I will try to explain the first equation. First assume the standard form of a Schwarzschild black hole. If you had an atomic clock which was prevented from changing its spatial coordinates you could write an equation that would describe the relationship between the atomic clocks time and the coordinate time (such a path is not a geodesic). The equation would match your first equation with a constant added to one side or the other.
In your equation T would be the coordinate time and T0 would be atomic clock time. At R->infinity they would differ only by the constant.
It is important to understand that the coordinate time is not physically significant by itself, it is determined by the form of the metric. The same thing can be said about all of the coordinates. You need to solve for the geodesics between pairs of events in order to describe the physics of the space (actually you need at least three events to describe anything really interesting). This physics describes the behavour of test particles with arbitrarily low energies and momentums. If the value of the right hand side of equation 1 from my link is positive the geodesic models a massive particle, if the value is 0 the geodesic models a mass less particle traveling at the speed of light (negative values are non-physical).

ChipperQ:
The equation is of no practical importance in describing velocity distribution of a galaxy, the effect is too small.
The finial observational result is the difference in Doppler shift caused by the difference between velocities of each visible region of the galaxy and galaxies average velocity. The differential velocity of some of the stars is away from us (relatively red shifted) for others it is toward us (relatively blue shifted).
The containing gratational field is calculated by assuming that the density and velocity distributions are on average changing slowly enough to be taken as static.

Something I should have added to (said differently in) my previous post. The left hand side of equation 1 from Schwarzschild Black Hole is usually set to 1, 0 or -1 (time like, null or space like) when calculating geodesics.

Thanks Mark – I have lots more to learn about, and I can't imagine any other kind of (nonlinear) relativistic explanation, then, for an observed slower rotation of stars... Since it may take only a “few” TeV to form a black hole, I don't think I'd have dismissed dark matter altogether, even if Cooperstock was right.

Now, the paper is mathematically beyond me, but I assume that "singular" is used here in a technical sense: a discontinuity.

Basically: A discontinuity in the metric implies a singular matter source (delta-function, or very thin and very dense source).
It comes from Cooperstock's assumption of a metric involving the absolute value of z. The Einstein equations say that the metric is such that a combination of derivatives (rates of change with respect to z and other coordinates) equals the matter source.

If you plot abs(z) versus z, it's shaped like a V. (The function of abs(z) Cooperstock uses looks different, but qualitatively it's the same.) At the bottom of the V (or more generally the kink in the curve) the rate of change with respect to z is not well defined. If you plot the rate of change vs z, it looks like a step. And if you plot the rate of change of that, it's zero everywhere except an infinitely tall, thin spike in the middle.

It's a classic undergrad result. I just gave a couple of problems like that on my quantum mechanics midterm (it happens in any field or wave equation, not just relativity) and most of my students got it.

Then again, Cooperstock has company in forgetting the extra singular term. Not so long ago a LIGO colleague of mine forgot such a term, but since there are many of us it got caught before appearing in a journal. They're easy to forget. Especially if they appear at the edge of the region you're looking at, so it's hard to see the kink.

The bottom line, of course, is that unless you're within 1AU of the center of the galaxy, Newtonian gravity works quite well.

Looks like I quoted the right person, and was at least right about the 'fast-moving' part. Thanks for clearing that up, Ben. I had totally missed the url to Cooperstock's paper (in the ABC Online Forum, from the 'source' link in the BAUT forum) when I first checked Ken's links, and now that I've read it, I wouldn't have known that Cooperstock's “disk of luminous matter” was actually a discontinuity in the metric, nor would I have known the implications, so your post is most helpful, indeed!

Thanks for those links, Ken – some very sharp minds also in the BAUT forum (nicked myself on a couple of the posts :))

You are very welcome. :)

Sorry for delay replying; so many boards so little time.

I should have made it clear that the link I posted was not to the "source" in the BAUT post, but the astro-ph paper. Sorry.

And Ben, thanks for saying what I would have wanted to say if I had only known what I was talking about! As always, your explanations are the best.

When I taught calculus, |x| was my favorite counterexample. I'll have to go looking for that "infinitely tall, thin spike" in the 2nd derivative though. :) Mathematicians and physicists will never get along, which is as it should be...

## ChipperQ: I will try to

)

ChipperQ:

I will try to explain the first equation. First assume the standard form of a Schwarzschild black hole. If you had an atomic clock which was prevented from changing its spatial coordinates you could write an equation that would describe the relationship between the atomic clocks time and the coordinate time (such a path is not a geodesic). The equation would match your first equation with a constant added to one side or the other.

In your equation T would be the coordinate time and T0 would be atomic clock time. At R->infinity they would differ only by the constant.

It is important to understand that the coordinate time is not physically significant by itself, it is determined by the form of the metric. The same thing can be said about all of the coordinates. You need to solve for the geodesics between pairs of events in order to describe the physics of the space (actually you need at least three events to describe anything really interesting). This physics describes the behavour of test particles with arbitrarily low energies and momentums. If the value of the right hand side of equation 1 from my link is positive the geodesic models a massive particle, if the value is 0 the geodesic models a mass less particle traveling at the speed of light (negative values are non-physical).

## ChipperQ: The equation is of

)

ChipperQ:

The equation is of no practical importance in describing velocity distribution of a galaxy, the effect is too small.

The finial observational result is the difference in Doppler shift caused by the difference between velocities of each visible region of the galaxy and galaxies average velocity. The differential velocity of some of the stars is away from us (relatively red shifted) for others it is toward us (relatively blue shifted).

The containing gratational field is calculated by assuming that the density and velocity distributions are on average changing slowly enough to be taken as static.

Something I should have added to (said differently in) my previous post. The left hand side of equation 1 from Schwarzschild Black Hole is usually set to 1, 0 or -1 (time like, null or space like) when calculating geodesics.

## Thanks Mark – I have lots

)

Thanks Mark – I have lots more to learn about, and I can't imagine any other kind of (nonlinear) relativistic explanation, then, for an observed slower rotation of stars... Since it may take only a “few” TeV to form a black hole, I don't think I'd have dismissed dark matter altogether, even if Cooperstock was right.

## Ken Vogt

)

Ken Vogt wrote:

Basically: A discontinuity in the metric implies a singular matter source (delta-function, or very thin and very dense source).

It comes from Cooperstock's assumption of a metric involving the absolute value of z. The Einstein equations say that the metric is such that a combination of derivatives (rates of change with respect to z and other coordinates) equals the matter source.

If you plot abs(z) versus z, it's shaped like a V. (The function of abs(z) Cooperstock uses looks different, but qualitatively it's the same.) At the bottom of the V (or more generally the kink in the curve) the rate of change with respect to z is not well defined. If you plot the rate of change vs z, it looks like a step. And if you plot the rate of change of that, it's zero everywhere except an infinitely tall, thin spike in the middle.

It's a classic undergrad result. I just gave a couple of problems like that on my quantum mechanics midterm (it happens in any field or wave equation, not just relativity) and most of my students got it.

Then again, Cooperstock has company in forgetting the extra singular term. Not so long ago a LIGO colleague of mine forgot such a term, but since there are many of us it got caught before appearing in a journal. They're easy to forget. Especially if they appear at the edge of the region you're looking at, so it's hard to see the kink.

The bottom line, of course, is that unless you're within 1AU of the center of the galaxy, Newtonian gravity works quite well.

Ben

## Looks like I quoted the right

)

Looks like I quoted the right person, and was at least right about the 'fast-moving' part. Thanks for clearing that up, Ben. I had totally missed the url to Cooperstock's paper (in the ABC Online Forum, from the 'source' link in the BAUT forum) when I first checked Ken's links, and now that I've read it, I wouldn't have known that Cooperstock's “disk of luminous matter” was actually a discontinuity in the metric, nor would I have known the implications, so your post is most helpful, indeed!

## Hi Chipper

)

Hi Chipper Q,

You are very welcome. :)

Sorry for delay replying; so many boards so little time.

I should have made it clear that the link I posted was not to the "source" in the BAUT post, but the astro-ph paper. Sorry.

And Ben, thanks for saying what I would have wanted to say if I had only known what I was talking about! As always, your explanations are the best.

When I taught calculus, |x| was my favorite counterexample. I'll have to go looking for that "infinitely tall, thin spike" in the 2nd derivative though. :) Mathematicians and physicists will never get along, which is as it should be...

Ken