Ben Niehoff said:
You seem to be clinging to Schwarzschild coordinates as though they are special; but I tell you, they are not special in any way. The following are all metrics that represent the Schwarzschild geometry:
Oh, certainly I think that Schwarzschild coordinates are special, because I've seen a common-sense argument
http://www.mathpages.com/rr/s8-09/8-09.htm" showing that
\left (\frac{\partial \tau}{\partial r} \right )^2\approx 1-\frac{G M}{r c^2}
The fact is, I have worked through this on my own time, and I know exactly what the variables are referring to. So when you make the claim that these variables are meaningless, you're telling me something I
know is not true; at least for the g
00 component.
On the other hand, I've not worked through any common-sense argument for the g
11 component of the Schwarzschild metric:
\left (\frac{\partial s}{\partial r} \right )^2\approx \frac{1}{1-\frac{G M}{r c^2}}
I can see there is a derivation in the http://www.blatword.co.uk/space-time/Carrol_GR_lectures.pdf" on pages 168-172. That remains a goal for me, to work through that derivation as well, but for now, I'm not comfortable with most of the concepts involved there. For instance, why does he start with the assumption that g
00 is the negative reciprocal of g
11?
But the point is, I didn't see the g
00 component by
- assuming g_{11}=-1/g_{00}
- finding the connection coefficients
- finding the nonvanishing components of the Reimann tensor
- taking the contraction (as usual?) to find the Ricci tensor
- setting all the components of the Ricci tensor to zero
- discovering that the g00 and g11 had to be functions of r, only,
- Setting R00=R11=0
- Doing some fancy differential equations with boundary conditions, and deriving the metric
Now, I am willing and (I think) able to go through all this; that's just a matter of time. But here's the point I want to make: If someone went through these steps, without going through the other derivation, they might be left with the impression that the coordinates involved were somehow, arbitrary, i.e. nonphysical. They might think that the physical interpretation of those variables were ambiguous, or even nonexistant. And that seems to be where you are coming from, Ben.
But where I'm coming from, you see, is the other derivation; where at all times, we're talking about clocks and rulers and measurements; figuring out how to modify the Rindler coordinates into the Schwarzschild coordinates. Even though I only figured out how to get the result for the time-time component, I trust the derivation to represent something meaningful.
Ben Niehoff said:
There is no overlaying cartesian coordinate system! Such a thing is impossible.
I wish I could find some way to argue this point with you. Let me try a few things.
(1) If you could imagine yourself in a space-ship; reasonably far from a gravitational mass, could you go around it? In that asymptotic region, far from the system, where the schwarzschild metric approaches the flat space-time. Can't you go around the planet? Don't you have a fairly firm concept of how far you went? Don't base it on your path, but take someone further away, who can see both your starting position, and your ending position. He can see that you've moved from one side of the planet to the other side of the planet. And even with the planet there, he can describe your current and final position according to known distances, and known angles, effectively figured in an "overlying cartesion coordinate system."
(2) As the space-ship operator, do you feel that you just now went around a spatial anomoly. You may feel that since distances are changed inside that anomoly that you have gone an "unmeasurable" distance. However, couldn't you also take another approach? That spatial anomoly is CONTAINED in a region from your perspective. It is an unusual feature in an otherwise cartesian space. In that space, the volume where the black-hole, large planet, star, etc is a finite region; even a small and insignificant region if you're far enough away.
If you can localize a spatial anomoly to a given region; and if you can go AROUND a planet or star, I would say, you are operating with an overlying cartesian coordinate system. And the coordinates of that coordinate system run explicitly
RIGHT THROUGH that spatial anomoly. They don't somehow take a break near the star and go all ambiguous on you. Every event that happens outside the schwarzchild radius, at least, is going to happen at a specific point in space and time in the overlying coordinate system.
I'm otherwise at a loss for how to explain this to you, but maybe you can identify what you think is in error.
Ben Niehoff said:
Then do tell us, what (t,x,y,z) variables are you talking about? Define them mathematically.
If you're completely convinced that an overlying cartesian coordinate system is impossible then there's hardly a point to explain this to you, because that's my exact definition.
That being said, (t,x,y,z) represent the coordinates of events in the overlying cartesian coordinate system (plus time). Naturally, unless you've been at least somewhat swayed by the arguments above, that definition won't help. But it's all I've got.
Contrast that with your own explanation:
Ben Niehoff said:
Coordinates do not carry any geometrical information at all! They are just labels. The metric carries the geometrical information. You must compute its Riemann curvature tensor to determine if it describes a curved manifold or a flat one.
(P.S. What criteria of the Riemann curvature determines whether a manifold is flat?)
In any case, it is my impression that coordinates DO carry geometrical information, and I'm rather at a loss for how you could argue otherwise.
Ben Niehoff said:
There is no ground or table. There is a black hole and we're floating in space. Can you give a more precise definition?
Mathematically, the Schwarzschild metric identifies the transformation:
c^2 {d \tau}^{2} = \left(1 - \frac{2 G M}{r c^2} \right) c^2 dt^2 - \left(1-\frac{2 G M}{r c^2}\right)^{-1} dr^2 - r^2 \left(d\theta^2 + \sin^2\theta \, d\varphi^2\right)
It's a differential equation, so you'll need some boundary values to define things explicitly.
But this isn't just a mapping of (dt,dr,d\theta,d\phi) \to ds
It is a mapping of (dt,dr,d\theta,d\phi) \to (d\tau,dR, d\theta',d\phi')
http://en.wikipedia.org/wiki/Coordinate_system In geometry, a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of a point or other geometric element.
My claim is that after applying boundary conditions, (\tau, R, \theta', \phi'), (t,r,\theta,\phi), \, \mathrm{and}\, (t,x,y,z) all represent coordinates, in that they all uniquely determine the position of events.
It should be obvious already that (t,r,\theta,\phi) represent coordinates. If not, please explain why.
These spherical coordinates can be mapped to cartesian coordinates in the standard way
\begin{matrix} t=t\\ z=r \cos(\theta)\\ x=r \sin(\theta)\cos(\phi)\\ y=r \sin(\theta)\sin(\phi) \end{matrix}
Which is also a unique mapping, except at r=0, there are several different values of phi all mapping to the same point.
The hard part is showing that (\tau, R, \theta', \phi') are coordinates.
First, a definition of their differentials:
\begin{matrix} dR \equiv ds|_{d\theta=d\phi=dt=0}=\left ( \frac{1}{\sqrt{1-\frac{G M}{r c^2}}} \right )dr\\ d\tau \equiv ds|_{d\theta=d\phi=dr=0}=\left ( \sqrt{1-\frac{G M}{r c^2}} \right )dr\\ r d\theta' \equiv ds|_{dt=d\phi=dr=0} =r d\theta\\ r sin(\theta)d\phi' \equiv ds|_{d\theta=dt=dr=0} =r sin(\theta)d\phi \\ \end{matrix}
Since d\phi'=d\phi\; \mathrm{and}\; d\theta'=d\theta, I use them interchangably.
Using boundary conditions of \begin{matrix} R(r=\frac{G M}{ c^2})=0\\ \tau(t=0)=0 \end{matrix}
We can calculate the definite integrals:
\begin{matrix} R(r)=\int_{G M/c^2}^{r}\frac{1}{\sqrt{1-\frac{G M}{\rho c^2}}}d\rho\\ \tau(t,r)=\int_{0}^{t}\sqrt{1-\frac{G M}{r c^2}}dt = \left (\sqrt{1-\frac{G M}{r c^2}} \right )t \\ \theta'(\theta)=\int_{0}^{\theta} d\theta=\theta\\ \phi'(\phi)=\int_{0}^{\phi} d\phi=\phi \end{matrix}
Ben Niehoff said:
There is no ground or table. There is a black hole and we're floating in space. Can you give a more precise definition?
If I am setting dr=0, I am saying, essentially there is no "drop" between the events. In a black hole, this might not be available; you'd have to be in a rocketship thrusting away from the center in order to maintain dr=0. But if you are on a solid planet, there is no problem. You just set your clock on a floor or table.
I can tell when you've used the letter tau, no problem with fonts. What I'm telling you is that you haven't used it as a coordinate. The quantity tau that you define here is not a coordinate. Do you see why?
No. I don't. Again, the definition I see says: "a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of a point or other geometric element."
It appears to me that tau is one of four coordinates that can uniquely determine the space-time location of an event.