But this seems to be the unexpected result. One should try to
understand why this is so...

Regards,

Murat Ozer


Steve Carlip