Conditions for spacetime to have flat spatial slices

[George Jones]

The Milne universe is an FRW universe that is empty of matter and energy. See

https://www.physicsforums.com/showthread.php?p=1757634&highlight=milne#post1757634

https://www.physicsforums.com/showthread.php?p=2872975#post2872975.

 

[JDoolin]

I'm not sure that what you're saying about the Friedmann-Walker diagram is correct. Physically, in a spatially infinite universe (which is what's required for an "infinite number of worldlines" in the sense you're using the term), I would *not* expect light reaching us now from the Big Bang event to have crossed *all* of that infinite number of worldlines, because only a finite amount of proper time has elapsed, and light can only cover a finite distance in a finite time--what you're suggesting would require the light to cover an infinite distance in a finite time. The "comoving particles" diagram (a more standard name for it would be a "conformal" diagram, as it is called on the Ned Wright page I linked to earlier) has the advantage that it makes the reasoning I just gave obvious.

As far as I can tell, your "Milne" diagram corresponds to the diagram on the Ned Wright page here...

http://www.astro.ucla.edu/~wright/cosmo_02.htm

...in what Wright calls "special relativistic" coordinates. This means to me that you have raised an interesting question about the light cones, since it certainly appears in your "Milne" diagram and Wright's "special relativistic" diagram that the light from the "Big Bang" *does* cross all the "infinite number" of worldlines. I'll have to think about this one some more.

At least on this page, I can see that I'm talking about the same thing as Ned Wright. He correctly notes that the observable universe is the entire universe. He also correctly notes that our past lightcone passes the most distant galaxies at $$x=c t_0/2$$

(He does not mention that the proper age of those asymptotically most distant galaxies is zero, so that past light-cone does cross the singularity. I imagine the era of hydrogen recombination to be a big hyperbolic arc in the diagram, and that represents the furthest distance we can actually see.)

Technically, the Friedmann Walker Diagram is ambiguous at the singularity. The question is what is the slope of the past light-cone at the point (0,0). If that slope is zero, then the light should cross all of the past light-cones. If the slope is nonzero, then it must cross only a finite number of light-cones.

(Sorry--in Wright's diagram (at http://www.astro.ucla.edu/~wright/cosmo_02.htm ), I looked again, and the speed of light, as drawn, is definitely a non-zero slope at the singularity. That means that it is not a conformal mapping with the image below it, because it does not cross every worldline.) As for me, I think that the speed of light at that singularity point should be faster than the speed of light for any particle at that point. So I think the line should be horizontal there.

 

[JDoolin]

I'm not sure that what you're saying about the Friedmann-Walker diagram is correct. Physically, in a spatially infinite universe (which is what's required for an "infinite number of worldlines" in the sense you're using the term), I would *not* expect light reaching us now from the Big Bang event to have crossed *all* of that infinite number of worldlines, because only a finite amount of proper time has elapsed, and light can only cover a finite distance in a finite time--what you're suggesting would require the light to cover an infinite distance in a finite time.

No, no. The light wouldn't have to cross an infinite distance; it would only need to cross an infinite number of particles. That's the problem Milne (and Epstein and I) have with the assumption of homogeneity. You're assuming homogeneity throughout the universe. You should be allowing for the particles to approach an infinite density, asymptotically as you go out to the edge of the universe.

We're not suggesting that the light has to cover an infinite distance, but that an infinite number of particles lie within a finite distance.

 

[JDoolin]

The Milne universe is an FRW universe that is empty of matter and energy. See

https://www.physicsforums.com/showthread.php?p=1757634&highlight=milne#post1757634

https://www.physicsforums.com/showthread.php?p=2872975#post2872975.

You are linking to links of you asserting that this is true, but as I told PeterDonis, this is one occasion, where you should really go back to the original text, because the secondary sources do not justly represent Milne's ideas.

 

[PeterDonis]

We're not suggesting that the light has to cover an infinite distance, but that an infinite number of particles lie within a finite distance.

A finite *coordinate* distance, yes. But have you integrated the metric over that finite coordinate distance to confirm that it covers a finite *proper* distance? The fact that the model indicates density going to infinity as you approach that finite coordinate distance is a big red flag to me that there's actually an infinite proper distance (meaning the actual physical density remains finite), and that the apparent infinite density is an artifact of the peculiar coordinate system you are using. Actual infinite physical density is not physically reasonable.

 

[JDoolin]

A finite *coordinate* distance, yes. But have you integrated the metric over that finite coordinate distance to confirm that it covers a finite *proper* distance? The fact that the model indicates density going to infinity as you approach that finite coordinate distance is a big red flag to me that there's actually an infinite proper distance (meaning the actual physical density remains finite), and that the apparent infinite density is an artifact of the peculiar coordinate system you are using. Actual infinite physical density is not physically reasonable.

Infinite physical density is physically reasonable, I think. For an infinite physical density, what is required is that several particles occupy the same place at the same time.

Certainly, you can't have any two particles sharing the same quantum numbers. But these particles have different momenta, and hence different quantum numbers. This situation can only exist for an instant.

However, since the proper time experienced by a particle traveling asymptotically approaching the speed of light is zero, that "instant" is effectively, the whole lightcone; it lasts forever in the frame of the central observer. Infinite physical density for an instant (a single event) is physically reasonable, and due to time dilation, that instant is effectively forever.

As for integrating the proper time, I hope you'll forgive me. As I mentioned before, the Einstein Field Equations need to be done over from scratch assuming this case where particles are diverging as v=d/t. So, no, I have not completed this task. That, I think, is something for better minds than mine. I just want it to be realized that the Einstein Field Equations were derived under the assumption of a finite number of comoving particles, and it does not cover the case of an infinite number of non-comoving particles. Then perhaps someone; (maybe you!) could re-do the EFE's from scratch, develop the theory and see what comes out of it.

 

[PeterDonis]

Infinite physical density is physically reasonable, I think. For an infinite physical density, what is required is that several particles occupy the same place at the same time.

To be precise, what's required is that an *infinite* number of particles occupy the same place at the same time. But I think you realize that--see below.

Certainly, you can't have any two particles sharing the same quantum numbers. But these particles have different momenta, and hence different quantum numbers. This situation can only exist for an instant.

However, since the proper time experienced by a particle traveling asymptotically approaching the speed of light is zero, that "instant" is effectively, the whole lightcone; it lasts forever in the frame of the central observer. Infinite physical density for an instant (a single event) is physically reasonable, and due to time dilation, that instant is effectively forever.

I can't be sure here whether or not you're committing a common confusion in relativity. An instant (event) is a single point. A null worldline--the path of a light ray--has "zero lapse of proper time", but it is *not* an instant--it is *not* a single point. It consists of a series of distinct events. Your language in the quote above is ambiguous; you need to be clear and explicit about whether the infinite density in your model occurs at a single point, or along a whole null worldline.

I suspect that what you mean is the former (the infinite density only occurs at a single point), in which case I would strongly discourage using the language "due to time dilation, that instant is effectively forever", since that language promotes exactly the confusion that I described. The instant of infinite density in your model, I suspect, is supposed to be a single point; it's only the particular coordinates you've chosen that distort that single point into what appears to be a null cone. Physically, it's a single point.

However, saying that the infinite density is only for a single point does *not* make it physically reasonable. It only means that your model "predicts its own downfall", just as GR does for spacetimes in which a singularity occurs. Your model is telling you that it can't cover that point--you need some new physics to tell you what happens there. The FRW models in GR handle that, as has been mentioned before, by *not* claiming to cover events all the way to the initial singularity; they only cover things back to (roughly) the beginning of the inflationary phase, where the universe was very small, hot, and dense, but not infinitely so. In your model, that would correspond (as you note) to drawing a hyperbola very *close* to the "null cone" boundary, corresponding to the (very small but not zero) proper time of the beginning of inflation (let's say), and saying that that's the actual physical limit of what your model covers; before that, new physics is needed to say what happened and how the initial state in your model came to be, just as with the FRW models.

As for integrating the proper time, I hope you'll forgive me. As I mentioned before, the Einstein Field Equations need to be done over from scratch assuming this case where particles are diverging as v=d/t.

No, the EFE doesn't need to be reworked, as I noted in a previous post. (Small nitpick: the usual term is "Einstein Field Equation", singular, even though it does have multiple tensor components. The tensors on each side of the EFE are considered single geometric objects, just as we normally write vector equations as relating single objects, even though vectors have multiple components.)

But that's actually not required anyway for what I was asking for. I said "integrate the metric", and you already have an expression for the metric; how that metric arises from the EFE is a separate issue. Integrate that expression over the appropriate range of coordinate values, and see what you come up with.

 

[JDoolin]

To be precise, what's required is that an *infinite* number of particles occupy the same place at the same time. But I think you realize that--see below.



I can't be sure here whether or not you're committing a common confusion in relativity. An instant (event) is a single point. A null worldline--the path of a light ray--has "zero lapse of proper time", but it is *not* an instant--it is *not* a single point. It consists of a series of distinct events. Your language in the quote above is ambiguous; you need to be clear and explicit about whether the infinite density in your model occurs at a single point, or along a whole null worldline.

See if I can make it a little less ambiguous without going into the math right now. If I take a "null world line" that comes straight from the big bang, then that world-line has a proper time of zero. You go out to that distance, and take away one meter, you're going to be in a region of incredibly high density. You go to 1 millimeter away from the null world-line, and you'll get many orders of magnitude higher of density. You go 1 micrometer, it will be may orders higher density still; you go to 1 nanometer, many more orders of magnitude. I could go on forever, but I hope you get the picture. The density is just a function that asymptotically approaches infinity as you approach the light ray. You can't say precisely what the density is AT the singularity.

I suspect that what you mean is the former (the infinite density only occurs at a single point), in which case I would strongly discourage using the language "due to time dilation, that instant is effectively forever", since that language promotes exactly the confusion that I described. The instant of infinite density in your model, I suspect, is supposed to be a single point; it's only the particular coordinates you've chosen that distort that single point into what appears to be a null cone. Physically, it's a single point.

I meant both. Why is it appropriate to map the singularity to a straight line, as is done in the "comoving" conformal mapping, but it is not appropriate to map the singularity to a light cone? The infinite density occurs at a single point in the Friedmann Walker Diagram (one single event), but in both the standard model, and in the Milne model, that single event is stretched out to infinity.

However, saying that the infinite density is only for a single point does *not* make it physically reasonable. It only means that your model "predicts its own downfall", just as GR does for spacetimes in which a singularity occurs. Your model is telling you that it can't cover that point--you need some new physics to tell you what happens there. The FRW models in GR handle that, as has been mentioned before, by *not* claiming to cover events all the way to the initial singularity; they only cover things back to (roughly) the beginning of the inflationary phase, where the universe was very small, hot, and dense, but not infinitely so. In your model, that would correspond (as you note) to drawing a hyperbola very *close* to the "null cone" boundary, corresponding to the (very small but not zero) proper time of the beginning of inflation (let's say), and saying that that's the actual physical limit of what your model covers; before that, new physics is needed to say what happened and how the initial state in your model came to be, just as with the FRW models.

I'm not sure if you're being entirely fair. The model I'm describing leaves out a singularity. The model you're describing leaves out a singularity plus some additional time including the inflationary stage.

You want me to stop at some finite time after the big bang. But I don't want to stop there. You name a time after the big bang, and I will name an earlier time after the big bang. No matter what time I choose, the universe has a density at that time. If you choose ZERO, then, I'm stuck. You're right. That is outside the scope of the theory. But it is also outside the scope of General Relativity, so what is the difference?

No, the EFE doesn't need to be reworked, as I noted in a previous post. (Small nitpick: the usual term is "Einstein Field Equation", singular, even though it does have multiple tensor components. The tensors on each side of the EFE are considered single geometric objects, just as we normally write vector equations as relating single objects, even though vectors have multiple components.)

But that's actually not required anyway for what I was asking for. I said "integrate the metric", and you already have an expression for the metric; how that metric arises from the EFE is a separate issue. Integrate that expression over the appropriate range of coordinate values, and see what you come up with.

You asserted that the EFE doesn't need to be reworked. I asserted that the EFE does need to be reworked, and backed up that statement, by pointing out that the EFE's assume a finite number of comoving particles. You asserted that the case I'm talking about has been tried, but I told you I couldn't find it anywhere in whatever you linked to. (By the way, I've been, off and on, looking since 2001 to try to find one "respectable" text on General Relativity that considered this case, and asking for it, and the only place I have found it is in Milne. So it is not a surprise to me if you can't find it on the Baez site.)

So, actually I don't have the metric. I would guess that the metric is {{-1,0,0,0},{0,1,0,0},{0,0,1,0},{0,0,0,1}} because of symmetry. And yes, it's quite easy to show that the proper distance in this case is finite.

 

[PeterDonis]

I meant both. Why is it appropriate to map the singularity to a straight line, as is done in the "comoving" conformal mapping, but it is not appropriate to map the singularity to a light cone? The infinite density occurs at a single point in the Friedmann Walker Diagram (one single event), but in both the standard model, and in the Milne model, that single event is stretched out to infinity.

In a Mercator projection of the Earth's surface, the North and South Poles are mapped to infinitely long lines. But physically, they're not lines; they're points. The "comoving" conformal mapping does something similar to the initial singularity: it takes what is physically a single point and maps it into an infinitely long line. As I understand it, your model does something similar, except that instead of taking what is physically a single point (like the Earth's North Pole) and mapping it into a line, it maps it into a null cone. I'm just trying to understand if you agree that that's what your model is doing, or if you intend it to be doing something else.

You want me to stop at some finite time after the big bang. But I don't want to stop there. You name a time after the big bang, and I will name an earlier time after the big bang. No matter what time I choose, the universe has a density at that time. If you choose ZERO, then, I'm stuck. You're right. That is outside the scope of the theory. But it is also outside the scope of General Relativity, so what is the difference?

Again, I'm just trying to get clear about whether or not your model actually claims that infinite density is physically reasonable. You were saying earlier that it was; now your comments here indicate that it is not--that your model can get arbitrarily close to the infinite density point (but at any such arbitrarily close point, the density will still be finite), but does not actually cover the infinite density point. As long as the latter is the case, I have no issue. (How close you actually want to get to the initial singularity in your model based on the data you have is a separate issue--the FRW models *can* be extended back arbitrarily close to the singularity, just as yours can; the reason they usually aren't is only because we estimate that the start of the inflation phase is where the new physics we need will actually end up coming in. We won't know for sure until we know what that new physics is.)

You asserted that the case I'm talking about has been tried, but I told you I couldn't find it anywhere in whatever you linked to.

Read my post #103 again. I described there how the page I linked to, in which you claimed you couldn't find anything relevant, *is* relevant. If you can't respond to the specifics I gave there, other than just saying "I can't find anything that applies", I don't think we can get any further on this particular point.

So, actually I don't have the metric. I would guess that the metric is {{-1,0,0,0},{0,1,0,0},{0,0,1,0},{0,0,0,1}} because of symmetry. And yes, it's quite easy to show that the proper distance in this case is finite.

The metric you just gave is a representation in matrix form of the line element:

$$ds^{2} = - dt^{2} + dx^{2} + dy^{2} + dz^{2}$$

This is the standard Minkowski metric, which covers a range of minus infinity to plus infinity for all four coordinates (t, x, y, z). That doesn't correspond to the diagram you drew, which only covers a limited range of coordinate values; so the above metric can't be the one that applies to your diagram.

 

[JDoolin]

Regarding Density:

You're asking whether infinite density is physically reasonable. Well, density is defined as a number of particles per unit area. Even though the local density at any point is finite by my description, If you take the average density of the universe,

$$\frac{\int \rho dV}{V}$$

at any given time, yes, the top will be infinite, and the bottom will be finite. Yes, an infinite density of the universe is reasonable. The average density of the universe is always infinite, because you always have an infinite number of particles within a finite distance.

So there are different meanings of an infinite density. You can have

(1) an infinite number of particles in a finite volume, (All the time in the Milne model)

or (2) a finite number of particles in a zero volume (never happens in the Milne model),

or (3) an infinite number of particles in a zero volume (approached asymptotically at r=c*t)


Regarding Mapping the Singularity to a Straight Line vs a Null Light Cone.

I erred, I think, which is leading to some confusion.

We have (1 FWD) the Friedmann Walker Diagram, (2 CPD) the comoving particle conformal mapping Diagram, and (3 MMD) the Milne/Minkowski conformal map.

Earlier I said

http://www.astro.ucla.edu/~wright/cosmo_02.htm[/url] ), I looked again, and the speed of light, as drawn, is definitely a non-zero slope at the singularity. That means that it is not a conformal mapping with the image below it, because it does not cross every worldline.) As for me, I think that the speed of light at that singularity point should be faster than the speed of light for any particle at that point. So I think the line should be horizontal there.
"

My error was in thinking that ONCE that line was horizontal at the origin, that it can turn. In fact, the mapping is not from the singularity to the light cone, but from this horizontal line to the lightcone.

You might well ask why this light does not bend back toward the infinite density behind it, which is a very good question, so my answer is going to seem glib. That lightcone surface is not causally connected to anything within the lightcone. If I send a beam of light to chase after a beam of light, it will never reach it.

You can receive light from any matter that is asymptotically close to the null light-cone, but the light-cone, itself is not "covered" by the model.
In other words, there is no straight line speed-of-light path from the big bang to any event inside the lightcone. Only events immediately following the big-bang can ever be observed within the lightcone.

I think this is actually a pretty reasonable conclusion, though it is contrary to the standard model. We shouldn't be able to see the Big Bang itself, since it already happened in our reference frame. But we can see events immediately after the Big Bang.


Regarding whether Baez considered this topic:

The page I linked to describes a ball B which is "expanding at t = 0". That ball corresponds to your particles moving apart with v = d/t. Baez says that we can't directly apply his equation (2) (which is his statement of the EFE) to ball B because its particles aren't at rest relative to each other at time t = 0. He then defines a second ball, B', which *does* have all its particles at rest relative to each other at time t = 0, and which has the same radius and the same acceleration as ball B at time t = 0. This allows him to apply his equation (2) to ball B', and then show that the equation he derives for the radius r of ball B' vs. time, *also* holds for the radius R of ball B vs. time. So he's showing that the EFE *does* apply to the type of ball you defined, where the particles are moving apart at time t = 0.

I'm sorry. I still don't see it on the page http://math.ucr.edu/home/baez/einstein/node7.html" . Maybe there's something in the equations that I'm missing. I kind of skimmed over them, looking for something in the text. If you see it is there in the math, or I am looking on the wrong page, I need more information.

 

[PeterDonis]

Well, density is defined as a number of particles per unit area. Even though the local density at any point is finite by my description, If you take the average density of the universe,

$$\frac{\int \rho dV}{V}$$

at any given time, yes, the top will be infinite, and the bottom will be finite.

I assume you mean "number of particles per unit volume" (not "area"), and that V is a spatial volume. How is the corresponding volume element dV related to a product of coordinate differentials (dx, dy, dz, or whatever spatial coordinates you are using) in your diagram? This question is related to my question about coordinate distance vs. proper distance and what metric you are using (since the metric and the volume element are related). Remember my earlier comment that the metric that applies to your diagram can't be the ordinary Minkowski metric, since whatever coordinates the diagram is drawn in do not have infinite ranges.

My error was in thinking that ONCE that line was horizontal at the origin, that it can turn. In fact, the mapping is not from the singularity to the light cone, but from this horizontal line to the lightcone.

This sounds basically the same as what I said, just with an extra mapping step, so to speak: the mapping from FWD to CPD maps a point (the initial singularity) into a horizontal line, and then you do a mapping from CPD to MMD that maps the horizontal line into a null cone.

Regarding whether Baez considered this topic:

Read the first two sentences from my post (that you quoted) again: The page I linked to describes a ball B which is "expanding at t = 0". That ball corresponds to your particles moving apart with v = d/t. Do you see how ball B here covers the case you were considering?

Now read these sentences again: He then defines a second ball, B', which *does* have all its particles at rest relative to each other at time t = 0, and which has the same radius and the same acceleration as ball B at time t = 0. This allows him to apply his equation (2) to ball B', and then show that the equation he derives for the radius r of ball B' vs. time, *also* holds for the radius R of ball B vs. time. Do you see how this shows that the EFE covers the case you were considering?

 

[JDoolin]

I assume you mean "number of particles per unit volume" (not "area"), and that V is a spatial volume. How is the corresponding volume element dV related to a product of coordinate differentials (dx, dy, dz, or whatever spatial coordinates you are using) in your diagram? This question is related to my question about coordinate distance vs. proper distance and what metric you are using (since the metric and the volume element are related). Remember my earlier comment that the metric that applies to your diagram can't be the ordinary Minkowski metric, since whatever coordinates the diagram is drawn in do not have infinite ranges.

From http://casa.colorado.edu/~ajsh/sr/wheel.html we have the lovely animation:

Andrew Hamilton has a nice diagram of the effect of Lorentz Transformation around the event at the origin.

This diagram is pure Minkowski diagram.

What you see in this diagram is events from the top light-cone being moved to other places in the top lightcone. What you never see is an event passing down from the future lightcone past the null light cone.

The transformation keeps on going and going and going forever, with each of an infinite number of worldlines blissfully thinking that it is at the center of that lightcone.

Now, there may or may not be anything outside that lightcone but if the model is correct, then it does not matter, because in order to get intothe lightcone from outside the lightcone, one would need to push past an infinite number of particles.

Still, this is the "ordinary Minkowski Metric." It's just that the entire observable universe lies within a single lightcone of the Minkowski Spacetime. The universe that we know only occupies a fraction of the Minkowski spacetime.

This sounds basically the same as what I said, just with an extra mapping step, so to speak: the mapping from FWD to CPD maps a point (the initial singularity) into a horizontal line, and then you do a mapping from CPD to MMD that maps the horizontal line into a null cone.

But there are differences in how that initial singularity is mapped.
Namely: The CPD maps the initial singularity (0,0) to a horizontal line, while the MMD maps the horizontal line through the singularity (0,0) to a light-cone. That represents the surface of $$\tau^2=t^2-r^2=0$$ The whole lightcone represents a singularity in the $$\tau$$ variable. In the MMD representation, the (0,0) point remains a single event.

If I'm not mistaken, the CPD simply does not recognize the horizontal line in the FWD. But how would the horizontal line be mapped, if it were? Would it become a vertical line "at infinity?"

Read the first two sentences from my post (that you quoted) again: The page I linked to describes a ball B which is "expanding at t = 0". That ball corresponds to your particles moving apart with v = d/t. Do you see how ball B here covers the case you were considering?

Now read these sentences again: He then defines a second ball, B', which *does* have all its particles at rest relative to each other at time t = 0, and which has the same radius and the same acceleration as ball B at time t = 0. This allows him to apply his equation (2) to ball B', and then show that the equation he derives for the radius r of ball B' vs. time, *also* holds for the radius R of ball B vs. time. Do you see how this shows that the EFE covers the case you were considering?

Thank you for clarifying what you saw.

http://math.ucr.edu/home/baez/einstein/node7.html] [/URL]
Suppose that, at some time t=0, she identifies a small ball B of test particles centered on her. Suppose this ball expands with the universe, remaining spherical as time passes because the universe is isotropic.

Okay, I see this now, but can you understand why I didn't recognize it as the same case? This example says the ball "expands with the universe." When I read that, I interpreted it to mean "It is a ball which stretches as the universe stretches." I see now that Baez actually meant to talk about something else, but I can't see how you get v = d/t out of this. He literally says later there is "nothing special" about time, t=0. If you have v=d/t, then you should have infinite density at t=0.

http://math.ucr.edu/home/baez/einstein/node7.html] [/URL]
Let R(t) stand for the radius of this ball as a function of time. The Einstein equation will give us an equation of motion for R(t). In other words, it will say how the expansion rate of the universe changes with time.

It is tempting to apply equation (2) to the ball , but we must take care.

See, he's already attempting to apply the EFE before he even gives the simple answer. He never says, $$R(t)= v_{max} t + R_0$$, so I would say he is not really considering the idea at all; not in any clear mathematical sense.

http://math.ucr.edu/home/baez/einstein/node7.html] [/URL]

This equation applies to a ball of particles that are initially at rest relative to one another -- that is, one whose radius is not changing at . However, the ball B is expanding at t=0. Thus, to apply our formulation of Einstein's equation, we must introduce a second small ball of test particles that are at rest relative to each other at t=0.

What has he done? He has said we can't simply apply the Einstein Field Equations. We have to define our coordinate system, THEN apply the Einstein Field Equations.

To me, nothing seems to have changed. He is still using the EFE's, but he gave a tiny bit of lip-service to the idea of not using them. He sure didn't start over from scratch, assuming d=v/t.

 

[PeterDonis]

Andrew Hamilton has a nice diagram of the effect of Lorentz Transformation around the event at the origin.

Yes, I understand how all this works. (I find all of the diagrams on Hamilton's pages to be very helpful, btw.) But the description you give based on this diagram...

It's just that the entire observable universe lies within a single lightcone of the Minkowski Spacetime. The universe that we know only occupies a fraction of the Minkowski spacetime.

...is *not* compatible with the description you give a little later concerning how the initial singularity is mapped:

Namely: The CPD maps the initial singularity (0,0) to a horizontal line, while the MMD maps the horizontal line through the singularity (0,0) to a light-cone. That represents the surface of $$\tau^2=t^2-r^2=0$$ The whole lightcone represents a singularity in the $$\tau$$ variable. In the MMD representation, the (0,0) point remains a single event.

If the observable universe is only a fraction of Minkowski spacetime, as you say above, then when you draw a diagram showing a light cone emerging from the initial singularity (the "big bang" event), the light cone is not a "mapping" of the big bang event, or anything else; it's just showing the limiting case of light rays going outward from the big bang event in opposite directions. In other words, the light cone is the boundary of the portion of Minkowski spacetime that is causally connected to the big bang event, but that certainly doesn't mean the light cone "is" the big bang event, or is a "mapping" of the big bang event into some other set of coordinates. You're using Minkowski coordinates throughout, and you're saying that we, now, on the Earth, lie on some timelike worldline in a Minkowski spacetime that, in the far past, met up with all the other timelike worldlines in the "universe" we observe, at an event called the "big bang". All those worldlines must lie within the future light cone of that big bang event, so you've just drawn in the future light cone to mark the spacetime boundary of "our universe", as opposed to the rest of the infinite Minkowski spacetime.

It is true that all points on the light cone have zero spacetime interval from the big bang event, but as I said in an earlier post, that does *not* make the entire light cone a single event! There are other ways of assigning coordinates in Minkowski spacetime that make it clear that there are distinct events on light cones. This is in contrast to the "mapping" that is done in, for example, the "CPD"--see next comment.

Also, in your model, there is nothing preventing other "big bang-like" events from happening *outside* the "big bang" light cone (after all, there is an infinite Minkowski spacetime for events to happen in), and sending signals (light rays or timelike worldlines) *into* the "big bang" light cone that represents our universe. As far as I can tell, your model doesn't account for that at all.

If I'm not mistaken, the CPD simply does not recognize the horizontal line in the FWD. But how would the horizontal line be mapped, if it were? Would it become a vertical line "at infinity?"

There actually isn't a "horizontal line" at the bottom of the FWD (I assume that's what you were asking about). There are lines going left and right that approach the horizontal as close as you like, but there are none that are exactly horizontal. In the CPD, the lines approaching the horizontal are mapped to vertical lines on the left and right sides, further and further out from the center. In the "special relativistic" diagram on Wright's page, those lines are mapped to "timelike" lines that get closer and closer to the bounding light cone, without ever reaching it.

The point about all three of these mappings is that the "singularities"--the single point at the "bottom" of the FWD, the line at the bottom of the CPD, and the bounding light cone of the "SR" diagram--are true *boundaries* of the entire spacetime: there is *no* spacetime at all outside of these boundaries. These diagrams are not embedded in any larger diagrams (for example, there is no "larger" Minkowski spacetime in which the "SR" diagram is embedded); they show the *entire* spacetime, *everything* that physically exists in the model.

Okay, I see this now, but can you understand why I didn't recognize it as the same case? This example says the ball "expands with the universe." When I read that, I interpreted it to mean "It is a ball which stretches as the universe stretches." I see now that Baez actually meant to talk about something else, but I can't see how you get v = d/t out of this. He literally says later there is "nothing special" about time, t=0. If you have v=d/t, then you should have infinite density at t=0.

First, a small clarification: by "t = 0" Baez did not mean the instant of the big bang, but just some arbitrarily defined "origin" of a time coordinate. He says "at some time, t = 0", not "at the time of the big bang, t = 0".

That said, you are correct that, at the actual "t = 0" of the big bang, the density (and hence the spacetime curvature) is infinite, which is why, as I've said before, General Relativity "predicts its own downfall" whenever a spacetime singularity--a point of infinite density, infinite curvature, etc.--occurs. The EFE becomes mathematically singular at those points, so we can't use it to predict what happens. But we can get as close to the singularity as we like, and everything will still be finite, so the EFE, mathematically, works just fine. (Whether it still gives predictions that are accurate physically is a separate question--we don't know at this point because we can't make any observations of regimes where the density, curvature, etc. gets large enough to be comparable to what the EFE says it would have to have been, for example, before the inflation phase started.)

See, he's already attempting to apply the EFE before he even gives the simple answer. He never says, $$R(t)= v_{max} t + R_0$$, so I would say he is not really considering the idea at all; not in any clear mathematical sense.

Yes, he is; you're missing the point of his argument. He is saying that (provided we are not at a spacetime singularity, as I noted above) *the initial velocities of the particles in the ball don't matter*, because the EFE relates the ratio of *acceleration* and *radius* of the ball to the density and pressure at the center--velocity doesn't enter into it at all. So we can apply the EFE to a ball with *any* distribution of initial velocities, using the method he describes.

 

[JDoolin]

When I said "A singularity in the tau variable," I think you over-interpreted my meaning. I meant that the value of tau is zero everywhere on the light-cone, but I did not mean that having the same tau represented the same event. $$\tau^2=t^2-r^2=0$$ means every event where t=r. That includes more than the big bang event.

The lightcone, even though tau is zero everywhere on it, still does have a distinct before and after. A photon has a source and a destination; a cause and an effect. (Actually, I don't know if these photons have a destination, but the Big Bang is the Source.)

The light cone is the set of events for whom the space-time-interval between them and the big bang is zero. It IS a mapping: the HORIZONTAL PLANE THROUGH (0,0) in the FWD is mapped to the LIGHT CONE in the MMD. It doesn't map an EVENT to the lightcone, it maps the WHOLE PLANE to the lightcone.

However, in the CPD (Comoving Particle Diagram), according to what (I think) you've been saying, the big bang itself is mapped to the horizontal line, and the horizontal line from the FWD is not mapped anywhere! In the CPD, you have a SINGLE EVENT from the FWD which is mapped to MULTIPLE LOCATIONS in the Comoving Particle Diagram. That means that you'll see light from the same event at multiple times. Can you see the difference? There are no "single event" mapped to multiple places in the MMD. There IS a "single event" mapped to multiple places in the CPD. In the CPD, there is always light on its way from the big bang, because this single event has been mapped to an infinite number of locations. It might be hard to switch gears between these two models. In your model, that horizontal line in the FRW metric simply doesn't exist. It's effectively, the vertical line "at" infinity in the CPD. In your model, the Big Bang wasn't a single event; it was a huge number of events which all occurred simultaneously throughout the universe.

In my model (The MMD) the Big Bang was a single event, that very likely, produced an infinite number of infinite intensity photons which mark the outside edge of the expanding sphere I call our universe. The horizontal line in the FRW metric does exist in my model, and it's that massive sphere of radiant energy exploding into a Minkowski Space.

In the MMD, the light from the big bang is gone... long gone. It might hit something OUTSIDE our universe, but it is not coming back in. What we are seeing is not light from the big bang, but light from matter billions of years after the big bang, (time dilated so that it seems instants after the big bang.) (Technically, we see light from the Hydrogen Recombination era, thousands of years after the big bang).

Also, in your model, there is nothing preventing other "big bang-like" events from happening *outside* the "big bang" light cone (after all, there is an infinite Minkowski spacetime for events to happen in), and sending signals (light rays or timelike worldlines) *into* the "big bang" light cone that represents our universe. As far as I can tell, your model doesn't account for that at all.

You're right, for the most part. Milne does bring this up if you have the e-book. He found it rather troubling, as do I, that there was nothing preventing other objects from being outside. Unless they are other Big Bang's though, there isn't much to worry about. The model really is infinite in mass and energy; it's not "really really big; so big we might as well call it infinite." in other words, before something from outside got to you, it would have to pass an infinite number of particles. And if you think of a particle right at the edge, then it has to pass an infinite number of particles before it gets to that particle at the edge, etc.

The only thing you could ever really touch from the outside would be the photon shell. You wouldn't see it coming, of course; it would just be happy pleasant day, and then BAM; instant annihilation.

That being said, I don't know what would happen if we had two Big Bangs in the same Minkowski Spacetime. It would be the unstoppable object meeting the unstoppable object. Just as in GR, you can't model the infinite curvature at t=0, I'm not sure how to model two planes of infinite density colliding at the speed of light.

(By the way, outside this sphere, there may well be infinite curvature, because you have the infinite density coming AT you from ONE direction, instead of the infinite density going AWAY from you in ALL directions. But inside, due to symmetry, I still argue that there is NO curvature.)

Yes, he is; you're missing the point of his argument. He is saying that (provided we are not at a spacetime singularity, as I noted above) *the initial velocities of the particles in the ball don't matter*, because the EFE relates the ratio of *acceleration* and *radius* of the ball to the density and pressure at the center--velocity doesn't enter into it at all. So we can apply the EFE to a ball with *any* distribution of initial velocities, using the method he describes.

Let me tell you what I think the point of his argument is, and then you can tell me I'm missing the point again. We can have a ball of particles that are moving apart. The point of his argument is that this ball he's talking about are the test particles. It doesn't matter whether these test particles are all comoving or if they are all moving apart. They are all going to follow the laws of physics. In regions where the Einstein Field Equations apply, they're going to follow the Einstein Field Equations. So, this should certainly cover situations like the Schwartszchild metric, or the Kerr Metric, or anyplace where we know the general matter distribution in the space around us; we can use that general matter distribution to figure out how to solve for the motions of the matter.

However, if we disagree on the general distribution of the matter around us, for instance, if I think the matter around us approaches an infinite density within a finite distance in all directions, while you think that the density of the universe is the same in all directions, but the majority of it is not yet causally connected to us, then we probably aren't going to agree on what the Einstein Field Equations say, or necessarily on whether they should even be applied.

 

[PeterDonis]

When I said "A singularity in the tau variable," I think you over-interpreted my meaning. I meant that the value of tau is zero everywhere on the light-cone, but I did not mean that having the same tau represented the same event. $$\tau^2=t^2-r^2=0$$ means every event where t=r. That includes more than the big bang event.

Ok, good, that makes it clear what the lightcone in your model means--except that this...

The light cone is the set of events for whom the space-time-interval between them and the big bang is zero. It IS a mapping: the HORIZONTAL PLANE THROUGH (0,0) in the FWD is mapped to the LIGHT CONE in the MMD. It doesn't map an EVENT to the lightcone, it maps the WHOLE PLANE to the lightcone.

...is *not* correct if what you said above is true. There is no "horizontal plane through (0, 0)" in the FWD. See my comments further below.

However, in the CPD (Comoving Particle Diagram), according to what (I think) you've been saying, the big bang itself is mapped to the horizontal line, and the horizontal line from the FWD is not mapped anywhere!

This is correct, because, as I just noted, there is *no* "horizontal line" in the FWD (I said this in a previous post as well). The FWD includes lines that get as close to "horizontal" as you like, but none that are exactly horizontal. Yes, that means that this...

In the CPD, you have a SINGLE EVENT from the FWD which is mapped to MULTIPLE LOCATIONS in the Comoving Particle Diagram.

...is true; that's usually how "conformal" diagrams work. (A Mercator projection of the Earth's surface, for example, is "conformal" in this sense--it maps the North and South poles to horizontal lines, not points.) But that does *not* mean that this...

That means that you'll see light from the same event at multiple times.

is true. Consider the Mercator projection again: the lines of longitude (great circles through the poles) are mapped to vertical lines, which *appear* to meet the poles at "different places". But that's an artifact of the "infinite distortion" that the projection makes at the poles. In the same way, the *apparent* "multiple light rays" from the initial singularity in the CPD are an artifact of the "infinite distortion" that this diagram makes at the initial singularity.

How do we deal with this? The only really consistent way is to accept that these "conformal" diagrams *cannot* actually represent the singularities (just as we don't actually use the Mercator projection at the poles). What the "multiple light rays" in the conformal diagram are actually indicating is that, at very short times after the initial singularity, worldlines which emerged from that singularity "in different directions" will be causally disconnected; the more "different" the initial directions are, the longer it will take for the worldlines to become causally connected again. The light that is reaching is now from "close to the big bang" is coming from worldlines that emerged from the big bang in a direction that was "more different" from ours than light that reached us from close to the big bang some time ago.

I realize the above is a somewhat vague description; when I have more time I can try to make it more precise if needed.

In your model, the Big Bang wasn't a single event; it was a huge number of events which all occurred simultaneously throughout the universe.

Strictly speaking, this is false, although it's sometimes used colloquially to describe what I was describing above, that events very close to the big bang happened throughout the universe (in the sense of causal disconnection I gave above).

In the MMD, the light from the big bang is gone... long gone. It might hit something OUTSIDE our universe, but it is not coming back in. What we are seeing is not light from the big bang, but light from matter billions of years after the big bang, (time dilated so that it seems instants after the big bang.) (Technically, we see light from the Hydrogen Recombination era, thousands of years after the big bang).

As I noted above, the FRW models do not claim that we can see light (or any other signal) "from the big bang" itself. The earliest *photons* we can see in the FRW models are, as you say, those from the time of recombination. However, the FRW model would predict that we could see other radiation from earlier times (e.g., neutrinos from the electroweak phase transition, or gravitational waves from even earlier times). We don't currently have any way of testing such predictions because of our poor ability to detect any kind of radiation other than electromagnetic.

You're right, for the most part. Milne does bring this up if you have the e-book. He found it rather troubling, as do I, that there was nothing preventing other objects from being outside. Unless they are other Big Bang's though, there isn't much to worry about. The model really is infinite in mass and energy; it's not "really really big; so big we might as well call it infinite." in other words, before something from outside got to you, it would have to pass an infinite number of particles. And if you think of a particle right at the edge, then it has to pass an infinite number of particles before it gets to that particle at the edge, etc.

I reached this point in the e-book today, as it happens. I still find the "infinite density" part of the model physically unreasonable, but I agree that *if* you stipulate that the density goes to infinity at the "photon shell", there would be no possibility of anything coming in from outside the shell.

That being said, I don't know what would happen if we had two Big Bangs in the same Minkowski Spacetime. It would be the unstoppable object meeting the unstoppable object. Just as in GR, you can't model the infinite curvature at t=0, I'm not sure how to model two planes of infinite density colliding at the speed of light.

This is one reason (but hardly the only reason) that I find the infinite density physically unreasonable.

(By the way, outside this sphere, there may well be infinite curvature, because you have the infinite density coming AT you from ONE direction, instead of the infinite density going AWAY from you in ALL directions. But inside, due to symmetry, I still argue that there is NO curvature.)

Another reason I find the infinite density physically unreasonable is that it should result in infinite curvature at the "photon shell", which, aside from any other objections, would contradict the initial assumption of a flat background Minkowski spacetime. Even if spacetime *inside* the shell were flat (which it could be since that's a general result for inside a symmetrical spherical shell even in Newtonian gravity), the *complete* spacetime in which everything is embedded could not be.

Let me tell you what I think the point of his argument is, and then you can tell me I'm missing the point again. We can have a ball of particles that are moving apart. The point of his argument is that this ball he's talking about are the test particles. It doesn't matter whether these test particles are all comoving or if they are all moving apart. They are all going to follow the laws of physics. In regions where the Einstein Field Equations apply, they're going to follow the Einstein Field Equations. So, this should certainly cover situations like the Schwartszchild metric, or the Kerr Metric, or anyplace where we know the general matter distribution in the space around us; we can use that general matter distribution to figure out how to solve for the motions of the matter.

However, if we disagree on the general distribution of the matter around us, for instance, if I think the matter around us approaches an infinite density within a finite distance in all directions, while you think that the density of the universe is the same in all directions, but the majority of it is not yet causally connected to us, then we probably aren't going to agree on what the Einstein Field Equations say, or necessarily on whether they should even be applied.

This is much closer, I think, but I still have a couple of comments:

(1) According to General Relativity, there are no situations where the EFE does not apply. It always does. There are certainly a wide variety of particular *solutions* to the EFE, among which are the various spacetimes we've been discussing (Schwarzschild, Kerr, FRW, etc.), and which specific solution applies in a particular case will depend on the distribution of matter. But the EFE, as the equation to be solved, applies in every case. So if we disagree on the distribution of matter, we may well disagree on which specific solution to the EFE to apply, but if we accept GR, we *have* to agree that the EFE applies. If you don't accept that, you don't accept GR.

(The only caveat to the above is what I've said before about spacetime singularities: there the EFE itself tells us it can't apply. But we can get as close to the singularities as we like and still apply the EFE.)

(2) Specifying a "matter distribution" in order to solve the EFE can be done in a variety of ways; it can, as you say, be "general", but you may not be appreciating just how general it can be. For example, to obtain the FRW solutions, we specify: "The matter distribution is a perfect fluid, and we'll write the solution in coordinates in which that fluid is isotropic." That's all. Similarly, to obtain the Schwarzschild solution, we specify: "There is no matter--the stress-energy tensor is identically zero--and the solution must be spherically symmetric." (As you can see, often our "specification" takes the form of symmetry properties that the solution must satisfy.)

 

[George Jones]

Heh, good phrase. Can you give a reference? I've read a fair amount of Penrose's writing (at least his writing for the lay reader) and I haven't come across this one.

Look at pages 189-190 in the hardcover edition of Penrose's Road to Reality.

The diagram below is Schwarzschild spacetime in Painleve(-Gullstrand) coodinates $\left(T,R\right)$, which are related to the usual Schwarzschild coordinates $\left(t,r\right)$ by

$$\begin{equation*} \begin{split} T &= t+4M\left( \sqrt{\frac{r}{2M}}+\frac{1}{2}\ln \left| \frac{\sqrt{\frac{r}{2M}}-1}{\sqrt{\frac{r}{2M}}+1}\right| \right)\\ R &= r. \end{split} \end{equation*}$$

On the diagram, the R-axis is horizontal and the T-axis is vertical, the black line is the worldline of an observer who freely falls radially from rest at infinity, and the event horizon is the vertical line R = 1. At three events on the observer's worldline (outside the horizon, on the horizon, and inside the horizon) I have plotted forward light cones in red. In green, I have plotted lines of constant Schwarzschild t that go through the worldline events inside and outside the horizon. These green lines would be horizontal lines on a Schwarzschild $\left(t,r\right)$ grid which has t as the vertical axis and r as the horizontal axis.The green lines both asymptotically approach the event horizon as $T \rightarrow -\infty$.

Of course, coordinate time in these coordinates is not actual proper time for any observer except at r=infinity. You used an interesting phrasing earlier:

"suppose we have a coordinate system where coordinate time directly represents the proper time of some family of observers"

Do you mean something other than equals for "directly represents"? Or are you thinking of some simple transform of the standard Schwarzschild coordinates that normalized t to equal tau ?

Suppose the observer's watch is set such that it reads zero when the observer "hits" the singularity $R = 0$. Then, the events on the observer's (black) worldline all have their T coordinates equal to the observer's watch readings.

Just to check that I've got this right, after some digestion, here's my grasp of these four items, slightly out of order:

2) The transformation between $t$ and $T$ doesn't "change the direction" of the integral curves; it just reparametrizes them.

Yes. Sorry, I going to be a bit pedantic. More pecisely, for every T-integral curve parametrized by T (T-integral curves don't have to be parametrized by T), there is a t-integral curve parametrized by t (t-integral curves don't have to be parametrized by t) such that the two integral curves differ by a constant shift of curve parameter. Even without shiftting parameter, however, every T-integral curve is already a t-integral curve (and vice versa).

This means that, in both coordinate systems, we can use the integral curves of "time" to uniquely label spatial points (each curve has constant values of $r$ = $R$, $\theta$, and $\varphi$).

Yes.

For the rest of this, I'll leave out the angular coordinates (assume them held constant) and only talk about "time" and "radius".

1) The lines of constant $T$, which are integral curves of $\partial_{R}$, "cut at a different angle" from the lines of constant $t$, which are integral curves of $\partial_{r}$. So even though the integral curves of "time" stay the same, the "spatial slices" cut through them can be different if they're cut at different angles.

Yes.

3) It looks to me like the 4-velocity you gave, which gives us 4), also gives us 3), since it makes it obvious that the 4-velocity is *not* just $\partial_{T}$, so the integral curves of the 4-velocity can't be the same as the integral curves of $\partial_{T}$. The integral curves of the 4-velocity are "tilted inward", while the integral curves of $\partial_{T}$ (and thus, of course, $\partial_{t}$) are "vertical".

Yes.

4) Since the 4-velocity "tilts inward", the integral curves of $\partial_{R}$, to be orthogonal to them, must "tilt downward" relative to the integral curves of $\partial_{r}$, which are "horizontal".

Oops, I think this should be "tilt upward", since this is spacetime, not space, so "orthogonal" works differently.

On a $\left(t,r\right)$ diagram, the nature of an integral curve of $\partial_{R}$ depends on which side of the horizon the curve lies. Here, I think you mean outside the horizon. On an $\left(t,r\right)$ diagram, integral curves of $\partial_{R}$ look like the natives of the green curves below.

5) The integral curves of $\partial_{R}$ are always spacelike (horizontal lines outside all lightcones below). Contrast this with the integral curves of $\partial_{r}$, which are spacelike outside the horizon (right green line outside lightcone) and timelike inside (left green line inside lightcone).

6) $\left(T,R\right)$ coodinates are defined on the horizon (unlike $\left(t,r\right)$ coodintes), with T lightlike on the horizon (part of red lightcone is vertical).

7) T is timelike outside the horizon, but all (four) Painleve coordinates are spacelike inside the horizon!

[PLAIN]http://img832.imageshack.us/img832/4121/painlevegullstrand.jpg

 

[PAllen]

Suppose the observer's watch is set such that it reads zero when the observer "hits" the singularity $R = 0$. Then, the events on the observer's (black) worldline all have their T coordinates equal to the observer's watch readings.

You responded this way to my observation:

"Of course, coordinate time in these coordinates is not actual proper time for any observer except at r=infinity. You used an interesting phrasing earlier:"

I should have said "any constant r observer" rather than "any observer". I thought the context was clear (implied by r=infinity, for example), but perhaps not. Anyway, thanks, it is useful to know this.

 

[PeterDonis]

Look at pages 189-190 in the hardcover edition of Penrose's Road to Reality.

Hmm, I have that book but I must have missed the quote. I'll see if I can dig it out of the pile of books awaiting shelving...

Here, I think you mean outside the horizon.

Yes, I did. Thanks for the detailed response and diagram, it makes things very clear.

7) T is timelike outside the horizon, but all (four) Painleve coordinates are spacelike inside the horizon!

I hadn't thought about this before, but it looks to me like this follows from the Painleve metric, since the coefficient $g_{TT}$ is the same as the coefficient $g_{tt}$ in the Schwarzschild metric, and changes sign for r < 2M.

This does bring up another question, which is whether such a coordinate system really "qualifies" as a coordinate system, since there is no timelike coordinate inside the horizon. Is there some way to tell "by inspection" that the system is still "OK" (where "OK" means something like "covers everything we want it to cover"--in this case that would be the exterior and the future interior)? Or does one just need to turn the crank and look at the integral curves of the coordinates in some other system, e.g., Kruskal, that is known to cover the entire manifold?

 

[JDoolin]

Ok, good, that makes it clear what the lightcone in your model means--except that this...



...is *not* correct if what you said above is true. There is no "horizontal plane through (0, 0)" in the FWD. See my comments further below.



This is correct, because, as I just noted, there is *no* "horizontal line" in the FWD (I said this in a previous post as well). The FWD includes lines that get as close to "horizontal" as you like, but none that are exactly horizontal. Yes, that means that this...



...is true; that's usually how "conformal" diagrams work. (A Mercator projection of the Earth's surface, for example, is "conformal" in this sense--it maps the North and South poles to horizontal lines, not points.) But that does *not* mean that this...



is true. Consider the Mercator projection again: the lines of longitude (great circles through the poles) are mapped to vertical lines, which *appear* to meet the poles at "different places". But that's an artifact of the "infinite distortion" that the projection makes at the poles. In the same way, the *apparent* "multiple light rays" from the initial singularity in the CPD are an artifact of the "infinite distortion" that this diagram makes at the initial singularity.

How do we deal with this? The only really consistent way is to accept that these "conformal" diagrams *cannot* actually represent the singularities (just as we don't actually use the Mercator projection at the poles). What the "multiple light rays" in the conformal diagram are actually indicating is that, at very short times after the initial singularity, worldlines which emerged from that singularity "in different directions" will be causally disconnected; the more "different" the initial directions are, the longer it will take for the worldlines to become causally connected again. The light that is reaching is now from "close to the big bang" is coming from worldlines that emerged from the big bang in a direction that was "more different" from ours than light that reached us from close to the big bang some time ago.

I realize the above is a somewhat vague description; when I have more time I can try to make it more precise if needed.



Strictly speaking, this is false, although it's sometimes used colloquially to describe what I was describing above, that events very close to the big bang happened throughout the universe (in the sense of causal disconnection I gave above).



As I noted above, the FRW models do not claim that we can see light (or any other signal) "from the big bang" itself. The earliest *photons* we can see in the FRW models are, as you say, those from the time of recombination. However, the FRW model would predict that we could see other radiation from earlier times (e.g., neutrinos from the electroweak phase transition, or gravitational waves from even earlier times). We don't currently have any way of testing such predictions because of our poor ability to detect any kind of radiation other than electromagnetic.



I reached this point in the e-book today, as it happens. I still find the "infinite density" part of the model physically unreasonable, but I agree that *if* you stipulate that the density goes to infinity at the "photon shell", there would be no possibility of anything coming in from outside the shell.



This is one reason (but hardly the only reason) that I find the infinite density physically unreasonable.



Another reason I find the infinite density physically unreasonable is that it should result in infinite curvature at the "photon shell", which, aside from any other objections, would contradict the initial assumption of a flat background Minkowski spacetime. Even if spacetime *inside* the shell were flat (which it could be since that's a general result for inside a symmetrical spherical shell even in Newtonian gravity), the *complete* spacetime in which everything is embedded could not be.



This is much closer, I think, but I still have a couple of comments:

(1) According to General Relativity, there are no situations where the EFE does not apply. It always does. There are certainly a wide variety of particular *solutions* to the EFE, among which are the various spacetimes we've been discussing (Schwarzschild, Kerr, FRW, etc.), and which specific solution applies in a particular case will depend on the distribution of matter. But the EFE, as the equation to be solved, applies in every case. So if we disagree on the distribution of matter, we may well disagree on which specific solution to the EFE to apply, but if we accept GR, we *have* to agree that the EFE applies. If you don't accept that, you don't accept GR.

(The only caveat to the above is what I've said before about spacetime singularities: there the EFE itself tells us it can't apply. But we can get as close to the singularities as we like and still apply the EFE.)

(2) Specifying a "matter distribution" in order to solve the EFE can be done in a variety of ways; it can, as you say, be "general", but you may not be appreciating just how general it can be. For example, to obtain the FRW solutions, we specify: "The matter distribution is a perfect fluid, and we'll write the solution in coordinates in which that fluid is isotropic." That's all. Similarly, to obtain the Schwarzschild solution, we specify: "There is no matter--the stress-energy tensor is identically zero--and the solution must be spherically symmetric." (As you can see, often our "specification" takes the form of symmetry properties that the solution must satisfy.)

This last post comes off as though you are correcting me, rather than acknowledging that we really are talking about two distinct theories. Do you understand that I am describing TWO DIFFERENT models yet?

The Milne Model (Milne Minkowski Diagram MMD), where the Big Bang is ONE EVENT, and the horizontal plane in the FWD is mapped to the light-cone.

The Standard Model (Comoving Particle Diagram CPD), where the Big Bang is MANY EVENTS, and the horizontal plane in the FWD does not exist.

I appreciate your objection to Milne Model, that it has "infinite density" and that is a problem for you. t least I can see that you are actually familiar with the model. But infinite density is a natural extension of lorentz contraction and time dilation as rapidities go from -infinity to infinity. It's not something that is simply assumed; it is something that follows logically from Special Relativity.

I also appreciate that the EFE's start with a symmetry. The Milne Model has the following symmetry: It is the ONLY distribution of matter which is invariant under Lorentz Transformation.

Finally, I wanted to ask whether this transformation (where a single event gets mapped to multiple locations) happens in other metrics; schwarszchild, Kerr, etc.? You seem comfortable with it, as though obviously, it happens at the big bang, but is there anything other than the imperfect analogy with the Mercator Projection which makes it follow naturally or inevitably?

 

[PeterDonis]

This last post comes off as though you are correcting me, rather than acknowledging that we really are talking about two distinct theories. Do you understand that I am describing TWO DIFFERENT models yet?

I haven't been sure, because some of the things you've been saying seem to imply that there is at least a correspondence between the models; for example, this:

The Milne Model (Milne Minkowski Diagram MMD), where the Big Bang is ONE EVENT, and the horizontal plane in the FWD is mapped to the light-cone.

The Standard Model (Comoving Particle Diagram CPD), where the Big Bang is MANY EVENTS, and the horizontal plane in the FWD does not exist.

If the Milne model is really supposed to be a "different model", then where does the "horizontal plane in the FWD is mapped to the light-cone" come from? That correspondence between the FWD and a "light cone" diagram, for example the "SR" diagram on Ned Wright's page, *only* applies if we are talking about the FRW model with k = -1 and zero density (as Wright says on his web page). In fact, that correspondence is the basis for the claim that the Milne model is a special case of the FRW models (the case with k = -1 and zero density), which you reject. If you are not talking about the FRW model but about some other model, then I don't see how you're coming up with a correspondence between your "light cone" diagram and the FWD.

But infinite density is a natural extension of lorentz contraction and time dilation as rapidities go from -infinity to infinity. It's not something that is simply assumed; it is something that follows logically from Special Relativity.

Only if you also assume that spacetime can be flat with a non-zero stress-energy tensor. See next comment.

I also appreciate that the EFE's start with a symmetry. The Milne Model has the following symmetry: It is the ONLY distribution of matter which is invariant under Lorentz Transformation.

First of all, I assume you mean "globally invariant under Lorentz Transformation," since any distribution which is proportional to the metric (e.g., a cosmological constant or a scalar field) will be *locally* invariant under Lorentz Transformation.

Second, and more important, your model assumes that you can have a non-zero distribution of matter without affecting the spacetime geometry. Or, if you insist on avoiding any "geometric" interpretation, your model assumes that you can have a non-zero distribution of matter without any tidal gravity effects--that any pair of freely moving particles in your model will have the same constant relative velocity (each one as seen by the other) for all time. This is known to be false--for example, because of the curvature in the Hubble diagram that we discussed before.

To be clear, I am not talking here about "local" effects such as bending of light by the Sun, but about "global" effects, about the relative velocity of "freely moving" particles on a cosmological scale (the ones whose worldlines are straight lines radiating out from the Big Bang event in your diagram). In GR, such particles, even though they are freely falling (they feel no acceleration), can have relative velocities that vary with time. This shows up in our observations as a variation in the "Hubble constant"--the slope of the curve in the Hubble diagram--with time. According to the Milne model, this is impossible--this should be obvious from the fact that, as you say, the Milne model is based on logical deductions from SR, since in SR there can be no such variation with time in the relative velocity of freely falling objects (i.e., objects moving on inertial worldlines). This is why GR was necessary--because in the presence of gravity (i.e., when the effect of mass-energy on the behavior of inertial worldlines is significant), freely falling objects can change their relative velocity with time (in other words, tidal gravity is present), and SR cannot account for that.

You can see this "curvature of freely falling worldlines" in Ned Wright's diagram of the "critical density" case (the "FPD" version). Notice that in that diagram, the worldlines radiating out from the Big Bang curve inward towards each other--unlike the "zero density" diagram, where they are straight. This is the effect of non-zero mass-energy (i.e., gravity) on freely falling worldlines (or "spacetime geometry" in the usual terminology). The usual pop-science way of describing this is that "the gravitational attraction of the mass-energy in the universe causes the expansion of the universe to slow down." (This terminology was invented before we discovered that, for the last few billion years or so, the expansion has actually been "speeding up", which is why dark energy has been added to the "standard" cosmological model--Ned Wright's diagrams don't cover that case, although I believe he discusses it elsewhere on his cosmology site.)

Finally, I wanted to ask whether this transformation (where a single event gets mapped to multiple locations) happens in other metrics; schwarszchild, Kerr, etc.? You seem comfortable with it, as though obviously, it happens at the big bang, but is there anything other than the imperfect analogy with the Mercator Projection which makes it follow naturally or inevitably?

Mathematically, it's fairly easy to construct transformations that do weird things like this. For example, the transformations used to construct Penrose diagrams map various points or lines at "infinity" to finite coordinate values (see the Wikipedia page here: http://en.wikipedia.org/wiki/Penrose_diagram). There's nothing inconsistent about them; you just have to get used to how they work.

As far as other metrics are concerned, yes, there are transformations often used in GR that have similar effects. For example, in Schwarzschild coordinates, there appears to be an entire infinite line at the horizon, r = 2M, t = minus infinity to plus infinity, that actually, physically, is just a point, as you can see by transforming to Kruskal coordinates, where that entire line becomes the single point at the center of the diagram. (Here I've been ignoring the angular coordinates; when we put them back in, the "point" is actually a 2-surface.) This transformation also maps the "point" at t = infinity in Schwarzschild coordinates to an entire null line (the 45-degree line between regions I and II in the diagram with a yellow background on the Wikipedia page here: http://en.wikipedia.org/wiki/Kruskal–Szekeres_coordinates); this null line, the "future horizon", is where all the interesting physics at the horizon actually happens, and it is "invisible" in Schwarzschild coordinates, which often leads to confusion if those coordinates are taken too literally.

The fact that stuff like this can happen is a big reason why physicists are hesitant to attribute too much meaning to coordinates; you always have to check the physical invariants to see what's really going on. For example, I asserted just now that the apparent "line" at the horizon in Schwarzschild coordinates is actually just a point--or, if we include the angular coordinates, what appears to be a 3-surface is actually just a 2-surface. How do I know this is right? (Put another way, how do I know that the description in Kruskal coordinates is the "right" one physically?) Because I can compute the physical 3-volume of the apparent 3-surface, using the metric, and find that it is zero (because the metric coefficient $g_{tt}$ is zero at r = 2M in Schwarzschild coordinates). A similar computation in FRW coordinates shows me that the initial singularity is, physically, a point (because a(t) is zero there, so the spatial metric vanishes), even though it looks like a line (actually a 3-surface, if we include the angular coordinates) in the "conformal" diagram. (Here I do really mean a literal point--zero dimensions--unlike the horizon of a black hole, which is physically a 2-surface--we can compute its area and find that it's non-zero, because the spatial part of the metric doesn't vanish completely. In the FRW case, the entire spatial metric vanishes at the initial singularity.)

 

[Passionflower]

For example, in Schwarzschild coordinates, there appears to be an entire infinite line at the horizon, r = 2M, t = minus infinity to plus infinity, that actually, physically, is just a point,...

A point, please explain? Are you saying that all travelers from different times on a given radial angle meet at the same time at this point?

For example, I asserted just now that the apparent "line" at the horizon in Schwarzschild coordinates is actually just a point--or, if we include the angular coordinates, what appears to be a 3-surface is actually just a 2-surface. How do I know this is right? (Put another way, how do I know that the description in Kruskal coordinates is the "right" one physically?) Because I can compute the physical 3-volume of the apparent 3-surface, using the metric, and find that it is zero (because the metric coefficient $g_{tt}$ is zero at r = 2M in Schwarzschild coordinates).

What volume are you computing?

Could you please explain a bit more what you mean here, I am not sure I can agree here, but I probably misunderstand.

 

[PeterDonis]

A point, please explain? Are you saying that all travelers from different times on a given radial angle meet at the same time at this point?

No, as I noted elsewhere in the post, it's actually a 2-surface when the angular coordinates are taken into account. Also, I was *not* saying that any traveler crossing the horizon passes through this point; as I noted further on, all the actual physics at the horizon is on the "future horizon" null line that runs at 45 degrees up and to the right from the center point in the Kruskal diagram. That's where worldlines crossing the horizon go, and they can cross at anyone of an infinite number of different events.

What volume are you computing?

The "3-volume" spanned by r = 2M, t = minus infinity to plus infinity, theta = 0 to pi, phi = 0 to 2 pi. Since the metric coefficient $g_{tt}$ is zero at r = 2M, the integral corresponding to this 3-volume vanishes, indicating that what looks like a 3-volume in Schwarzschild coordinates is actually, at most, a 2-surface. (We can verify that it is, in fact, a 2-surface and not something with even fewer dimensions by, for example, integrating over the full range of angular coordinates at the "point" at the center of the Kruskal diagram, which gives the nonzero area of the horizon.)

 

[Passionflower]

That's where worldlines crossing the horizon go, and they can cross at anyone of an infinite number of different events.

If the worldlines don't cross there then there must be a dimension to separate them right? How do you explain that?

The "3-volume" spanned by r = 2M, t = minus infinity to plus infinity, theta = 0 to pi, phi = 0 to 2 pi. Since the metric coefficient $g_{tt}$ is zero at r = 2M, the integral corresponding to this 3-volume vanishes, indicating that what looks like a 3-volume in Schwarzschild coordinates is actually, at most, a 2-surface. (We can verify that it is, in fact, a 2-surface and not something with even fewer dimensions by, for example, integrating over the full range of angular coordinates at the "point" at the center of the Kruskal diagram, which gives the nonzero area of the horizon.)

Perhaps you could show me the formulas you use, let's say we take the "volume" (I am still not sure what physical volume you are talking about) of r=4M and r=2M so we can all see the difference between a surface at r=4M and r=2M.

Just to remind everybody, the r-coordinate is not a radius or a measure of distance, the r-coordinate is instead a function of an area!

So r is defined as:

$$r = {1 \over 2}\,\sqrt {{\frac {A}{\pi }}}$$

Let's keep that in mind.

 

[JDoolin]

I haven't been sure, because some of the things you've been saying seem to imply that there is at least a correspondence between the models; for example, this:



If the Milne model is really supposed to be a "different model", then where does the "horizontal plane in the FWD is mapped to the light-cone" come from? That correspondence between the FWD and a "light cone" diagram, for example the "SR" diagram on Ned Wright's page, *only* applies if we are talking about the FRW model with k = -1 and zero density (as Wright says on his web page). In fact, that correspondence is the basis for the claim that the Milne model is a special case of the FRW models (the case with k = -1 and zero density), which you reject. If you are not talking about the FRW model but about some other model, then I don't see how you're coming up with a correspondence between your "light cone" diagram and the FWD.

If I recall correctly, yes, the Milne model has k=0, a=1. It is Minkowski Spacetime with a density that approaches infinity as you go out from the center toward r=c*t, and it has a density that approaches infinity as to go toward t=0. That is not at all compatible with a zero density.

If anyone thinks they have correctly modeled the Milne universe with a zero density, they are just fooling themselves, and using some kind of circular reasoning or a straw-man argument.

There are two variables in the Friedmann Walker Diagram, the horizontal variable is a "space-like" variable, and the vertical is a "time-like" variable. To map to the Comoving Particle Diagram, I'm not sure exactly how it is done, but I think the vertical "time component" is just mapped straight over, while the horizontal "space component" is some form of velocity * distance. I may be wrong, but I *think* it is the integral of the changing scale factor with respect to the "cosmological time."

The horizontal variable is SPACE, and the vertical variable is TIME; some kind of "Absolute" or "cosmological" time, which really doesn't exist in the realm of Special Relativity.

What the Milne model does is treats the horizontal variable in the FWD as sort of a "rapidity-space" The mapping from the FWD to the MMD assumes that the meaning of the FWD "space-like" variable is distance = rapidity * proper time. For a set of particles all coming from an event (0,0), giving the rapidity and proper time for a particle uniquely defines its position in space and time. To map from the FWD to the MMD, you are simply mapping: (d'=rapidity*proper time,t'=proper time) to (d=space, t=time).

To map from the FWD to the CPD, you are mapping (d'="Stretchy" Velocity * Cosmological Time, t'=Cosmological Time) to (d=Space,t=Cosmological Time).

I'll see if I can express this as mathematically and unambiguously as I can, so that if I'm wrong it can be corrected.

$$\begin{matrix} FWD \mapsto CPD \text{ as }(d\int a(\tau)d\tau,\tau)\mapsto(d,\tau) \\ d=Proper Distance = Cosmological Distance \\ \tau=Proper Time=CosmologicalTime \\ a(\tau)=ScaleFactor \end{matrix}$$​

On the other hand, the Milne mapping looks like this:

$$\begin{matrix} FWD \mapsto MMD \text{ as }(\varphi \cdot\tau,\tau)\mapsto(v \cdot t,t) \\ \varphi=rapidity \\ \tau=proper time \\ v = velocity \\ t = time \end{matrix}$$​

As you can see, the Milne mapping is linear; there's no changing scale factor. The relation between rapidity and velocity and distance, time, and proper time is the same as is usually given in Special Relativity.

Rapidities between -infinity and +infinity map to velocities between -c and +c. So the horizontal plane (representing infinite rapidity) in the Friedmann Walker Diagram maps to the light-cone in the Milne Minkowski Diagram.

So after any corrections to the FWD to CPD mapping, I wonder if you are yet convinced that we are talking about two different mappings? Can you see that the rapidity=infinity line is included in the Milne Model? Can you see that the distance vs. time relation is fundamentally different? Can you see how an infinite density naturally results from this mapping? Can you see how the Milne Model has only one event at x=0,t=0, while the CPD has an infinite number of events at x=0,t=0?

Only if you also assume that spacetime can be flat with a non-zero stress-energy tensor. See next comment.

First of all, I assume you mean "globally invariant under Lorentz Transformation," since any distribution which is proportional to the metric (e.g., a cosmological constant or a scalar field) will be *locally* invariant under Lorentz Transformation.

Yes, I mean "globally invariant"

Second, and more important, your model assumes that you can have a non-zero distribution of matter without affecting the spacetime geometry. Or, if you insist on avoiding any "geometric" interpretation, your model assumes that you can have a non-zero distribution of matter without any tidal gravity effects--that any pair of freely moving particles in your model will have the same constant relative velocity (each one as seen by the other) for all time. This is known to be false--for example, because of the curvature in the Hubble diagram that we discussed before.

I may have implied that "any pair" of freely moving particles would have the same constant relative velocity, but I must back off on that. But it is not "any pair." It is a specific set of particles which are completely undisturbed after the moment of the Big Bang which will maintain a constant relative velocity.

You have the "Relativity, Gravitation, and World Structure" e-book; check section 112, the list of properties, item 10, Milne says:

"Every particle of the system is in uniform radial motion outward from any arbitrary particle O of the system, and the acceleration of every particle in the system is zero. But the acceleration of a freely projected particle, other than the given particles, is not zero."

I think what is happening is that every particle that is at the center of mass remains at the center of mass, but if you move away from that center of mass, you'll be attracted toward it.

The unusual thing is that accelerating toward the center of mass actually causes the center to move away from you rather than toward you. That results in inflation.

To be clear, I am not talking here about "local" effects such as bending of light by the Sun, but about "global" effects, about the relative velocity of "freely moving" particles on a cosmological scale (the ones whose worldlines are straight lines radiating out from the Big Bang event in your diagram). In GR, such particles, even though they are freely falling (they feel no acceleration), can have relative velocities that vary with time. This shows up in our observations as a variation in the "Hubble constant"--the slope of the curve in the Hubble diagram--with time. According to the Milne model, this is impossible--this should be obvious from the fact that, as you say, the Milne model is based on logical deductions from SR, since in SR there can be no such variation with time in the relative velocity of freely falling objects (i.e., objects moving on inertial worldlines).

This is the elephant in the room that I wanted to talk about earlier. Acceleration.
The Big Bang is one explosion, but particle decay should lead to secondary explosions. If you have secondary (and tertiary, etc.) explosions of matter, this naturally leads to variations in the "Hubble Constant" based on logical deductions from SR. Our local Hubble Constant is based not on the age of the universe, but on the time since the most recent explosion. Only the most distant Hubble Constant tells you the age of the universe.

This is why GR was necessary--because in the presence of gravity (i.e., when the effect of mass-energy on the behavior of inertial worldlines is significant), freely falling objects can change their relative velocity with time (in other words, tidal gravity is present), and SR cannot account for that.

You can see this "curvature of freely falling worldlines" in Ned Wright's diagram of the "critical density" case (the "FPD" version). Notice that in that diagram, the worldlines radiating out from the Big Bang curve inward towards each other--unlike the "zero density" diagram, where they are straight. This is the effect of non-zero mass-energy (i.e., gravity) on freely falling worldlines (or "spacetime geometry" in the usual terminology). The usual pop-science way of describing this is that "the gravitational attraction of the mass-energy in the universe causes the expansion of the universe to slow down." (This terminology was invented before we discovered that, for the last few billion years or so, the expansion has actually been "speeding up", which is why dark energy has been added to the "standard" cosmological model--Ned Wright's diagrams don't cover that case, although I believe he discusses it elsewhere on his cosmology site.)

The pop-science way of describing these things is not nearly sufficient to give any hint to me about what they are actually measuring. I'm interested in whatever data they've gathered to determine that the "expansion of the universe is speeding up" but I think it more likely that whatever the effect, the cause is much more likely to be our galaxy's acceleration toward the receding center of mass, rather than some universal cosmological scale factor increasing at a faster rate.

Mathematically, it's fairly easy to construct transformations that do weird things like this. For example, the transformations used to construct Penrose diagrams map various points or lines at "infinity" to finite coordinate values (see the Wikipedia page here: http://en.wikipedia.org/wiki/Penrose_diagram). There's nothing inconsistent about them; you just have to get used to how they work.

As far as other metrics are concerned, yes, there are transformations often used in GR that have similar effects. For example, in Schwarzschild coordinates, there appears to be an entire infinite line at the horizon, r = 2M, t = minus infinity to plus infinity, that actually, physically, is just a point, as you can see by transforming to Kruskal coordinates, where that entire line becomes the single point at the center of the diagram. (Here I've been ignoring the angular coordinates; when we put them back in, the "point" is actually a 2-surface.) This transformation also maps the "point" at t = infinity in Schwarzschild coordinates to an entire null line (the 45-degree line between regions I and II in the diagram with a yellow background on the Wikipedia page here: http://en.wikipedia.org/wiki/Kruskal–Szekeres_coordinates); this null line, the "future horizon", is where all the interesting physics at the horizon actually happens, and it is "invisible" in Schwarzschild coordinates, which often leads to confusion if those coordinates are taken too literally.

The fact that stuff like this can happen is a big reason why physicists are hesitant to attribute too much meaning to coordinates; you always have to check the physical invariants to see what's really going on. For example, I asserted just now that the apparent "line" at the horizon in Schwarzschild coordinates is actually just a point--or, if we include the angular coordinates, what appears to be a 3-surface is actually just a 2-surface. How do I know this is right? (Put another way, how do I know that the description in Kruskal coordinates is the "right" one physically?) Because I can compute the physical 3-volume of the apparent 3-surface, using the metric, and find that it is zero (because the metric coefficient $g_{tt}$ is zero at r = 2M in Schwarzschild coordinates). A similar computation in FRW coordinates shows me that the initial singularity is, physically, a point (because a(t) is zero there, so the spatial metric vanishes), even though it looks like a line (actually a 3-surface, if we include the angular coordinates) in the "conformal" diagram. (Here I do really mean a literal point--zero dimensions--unlike the horizon of a black hole, which is physically a 2-surface--we can compute its area and find that it's non-zero, because the spatial part of the metric doesn't vanish completely. In the FRW case, the entire spatial metric vanishes at the initial singularity.)

I'm afraid I'm at a loss for following most of this. I'm not entirely convinced you have anywhere a single event which is mapped to multiple places (except for the FRW case where a(t) shrinks to zero).

If we could focus on the nature of the Schwarzschild coordinates; I'm afraid you'll have to go completely remedial to explain it to me, because I don't know the equations for the Schwarzschild coordinates or the reasoning. All I know is that time slows down as you get near the surface, and this causes light to turn toward the planet and get a higher frequency. When you're talking about the "line" at the horizon which is actually just a point, or the three-surface which is really just a two-surface, I'm not sure precisely what is meant.

 

[PeterDonis]

If the worldlines don't cross there then there must be a dimension to separate them right? How do you explain that?

There is--it's the Kruskal V coordinate (in the null U, V formulation of Kruskal coordinates, which uses V to label events along the future horizon).

Perhaps you could show me the formulas you use, let's say we take the "volume" (I am still not sure what physical volume you are talking about) of r=4M and r=2M so we can all see the difference between a surface at r=4M and r=2M.

The 3-volume is a 3-volume in 4-dimensional spacetime; the only difference between it and a "normal" spatial 3-volume is that one of the dimensions is the time coordinate. All you do is integrate the appropriate volume measure over the appropriate range of coordinates. The volume measure is just the product of the distance measures along each coordinate; each distance measure is the square root of the appropriate metric coefficient times the coordinate differential. (Some of this may be simpler because the Schwarzschild metric is diagonal; I'd have to go back and do some review to remind myself of what, if any, complications arise when dealing with a non-diagonal metric such as the Kerr metric.) So for a constant value of r, the "volume integral" looks like:

$$V = \int_{t_{1}}^{t_{2}} \sqrt{- g_{tt}} dt \int_{0}^{\pi} \sqrt{g_{\theta \theta}} d\theta \int_{0}^{2 \pi} \sqrt{g_{\varphi \varphi}} d\varphi = 4 \pi r^{2} \sqrt{1 - \frac{2M}{r}} \left( t_{2} - t_{1} \right)$$

where I've included the minus sign in front of ${g_{tt}}$ because of the opposite sign of that metric coefficient. At r = 4M, this integral gives a positive value (how large a value depends on the range of t we choose--we can make it infinite by letting t cover its full range of $- \infty$ to $\infty$), but at r = 2M, the integral vanishes identically.

Just to remind everybody, the r-coordinate is not a radius or a measure of distance, the r-coordinate is instead a function of an area!

Yes, it is, which makes the integral over the angular coordinates very easy to do at a constant value of r (that's why there's just $4 \pi r^{2}$ in the integral above, instead of some more complicated function of r). But the r coordinate is still being used to *label* events along the radial dimension, which is different from the two angular dimensions. It's not a direct measure of radial distance because the metric coefficient $g_{rr}$ varies, but it's still a coordinate in the radial direction.

 

[Passionflower]

There is--it's the Kruskal V coordinate (in the null U, V formulation of Kruskal coordinates, which uses V to label events along the future horizon).

So then if you agree that different observers at different times can pass the event horizon at the same physical location then clearly there must be a line and not a point for a given theta and phi?

I have to get back on your volume calculation, I am not very encouraged by what I see (but clearly that must be my shortcoming).

 

[PeterDonis]

If I recall correctly, yes, the Milne model has k=0, a=1.

Not if it's a different model from the FRW models; the k and a parameters apply to those models. The FRW model that is claimed to be an alternate formulation of the Milne model has k = -1, a(t) = t (linear variation, starting with a = 0 at t = 0, the Big Bang) in the FRW parameter system; but you don't agree that that model actually is an alternate formulation of the Milne model.

It is Minkowski Spacetime with a density that approaches infinity as you go out from the center toward r=c*t, and it has a density that approaches infinity as to go toward t=0. That is not at all compatible with a zero density.

If anyone thinks they have correctly modeled the Milne universe with a zero density, they are just fooling themselves, and using some kind of circular reasoning or a straw-man argument.

No, they are pointing out the same physical issue that I pointed out in my last post, that a nonzero density will curve freely falling worldlines, so you can't have Minkowski spacetime with a non-zero density. See below.

So after any corrections to the FWD to CPD mapping, I wonder if you are yet convinced that we are talking about two different mappings? Can you see that the rapidity=infinity line is included in the Milne Model? Can you see that the distance vs. time relation is fundamentally different? Can you see how an infinite density naturally results from this mapping? Can you see how the Milne Model has only one event at x=0,t=0, while the CPD has an infinite number of events at x=0,t=0?

I understand everything you're saying about your diagram of the Milne model. You're not correct that the CPD has an infinite number of events at x = 0, t = 0; it's only one event., and it has coordinates t = 0, but x, y, z undefined (in the same way the longitude of the South Pole is undefined), and it looks like a line in the diagram only because we need to have a reference for the "limit point" of all the vertical worldlines. As you'll see from my comments near the end of this post, I'm afraid I can't really do justice to the mathematical side of this, but I'll try to briefly explain what I think it's supposed to mean physically.

The "conformal" diagram of the FRW model (your "CPD") is intended to make it easy to see the causal structure of the spacetime, and that's all--it highly distorts everything else in order to ensure that light rays always travel on 45 degree lines, so it's easy to see the boundaries of light cones. Pick two worldlines that are widely separated in the conformal diagram, and follow them back towards the "initial singularity" line at the bottom. You'll see that the past light cones of those two worldlines stop overlapping while they're still a fair distance away from the initial singularity. If you pick two worldlines that are closer, their past light cones stop overlapping closer to the singularity. As worldlines get closer and closer, their past light cones get closer and closer to the singularity before they stop overlapping. But for any time after the initial singularity, there will be some "conformal separation" (separation horizontally in the conformal diagram) of worldlines for which the past light cones have stopped overlapping. The conformal diagram was constructed to make all this visually obvious, but the price you pay is making the single event at t = 0 (all space coordinates undefined) look like a line. Actually, the boundary line itself is *not* part of the spacetime; it's just a convenient way of showing, visually, that the vertical worldlines "end" at the initial singularity (which, technically, is not part of the spacetime either, as I've said before--the density and curvature are infinite there, so physically we expect new physics, perhaps quantum gravity, to take over before that point is reached).

I may have implied that "any pair" of freely moving particles would have the same constant relative velocity, but I must back off on that. But it is not "any pair." It is a specific set of particles which are completely undisturbed after the moment of the Big Bang which will maintain a constant relative velocity.

You have the "Relativity, Gravitation, and World Structure" e-book; check section 112, the list of properties, item 10, Milne says:

"Every particle of the system is in uniform radial motion outward from any arbitrary particle O of the system, and the acceleration of every particle in the system is zero. But the acceleration of a freely projected particle, other than the given particles, is not zero."

Yes, thanks for including that quote as it clarifies the point I was trying to make. The "specific set of particles" you mention is the set of "given particles" in Milne's terminology. In your diagram, they appear as straight lines radiating out from the big bang event, all within the "envelope" of the light cone. The point I am making is that in the presence of gravity (i.e., with a non-zero density), GR predicts that those lines cannot be straight lines! As I noted before, in the diagram of the "critical density" FRW model on Ned Wright's page, you see the corresponding lines curve inward towards each other, whereas in the "zero density" model, they are straight.

So here we have a clear case where standard GR and the Milne model make different physical predictions: Milne's model says that we can have a non-zero density in the universe but still have a family of "given particles" that maintain constant relative velocity for all time; GR says that's not possible and predicts that with non-zero density we will see the relative velocities even of "comoving" particles (the ones that correspond to Milne's "given" particles) vary with time.

The Hubble diagram provides a way to test this; but since you commented on that with regard to "inflation", I'll comment further on it below.

The unusual thing is that accelerating toward the center of mass actually causes the center to move away from you rather than toward you. That results in inflation.

This is the elephant in the room that I wanted to talk about earlier. Acceleration. The Big Bang is one explosion, but particle decay should lead to secondary explosions. If you have secondary (and tertiary, etc.) explosions of matter, this naturally leads to variations in the "Hubble Constant" based on logical deductions from SR. Our local Hubble Constant is based not on the age of the universe, but on the time since the most recent explosion. Only the most distant Hubble Constant tells you the age of the universe.

Saying that "our local Hubble constant is based not on the age of the universe, but on the time since the most recent explosion" is not based on direct observation but on a conclusion from your model; you need to give me some way of directly testing this assertion vs. the contrary assertion of standard GR, which is that the "Hubble Constant" at any distance indicates the general "expansion of the universe" at the epoch corresponding to that distance. Since we observe variation in the observed Hubble constant with distance (the curvature in the Hubble diagram), the standard GR model (the FRW model) infers that the relative velocity of "comoving" observers can change with time.

It seems as though you're saying that the Milne model explains the curvature in the Hubble diagram by "secondary explosions". The main problem I see with this is that it should not produce curvature in the diagram; it should produce anisotropy--different slopes of the Hubble plot in different directions (corresponding to different epochs for each "secondary explosion"), but each plot should be a straight line, no curvature (because once the explosion happens, the debris, so to speak, travels in straight lines, with each piece having a constant relative velocity with respect to all other pieces--each "piece" being one of the set of "given particles" created by that particular explosion). We don't see anything like this; we see curvature in the diagram, but it's isotropic.

Even if you push all the secondary explosions back into the "inflation" era, they should still produce anisotropies in, for example, the CMBR, that would have to be larger than those we've observed--at least it seems that way to me, but since your model isn't quantified, it's hard to tell what it predicts for this. But in any case, if all the secondary explosions occurred during the inflation era, they wouldn't affect the Hubble diagram at all, since that only goes back as far as we can see galaxies, quasars, and other objects that have measurable redshifts. So the Hubble diagram, in this case, would indicate the general expansion of the universe, and should *not* have any curvature according to the Milne model.

I'm afraid I'm at a loss for following most of this. I'm not entirely convinced you have anywhere a single event which is mapped to multiple places (except for the FRW case where a(t) shrinks to zero).

If we could focus on the nature of the Schwarzschild coordinates; I'm afraid you'll have to go completely remedial to explain it to me, because I don't know the equations for the Schwarzschild coordinates or the reasoning. All I know is that time slows down as you get near the surface, and this causes light to turn toward the planet and get a higher frequency.

That would take us pretty far afield, and there's a lot of material out there already about Schwarzschild spacetime. (Sorry to punt, more or less, but I just don't think I can do justice to the topic or give you a proper treatment of it in a discussion thread; it really does need to be studied carefully offline, at your own pace and following your own thread of inquiry.) Instead, let me just post a couple of links:

http://casa.colorado.edu/~ajsh/schwp.html

http://www.phy.syr.edu/courses/modules/LIGHTCONE/schwarzschild.html

Unfortunately, the PhysicsForums library doesn't appear to have anything on Schwarzschild spacetime (at least, not anything I could find with a quick search).

Also, all the textbooks on GR spend a fair bit of time talking about Schwarzschild spacetime because it's used so much.

(I'm also pretty much punting at this point, I'm afraid, on the general question of transformations that do weird things like map points to lines, lines to points, etc. As I said in the previous post you quoted, I think it's a mistake to get too wrapped up in what a geometry "looks like" in a particular coordinate system. I'd rather focus on the physics and what is actually observed, which is what I was trying to do with my comments about tidal gravity, curvature in the Hubble diagram, etc. There is a vast body of mathematics behind these various coordinate transformations, but again, I just don't think I can do justice to it here; it's something that really has to be studied at your own pace. Unfortunately I can't point you to any particular texts in this area--the GR textbooks go into this somewhat, but they don't really do it with the sort of rigor that mathematicians do.)

When you're talking about the "line" at the horizon which is actually just a point, or the three-surface which is really just a two-surface, I'm not sure precisely what is meant.

See my previous post in response to Passionflower for a little more info about this point.

 

[PeterDonis]

So then if you agree that different observers at different times can pass the event horizon at the same physical location then clearly there must be a line and not a point for a given theta and phi?

Yes--it's the 45 degree line that goes up and to the right from the center point of the Kruskal diagram. (This line has U = constant, V > 0 in the null Kruskal coordinates.) All worldlines that cross the horizon into the black hole cross somewhere on that line, at some positive V coordinate.

 

[Passionflower]

Yes--it's the 45 degree line that goes up and to the right from the center point of the Kruskal diagram. (This line has U = constant, V > 0 in the null Kruskal coordinates.) All worldlines that cross the horizon into the black hole cross somewhere on that line, at some positive V coordinate.

Ok so you are saying that for a given theta and phi that an observer A can cross the EH before observer an B. That implies that r=rs, phi=0, theta=0 must be a line not a point. How else would you explain it?

 

[PeterDonis]

Ok so you are saying that for a given theta and phi that an observer A can cross the EH before observer an B. That implies that r=rs, phi=0, theta=0 must be a line not a point. How else would you explain it?

It is a line, but the line does not appear in the Schwarzschild chart (either exterior or interior). The line is the one I described; U = 0, V > 0 in the null Kruskal coordinates. (The full "horizon" is actually the pair of crossing lines U = 0 and V = 0, but the portion I've described is where all the worldlines going into the black hole from "region I", the "normal" exterior, cross.) This line also has the r coordinate 2M, as can be seen from the implicit equation for r in terms of U and V, as given for example on the Wikipedia page:

http://en.wikipedia.org/wiki/Kruskal–Szekeres_coordinates.

I'm using U, V here to refer to what the Wiki page calls the "light cone variant" of the Kruskal coordinates (this is its term for null coordinates, apparently--they put tildes over U, V for these coordinates and use U, V without tildes to denote the spacelike and timelike Kruskal coordinates, which I'm more used to seeing as X, T, or sometimes R, T, as the Wiki page briefly comments). In these coordinates, r is given implicitly by

$$UV = \left( 1 - \frac{r}{2M} \right) e^{\frac{r}{2M}}$$

So if either U = 0 or V = 0, r = 2M. But this pair of crossing lines (U = 0 and V = 0) does *not* have a well-defined Schwarzschild t coordinate. The equation for t is either

$$tanh \left( \frac{t}{4M} \right) = \frac{V + U}{V - U}$$

outside the horizon, or

$$tanh \left( \frac{t}{4M} \right) = \frac{V - U}{V + U}$$

inside the horizon; but if either U = 0 or V = 0 (i.e., on the horizon), both of these equations have no solution (because the tanh function would have to have a value of +/- 1, and it never does, it only asymptotes to those values).

 

[Passionflower]

It is a line

Well good I agree with that. So now we extend this line in the direction of phi and theta what do we get?

Now earlier you wrote:

For example, in Schwarzschild coordinates, there appears to be an entire infinite line at the horizon, r = 2M, t = minus infinity to plus infinity, that actually, physically, is just a point,

So do you agree it is a line or do I misunderstand here?

 

[PeterDonis]

if either U = 0 or V = 0 (i.e., on the horizon), both of these equations have no solution (because the tanh function would have to have a value of +/- 1, and it never does, it only asymptotes to those values).

I should add, to clarify, that if both U = 0 *and* V = 0 (i.e, at the "center point" of the diagram"), both expressions give 0 / 0 on the RHS, and we would have to take a limit somehow to see if the tanh function could satisfy either equation. The usual assumption seems to be that this can somehow be done in such a way as to be able to assign *any* Schwarzschild t value we like to the center point, from - infinity to infinity (but still at r = 2M).

I haven't seen anything specific in the books I've read on how this could be done, but here's one possible way I can think of: if we assume that either V is some function of U that's linear in U, or U is some function of V that's linear in V, then we can apply L'Hopital's rule and differentiate top and bottom to obtain some constant value on the RHS; and by suitable definition of the functions we can make that constant value basically be anything from -1 to 1, so in effect the center point can map to *any* Schwarzschild t value from - infinity to infinity (at r = 2M).

 

[PeterDonis]

Well good I agree with that. So now we extend this line in the direction of phi and theta what do we get?

A spacetime 3-volume. But again, this 3-volume does not appear in the Schwarzschild chart; only a single 2-surface picked out of it does (the one corresponding to the "center point" of the Kruskal diagram). To see the full 3-volume you have to use a chart that covers the "future horizon" line, such as the Kruskal chart. (The ingoing Eddington-Finkelstein chart or the Painleve chart would also work.)

 

[Passionflower]

A spacetime 3-volume. But again, this 3-volume does not appear in the Schwarzschild chart; only a single 2-surface picked out of it does (the one corresponding to the "center point" of the Kruskal diagram).

Yes, but did you claim just the opposite?

 

[PeterDonis]

Yes, but did you claim just the opposite?

No. Here's everything I said in my post #127 that first responded to this question from you:

No, as I noted elsewhere in the post, it's actually a 2-surface when the angular coordinates are taken into account. Also, I was *not* saying that any traveler crossing the horizon passes through this point; as I noted further on, all the actual physics at the horizon is on the "future horizon" null line that runs at 45 degrees up and to the right from the center point in the Kruskal diagram. That's where worldlines crossing the horizon go, and they can cross at anyone of an infinite number of different events.

The "3-volume" spanned by r = 2M, t = minus infinity to plus infinity, theta = 0 to pi, phi = 0 to 2 pi. Since the metric coefficient $g_{tt}$ is zero at r = 2M, the integral corresponding to this 3-volume vanishes, indicating that what looks like a 3-volume in Schwarzschild coordinates is actually, at most, a 2-surface. (We can verify that it is, in fact, a 2-surface and not something with even fewer dimensions by, for example, integrating over the full range of angular coordinates at the "point" at the center of the Kruskal diagram, which gives the nonzero area of the horizon.)

And I've been repeating ever since that yes, there is a line, but the line is *not* covered by the Schwarzschild chart--what looks like a line in the Schwarzschild chart is actually just a single point, which is at the center of the Kruskal diagram. The line, the "real" one that worldlines going into the hole actually cross, is the line going up and to the right at 45 degrees from the center point. And when you add in the angular coordinates, "point" becomes "2-surface" and "line" becomes "3-volume".