W.D.Clinger
Philosopher
You continue to speak as though you were completely ignorant of the relevant mathematics and physics.
You continue to speak as though you were completely ignorant of the relevant mathematics and physics.
In this metric though:
http://latex.codecogs.com/gif.latex?ds^2 = (1-\frac{r}{R})^{-2} dt^2 - dx^2 - dy^2 - dz^2
The height of the chart would be infinity nearly everywhere, except around the observer, where it dips down to 1.
Are we to assume r = sqrt (x2 + y2 + z2) ?
Let's say there's an event at (t, x, y, z). The metric tensor, being a tensor field, places a tensor at the point (t, x, y, z).
Let's say we change coordinates, using a different origin, and now that event lies at new coordinates (t', x', y', z').
The metric tensor should place a tensor there, and am I right that it should be the same tensor (have the same scalar values) as before?
It should be the same tensor, but the part I highlighted in red doesn't make sense as written, and would be incorrect if rewritten to convey what I think you were trying to say.
The reason that highlighted phrase doesn't make sense is that it doesn't really make sense to ask whether a tensor has "the same scalar values". I suspect you were trying to say the numerical components of the tensor are unchanged by the coordinate transformation, but that is flat-out false. Although it's the same tensor as before, its coordinate-dependent components will be different in the new coordinate system.
In my previous message, I was going to give you an example of that, but I got a minus sign wrong so I deleted the example. Here is the example:
ds2 = − (ρ + 1)2 c2 dt2 + (ρ + 1)−4 dρ2 + (ρ / (ρ + 1))2 [ dθ2 + sin2 θ dφ2 ]
In that metric form, the radial coordinate ρ (which is the Greek letter rho) ranges over the non-negative real numbers.
Do you recognize the tensor field expressed by that metric form?
Your discussion is insufficiently precise. It is unclear, for example, what you mean by moving the origin to a new point. Are you transforming θ as well as r? If so, then your metric form is going to be a lot more complicated in the new coordinates than in the original coordinates. If you aren't transforming θ, then you are continuing to calculate the angle θ with respect to the original origin, not with respect to the new origin.It should be the same tensor, but the part I highlighted in red doesn't make sense as written, and would be incorrect if rewritten to convey what I think you were trying to say.
The reason that highlighted phrase doesn't make sense is that it doesn't really make sense to ask whether a tensor has "the same scalar values". I suspect you were trying to say the numerical components of the tensor are unchanged by the coordinate transformation, but that is flat-out false. Although it's the same tensor as before, its coordinate-dependent components will be different in the new coordinate system.
What if you kept the units and coordinate system (Cartesian or polar) and units, but just changing the origin.
I'm using the 2D space + 1D of time because it's easier to visualize than 4D spacetime and a bit richer than 1D space + 1D time.
So, I'll try to stick with radial coordinates, so we have (t, r, θ). Piece of paper on the table. Place a dot, that's r0, where r=0. Now any point on the paper can be described by an r and θ using the point.
At all points on the paper there is a tensor with a gtt component.
At r=0, gtt = 1 (ignoring the -c2 for now).
r0 is the original origin. Now let's move it to a new point, say θ=0, r=0.5 from the original origin. That's r1, and now r=0 there.
If the origin is moved via a coordinate transformation analogous to the transformation I will give below, the value of gtt at the original origin remains unchanged.Now the metric tensor gives a different gtt component (using the 1/(1-r) metric) for r0.
Now the gtt for r0, which is now at r=0.5 from the new origin, comes out to gtt = 22 (eta the square).
I doubt whether you understand what I was saying. I was certainly not saying "we've already established gtt = 1 for that point." We had established that gtt = 1 for that point in your original coordinates, and it is possible that gtt = 1 for that point in the new coordinates as well, but whether that is true depends on the specific coordinate transformation, and your description of your coordinate transformation was insufficiently precise for anyone (including yourself) to draw a firm conclusion here.That's what you're saying you can't do. That's trying to change the territory because we've already established gtt = 1 for that point.
Is that correct?
Nope.In my previous message, I was going to give you an example of that, but I got a minus sign wrong so I deleted the example. Here is the example:
ds2 = − (ρ + 1)2 c2 dt2 + (ρ + 1)−4 dρ2 + (ρ / (ρ + 1))2 [ dθ2 + sin2 θ dφ2 ]
In that metric form, the radial coordinate ρ (which is the Greek letter rho) ranges over the non-negative real numbers.
Do you recognize the tensor field expressed by that metric form?
Assuming ρ = z, that's FLRW,
I think.
The metric tensor field of your geocentric Helland universe is then expressed by both of these line elements:
ds2 = − (1 − r)−2 c2 dt2 + dr2 + r2 [ dθ2 + sin2 θ dφ2 ]
= − (½ − q)−2 c2 dt2 + dq2 + (q + ½)2 [ dθ2 + sin2 θ dφ2 ]
Letting E0 be the event whose t, r, θ, and φ coordinates are (0, 0, 0, 0), it's easy to see that the coordinates of E0 in this new coordinate system are (0, −½, 0, 0). The event E1 whose t, r, θ, and φ coordinates are (0, ½, 0, 0) has coordinates (0, 0, 0, 0) in this new coordinate system.
It is therefore possible to say that the transformation from r to q moves the origin from E0 to E1, but I don't know whether that's what you meant by moving the origin to a different point. If it isn't what you meant, then this transformation from r to q should help you to understand why you need to be more precise when describing a coordinate transformation.
I strongly recommend the following exercise: Prove that all three of those metric forms describe exactly the same tensor field.
Moving the origin is a coordinate transformation. Coordinate transformations do not change the spacetime manifold. The map is not the territory.Huh.
Well, I thought the issue was the metric was "geocentric". That there is a special place. Specifically, there is a special observer.
So I said, you just move the origin to a new observer at a new place, and now, r=0 there, so it's not special.
You seem to be talking about coordinate transformations in a more general sense, where I specifically had in mind changing frames of reference to another observer.
But it's the same problem either way.
As Mike Helland has been told on several occasions, using that particular coordinate system to express the metric of de Sitter space is misleading. For one thing, those coordinates lead to an artificial horizon at r=α.It seems this would also be a problem for de Sitter's original coordinates:
http://latex.codecogs.com/gif.latex?ds^2 = -\left(1-\frac{r^2}{\alpha^2}\right)dt^2 + \left(1-\frac{r^2}{\alpha^2}\right)^{-1}dr^2 + r^2 d\Omega_{2}^2.
I assumed since this was valid (at least mathematically) then the "geocentrism" isn't a huge deal.
The metric tensor field is a mathematical object. You can't change it by talking rubbish about its "intention".I think we have to go back to what the intention was. The intention of the metric is to describe the past light cone of an observer. The relevant "territory" in this case is not the universe as a whole, but the past light cone of an observer.
The Helland universe is the same for all observers. You can use different coordinate systems to describe it, such as the three coordinate systems I used near the beginning of this post, but all of those coordinate systems describe exactly the same territory.So, the territory of one observer differs from the territory of another observer.
The metric tensor field defines the Helland universe and determines the light cones for all observers.That's fine. All that means is the past light cone for one observer is different than the past light cone of another observer. That's basically a given.
It's hard to read that as anything other than the confused rambling of someone who doesn't actually understand general relativity.In which case, I suppose they both need to be thought of as sub-manifolds of a greater manifold.
It seems the technique would be to start at the t=0 (present) and r=0 (position of the observer of the light cone) and then we can work our way back in time along a null path.
And any given place, change direction in space and time. When you get back to t=0 you'll be at a different observer.
The highlighted phrase is nonsense, and the entire paragraph is risible.Essentially, the universe as a whole doesn't have a particular metric (as of yet) , but each observer in the universe has a past light cone, and you can use where they intersect to move between observers.
Yep.Sounds good. Before I get too far... is that -4 supposed to be there?
In the new Helland universe
Mike are you feeling ok?
Sure.
Hypothetical scenario for you or anyone else reading this that would like to take a crack at it.
Let's assume I spoke perfect English and had no significant mental impairments.
Let's say I told you, where I'm from, we believe in rocks, and we believe in clouds, and stars and galaxies, and that these things are real and in reality, and reality is real.
All of that is just to preface, this isn't solipsism, which most people just assume it is.
Then I told you, we don't believe in a universe.
Why should I?
Why should I believe, beyond or in addition to reality and the things in it, I should believe in this relatively modern idea that is "the universe", which is apparently all of existence as we know it... and it's a whopping 3.5 times older than the rock we live on?
Go ahead.
For a starter, I’d ask you to look through a telescope, then gradually build upon that going through the history of astronomy and cosmology without the detours, and end up with what we know today.
What reason would there be to think that the universe does not exist, when you can observe it in every possible wavelength?
When I look through a telescope I observe stars and galaxies.
What direction do I point the telescope to see "the universe" and what should I be looking for?
When I look through a telescope I observe stars and galaxies.
What direction do I point the telescope to see "the universe" and what should I be looking for?
"What the hell is this 'forest' you keep going on about. All I can see is trees!"
There are these two young fish swimming along, and they happen to meet an older fish swimming the other way, who nods at them and says, “Morning, boys. How’s the water?” And the two young fish swim on for a bit, and then eventually one of them looks over at the other and goes, “What the hell is water?” - David Foster Wallace
I get your point.
But my name for the forest is "reality".
Adding "the universe" to the lexicon has provided nothing except a ridiculous origin story and the necessity of the multiverse.
While removing "the universe" from the lexicon neither eliminates the evidence of an origin (a very hot dense period of almost, if, no spatial extent) nor the possibility (not "necessity") of a multiverse, in reality. Getting rid of the word doesn't do what you want because simply having it didn't do what you just claimed it did. It's the other way 'round, that evidence supports a very hot dense period of almost, if, no spatial extent (an origin for the perceived universe)
and having the mere possibility of a multiverse extending the possible universe beyond what we can directly perceive. The word didn't create the evidence or implications but simply came and is defined as a result of them.
Similarly, just getting rid of the word "cancer" doesn't eliminate a condition where cells reproduce rapidly and out of control perhaps spreading throughout the body. The word didn't cause or drive the condition, in reality the condition simply spawned the word.