What Are the Key Questions About Supernova Data and the Modified Milne Model?

[JDoolin]

This leads to the Friedmann equations, which describe how the rate of expansion relates to the energy density and pressure of the contents of the fluid.

What I would prefer to see are the actual logical steps coming from the assumptions of homogeneity and isotropy which lead to the Friedmann equations.

For instance, I seem to recall reading an article where, to calculate the force on a particle, the author picked an arbitrary distant particle, imagined a sphere around it, and used http://en.wikipedia.org/wiki/Gauss%27_law_for_gravity" . I also recall being rather dismayed at the author's choice of using symmetry around an arbitrary distant particle, instead of using symmetry around the point of interest.

Perhaps, a more worthy analysis of the uniform perfect fluid is to ask what happens when you have a minor perturbation in the uniformity. For instance, if one particle is removed, or pushed away from it's position, all of the adjacent particles are affected. I believe it may even result in an expanding hole, since all of the adjacent particles would then be pulled away from that opening.

In the uniform model, the effect of distant particles goes down as 1/r^2, while the density remains constant. In the Minkowski-Milne model,the density is NOT constant; the density tends to infinity at a finite distance.

The question does beg for some kind of vector volume integral, and this deserves more thought.

 

[Chalnoth]

What I would prefer to see are the actual logical steps coming from the assumptions of homogeneity and isotropy which lead to the Friedmann equations.

1. Construct a homogeneous, isotropic stress-energy tensor. This isn't terribly difficult: if we start with Euclidean coordinates, it must be diagonal and the diagonal spatial components must be the same. So we end up with just two degrees of freedom: energy density and pressure. The energy density tells us how much of the stuff there is, and the pressure is then determined from the energy density based upon what kind of stuff we have.

2. Construct a homogeneous, isotropic metric. I already showed you this part of it. It ends up depending on two parameters: a function of time (by convention, a(t)), and the spatial curvature (k).

3. From the homogeneous, isotropic metric we can calculate the Einstein tensor. The exact steps here are a bit hairy, but suffice it to say you end up with a tensor that only has diagonal components, and those components depend upon a(t) and k.

4. The Einstein Field equations now equate the Einstein tensor (which depends upon a(t) and k) to the stress-energy tensor we constructed earlier (which depends upon \rho and p). In principle this gives us four equations, but all of the spatial equations are identical, so there's really just two independent equations. The time-time equation can be reduced to the first Friedmann equation. The second equation, by convention again, comes from the sum of all four equations.

This is how it works in General Relativity, of course. You can do the same thing in Newtonian physics just by assuming a homogeneous, isotropic fluid that has some energy density (which equates to the mass density in Newtonian physics) and pressure.

For instance, I seem to recall reading an article where, to calculate the force on a particle, the author picked an arbitrary distant particle, imagined a sphere around it, and used http://en.wikipedia.org/wiki/Gauss%27_law_for_gravity" . I also recall being rather dismayed at the author's choice of using symmetry around an arbitrary distant particle, instead of using symmetry around the point of interest.

Well, in general when you want to exploit the symmetry of the system, the symmetry has to actually exist for it to be valid. You don't have complete freedom to choose the symmetry.

In Newtonian physics, when you compute the force between two particles, you only consider the gravitational field around one of them (basically, a particle's own gravitational field doesn't contribute to the force that particle feels, so it's irrelevant). Thus the correct point of symmetry is not the particle on which you're calculating the force, but the particle that is the source of the force you're calculating.

Perhaps, a more worthy analysis of the uniform perfect fluid is to ask what happens when you have a minor perturbation in the uniformity. For instance, if one particle is removed, or pushed away from it's position, all of the adjacent particles are affected. I believe it may even result in an expanding hole, since all of the adjacent particles would then be pulled away from that opening.

This is a whole topic in cosmology, called perturbation theory. The basic idea is you start with a uniform fluid, and allow there to be deviations from uniformity. You then calculate the effects of those deviations. In general this is a very difficult thing to do, but there are approximations you can make that allow you to calculate the behavior under certain constraints. For our universe, those constraints mean that you can use perturbation theory to accurately calculate the formation of structure in our universe at very large scales. At smaller scales things get much messier and we have to use N-body simulations.

 

[JDoolin]

This is how it works in General Relativity, of course. You can do the same thing in Newtonian physics just by assuming a homogeneous, isotropic fluid that has some energy density (which equates to the mass density in Newtonian physics) and pressure.

If possible, I would like to approach the problem with a basically Newtonian physics. My first thought is that one could perform a volume integral around the affected particle.

integrate (Gravitational function*density*differential Volume element)

The two hard parts are figuring out the density and figuring out the gravitational function.

The density will not be symmetrical around the "observer particle" but will be symmetrical around the world-line through the Big Bang event, parallel to the tangent of the observer-particle's world-curve. There should be a polar symmetry, since the end result of all the acceleration must be a single velocity. We should be able to express the final density as a function of (r,theta).

Also, the density should not be calculated as it is "now" in the observer's frame, but I would presume that the speed of gravity is the same as the speed of light. So if we take Milne's density function as given, we still have to take into account this delay.

Also, the gravitational field of a receding body is going to be less than the gravitational field of an oncoming body. I can see it in the demonstration of the linear motion of the point charge
http://www.its.caltech.edu/~phys1/java/phys1/MovingCharge/MovingCharge.html" , by clicking "Go" and then adjusting the velocity slider, but I need some way of expressing this mathematically.

I think, after having precise mathematical answers to these questions, I can give a legitimate answer to the question I posed in post 12.

Okay. Which way is the particle at (r,t) pulled by gravity? Toward the center, or away from the center, and why? (and by how much?)

As a hint, Milne claimed, a particle at rest (v=0) in this reference frame would be pulled toward the center. I do not recall how he reasoned this out, though. It was not entirely clear. I would have expected there to be no pull in either direction. Because a particle in the same position, but with v=r/t would be in the center, in its own reference frame, so would feel no such pull.

 

[Chalnoth]

If possible, I would like to approach the problem with a basically Newtonian physics. My first thought is that one could perform a volume integral around the affected particle.

integrate (Gravitational function*density*differential Volume element)

The two hard parts are figuring out the density and figuring out the gravitational function.

Yes, this is relatively difficult. For some help, you can try reading this Wikipedia article section.

Also, the gravitational field of a receding body is going to be less than the gravitational field of an oncoming body.

This is wrong. In Newtonian physics it's obvious that it's wrong, because there is no velocity dependence of gravity at all. In General Relativity, it's also wrong but the argument gets a bit more subtle. Basically, because velocity is arbitrary, the gravitational field of a moving particle is just the gravitational field of a stationary particle in a coordinate system moving with respect to the particle.

 

[JDoolin]

Yes, this is relatively difficult. For some help, you can try reading this Wikipedia article section.


This is wrong. In Newtonian physics it's obvious that it's wrong, because there is no velocity dependence of gravity at all. In General Relativity, it's also wrong but the argument gets a bit more subtle. Basically, because velocity is arbitrary, the gravitational field of a moving particle is just the gravitational field of a stationary particle in a coordinate system moving with respect to the particle.

In Special Relativity, there's another problem--simultaneity. If I try to use the gravitational field of the distant, receding particle in its "current" rest frame, then I would be talking about an event that happened long in the past on earth.

If I want to talk about the event that Earth is experiencing now in the receding particle's rest frame, that event is far far in the future in the frame of the receding particle.

 

[Chalnoth]

In Special Relativity, there's another problem--simultaneity. If I try to use the gravitational field of the distant, receding particle in its "current" rest frame, then I would be talking about an event that happened long in the past on earth.

If I want to talk about the event that Earth is experiencing now in the receding particle's rest frame, that event is far far in the future in the frame of the receding particle.

I think you're somewhat misunderstanding simultaneity in relativity. The issue here is that if two events are separated by a space-like distance, then some hypothetical observer will see those two events as being simultaneous. This means that there is no "true" simultaneity at all: any simultaneity that we observe is purely imposed by the coordinate system we are using.

To properly deal with how this arbitrariness interacts with gravity, you really need to use General Relativity. Otherwise there's a chance you won't properly account for the differences in different coordinate systems, and may end up making a mistake without realizing it.

 

[JDoolin]

I think you're somewhat misunderstanding simultaneity in relativity. The issue here is that if two events are separated by a space-like distance, then some hypothetical observer will see those two events as being simultaneous. This means that there is no "true" simultaneity at all: any simultaneity that we observe is purely imposed by the coordinate system we are using.

To properly deal with how this arbitrariness interacts with gravity, you really need to use General Relativity. Otherwise there's a chance you won't properly account for the differences in different coordinate systems, and may end up making a mistake without realizing it.

For now, let me point out that you claim there is no "true" simultaneity at all. Then you "impose" a definition of simultaneity based on a metric.

I feel that a definition of simultaneity based on events of equal t in the Minkowski metric, in fact, is "true" simultaneity. You may disagree on the basis of your opinion.

However, even if you differ in opinion about this, the fact still remains that simultaneity is unambiguously defined in Minkowski Space. Though it is a function of the observer's velocity, it is an explicitly defined function.

 

[Chalnoth]

I feel that a definition of simultaneity based on events of equal t in the Minkowski metric, in fact, is "true" simultaneity. You may disagree on the basis of your opinion.

This doesn't work in General Relativity. The problem is that while you can always use the Minkowski metric for local space-time, in general curvature prevents this metric from working over significant distances (the exact distance is determined by the amount of curvature).

Because of the space-time curvature, simultaneity becomes significantly more arbitrary than in Special Relativity. Furthermore, there is no a priori reason why we should always identify an observer's rest frame with a Minkowski metric that is only valid locally. We might define it as such, but there's no reason we have to. In fact, if we do try, we'll just end up getting the wrong behavior for locations too far away.

However, even if you differ in opinion about this, the fact still remains that simultaneity is unambiguously defined in Minkowski Space. Though it is a function of the observer's velocity, it is an explicitly defined function.

Even in this case, however, Newtonian mechanics has no knowledge of this distinction, and thus there is the chance that naively coupling Newtonian mechanics to special relativity will just give you the wrong answer. The only way to be sure you're right is to use General Relativity, which fully takes into account the ambiguity of simultaneity.

 

[JDoolin]

Even in this case, however, Newtonian mechanics has no knowledge of this distinction, and thus there is the chance that naively coupling Newtonian mechanics to special relativity will just give you the wrong answer.

That is very fatalistic. What I am suggesting is that we consider a very careful (not naive) approach to coupling Newtonian Mechanics to special Relativity. In particular, taking into account Penrose-Terrell rotation, and all the phenomena http://www.wiu.edu/users/jdd109/stuff/relativity/gardner.swf".

There are such things as mathematical proofs, and reason actually does have a place in physics. It's not all just about assertion.

The only way to be sure you're right is to use General Relativity, which fully takes into account the ambiguity of simultaneity.

I do wish you would stop using the blanket term, "General Relativity" what you are discussing here is the Friedmann Metric, or the Lambda CDM model, or the Standard Model or some such thing.

In any case there is no ambiguity of simultaneity. And your theory doesn't "take into account" simultaneity. It assumes comoving particles specifically so that it can ignore simultaneity.

 

[Chalnoth]

I do wish you would stop using the blanket term, "General Relativity" what you are discussing here is the Friedmann Metric, or the Lambda CDM model, or the Standard Model or some such thing.

Er, blanket term? What? General Relativity is the only relativistic theory of gravity we have, and is thus absolutely essential for properly taking into account relativistic effects in systems where gravity is important. The Friedmann equations are, it is true, a very specific solution to the Einstein field equations, but they are the only solution that is consistent with a homogeneous, isotropic universe (meaning anything else you might think is a solution can be related to the Friedmann equations through a simple coordinate transformation).

In any case there is no ambiguity of simultaneity. And your theory doesn't "take into account" simultaneity. It assumes comoving particles specifically so that it can ignore simultaneity.

This is just flat wrong. The ambiguity of simultaneity is dealt with in General Relativity by virtue of the fact that General Relativity propagates the full effect of any coordinate transforms you may wish to perform. Each coordinate system you choose will have a set of simultaneous points, but this is irrelevant to the behavior of the system in GR because the change to any other sort of coordinate system is fully taken into account (meaning that the behavior of a system in GR cannot change just because you describe the system using different coordinates).

You can, if you like, make a transformation to a different coordinate system than that used in the FRW metric where comoving points are no longer simultaneous, and it will all still work. The math may be a bit more complicated, but it will still give the right answer (provided you're careful about dividing by zero and that the new coordinates have a one-to-one relationship with the old ones).

 

[JDoolin]

Er, blanket term? What? General Relativity is the only relativistic theory of gravity we have, and is thus absolutely essential for properly taking into account relativistic effects in systems where gravity is important. The Friedmann equations are, it is true, a very specific solution to the Einstein field equations, but they are the only solution that is consistent with a homogeneous, isotropic universe (meaning anything else you might think is a solution can be related to the Friedmann equations through a simple coordinate transformation).


This is just flat wrong. The ambiguity of simultaneity is dealt with in General Relativity by virtue of the fact that General Relativity propagates the full effect of any coordinate transforms you may wish to perform. Each coordinate system you choose will have a set of simultaneous points, but this is irrelevant to the behavior of the system in GR because the change to any other sort of coordinate system is fully taken into account (meaning that the behavior of a system in GR cannot change just because you describe the system using different coordinates).

You can, if you like, make a transformation to a different coordinate system than that used in the FRW metric where comoving points are no longer simultaneous, and it will all still work. The math may be a bit more complicated, but it will still give the right answer (provided you're careful about dividing by zero and that the new coordinates have a one-to-one relationship with the old ones).

Okay. In Special Relativity, all measurements are strictly source dependent. Are you saying that in General Relativity; in the theory of gravity, you actually go to the inertial frame of the gravitating body to calculate the force on distant points?

And then what do you do if these bodies are traveling apart relativistically? What distance would you use, the distance according to the gravitating body, or the distance according to the body which is pulled by the force?

It seems to me that the gravitational force lines must be somehow transformed into the reference frame of the body which is pulled by the force... i.e. the body which is observing the force.

 

[Chalnoth]

Okay. In Special Relativity, all measurements are strictly source dependent. Are you saying that in General Relativity; in the theory of gravity, you actually go to the inertial frame of the gravitating body to calculate the force on distant points?

No, I'm saying that in General Relativity, you get the exact same answer for any observable quantity no matter what coordinate system you use, provided you're careful to use sensible coordinates that don't involve division by zero over the range you're measuring.

This means, for instance, I'll get the right answer for the orbital period of Earth as seen from somebody on Earth whether I use coordinates centered on the Earth, centered on the Sun, centered on Jupiter, or centered on my head. It doesn't matter.

And then what do you do if these bodies are traveling apart relativistically? What distance would you use, the distance according to the gravitating body, or the distance according to the body which is pulled by the force?

Distance isn't a well-defined quantity in General Relativity. So it isn't that simple. For general sorts of behavior, you typically do one of two things:
1. Solve for the relationship between the metric and the particular matter distribution you're considering using Einstein's field equations. This allows you to talk sensibly, for instance, about how a physical system changes in time if you know its configuration at a specific time.

2. Take a known metric (perhaps a solution to the above), and compute how particles which are light compared to the gravitational sources of the metric behave. This involves calculating shortest space-time paths between events (called geodesics). All objects in General Relativity follow such geodesics.

It seems to me that the gravitational force lines must be somehow transformed into the reference frame of the body which is pulled by the force... i.e. the body which is observing the force.

Usually we don't think of things as "force" in General Relativity. We just calculate paths based upon the shortest space-time distance between events. An example of events would be a photon emitted by a laser, reflected off a mirror some time later, then seen by the person holding the laser after that. GR says that the photon will take the shortest space-time path available between each step of that journey.

 

[JDoolin]

No, I'm saying that in General Relativity, you get the exact same answer for any observable quantity no matter what coordinate system you use, provided you're careful to use sensible coordinates that don't involve division by zero over the range you're measuring.

This means, for instance, I'll get the right answer for the orbital period of Earth as seen from somebody on Earth whether I use coordinates centered on the Earth, centered on the Sun, centered on Jupiter, or centered on my head. It doesn't matter.

This still confuses me, because within special relativity, it is trivial to go to a reference frame where the events haven't happened yet.. Click http://www.wiu.edu/users/jdd109/stuff/relativity/timetravel.swf"*, for instance.

Distance isn't a well-defined quantity in General Relativity. So it isn't that simple. For general sorts of behavior, you typically do one of two things:
1. Solve for the relationship between the metric and the particular matter distribution you're considering using Einstein's field equations. This allows you to talk sensibly, for instance, about how a physical system changes in time if you know its configuration at a specific time.

2. Take a known metric (perhaps a solution to the above), and compute how particles which are light compared to the gravitational sources of the metric behave. This involves calculating shortest space-time paths between events (called geodesics). All objects in General Relativity follow such geodesics.


Usually we don't think of things as "force" in General Relativity. We just calculate paths based upon the shortest space-time distance between events. An example of events would be a photon emitted by a laser, reflected off a mirror some time later, then seen by the person holding the laser after that. GR says that the photon will take the shortest space-time path available between each step of that journey.

Hmmmm. So I should probably be looking toward the Least Action Principle. And I expect, that is fine for considering low-velocity, local behavior. But we are talking about the entire Cosmos here.

If you consider far, far distant masses, you can't consider simultaneity based on them. A changing plane of simultaneity defined by the motions of a distant mass, will see-saw on a fulcrum located at that distant mass. In the view of the particle affected, the force would be sometimes hitting it in the future, sometimes hitting it in the past.

At long distances, even minor motions back and forth result in major changes in synchronicity. I really don't think it will work--there's no way to compensate for these wild fluctuations in synchronicity. Have a look, for instance, at Tom Fontenot's posts about Current Age of Distant Objects; CADO. I think he has published a paper on it as well.

If the idea of a truly source-dependent gravitational simultaneity is to be compatible with SR at all, then at the very least, it would result in time-traveling gravitational fields.

*I was actually working on this demo in September 2001. I remember, when I made this demo, and calculating that an instantaneous FTL signal would reach its destination on September 12, 3001, thinking how coincidental that date seemed to be... just off by one millenium and one day too late to send a warning.

 

[Chalnoth]

This still confuses me, because within special relativity, it is trivial to go to a reference frame where the events haven't happened yet.. Click http://www.wiu.edu/users/jdd109/stuff/relativity/timetravel.swf"*, for instance.

That's not a problem within General Relativity because you describe the entire relevant region of space-time at once. You can assign numbers to the different points in this space-time however you want, and the differential geometry used in General Relativity allows you to carry all of the effects of that choice through and get the right answer no matter what choice you make (again, provided you're careful not to divide by zero).

Hmmmm. So I should probably be looking toward the Least Action Principle. And I expect, that is fine for considering low-velocity, local behavior. But we are talking about the entire Cosmos here.

It's similar. The Einstein equations themselves can be derived from a least action principle. The mathematics of calculating a geodesic are similar, but it is a space-time distance being minimized, not an action.

Anyway, no, geodesics are valid in any situation where you have a particle whose effect on the gravitational field is minimal. High speeds and long distances do not decrease the accuracy of such estimates.

If you consider far, far distant masses, you can't consider simultaneity based on them. A changing plane of simultaneity defined by the motions of a distant mass, will see-saw on a fulcrum located at that distant mass. In the view of the particle affected, the force would be sometimes hitting it in the future, sometimes hitting it in the past.

Well, as I said before, in General Relativity this sort of consideration is irrelevant. Remember, in GR we're dealing with an entire region of space-time. We don't need to worry at all about simultaneity, as those considerations are taken care of automatically. When we solve for the space-time curvature, for instance, we do it not just for some bit of space at a particular time, but do it for an entire 4D volume of space-time. The geodesics we solve are then perfectly valid within that volume of space-time provided the matter that follows the geodesics doesn't significantly perturb the space-time itself.

 

[JDoolin]

That's not a problem within General Relativity because you describe the entire relevant region of space-time at once. You can assign numbers to the different points in this space-time however you want, and the differential geometry used in General Relativity allows you to carry all of the effects of that choice through and get the right answer no matter what choice you make (again, provided you're careful not to divide by zero).

It's similar. The Einstein equations themselves can be derived from a least action principle. The mathematics of calculating a geodesic are similar, but it is a space-time distance being minimized, not an action.

Anyway, no, geodesics are valid in any situation where you have a particle whose effect on the gravitational field is minimal. High speeds and long distances do not decrease the accuracy of such estimates.

Well, as I said before, in General Relativity this sort of consideration is irrelevant. Remember, in GR we're dealing with an entire region of space-time. We don't need to worry at all about simultaneity, as those considerations are taken care of automatically. When we solve for the space-time curvature, for instance, we do it not just for some bit of space at a particular time, but do it for an entire 4D volume of space-time. The geodesics we solve are then perfectly valid within that volume of space-time provided the matter that follows the geodesics doesn't significantly perturb the space-time itself.

While General Relativity may well do what you say ("allows you to carry all of the effects of that choice through and get the right answer no matter what choice you make"), Friedmann GR is a mutant stepchild of General Relativity, which invokes a shared concept of simultaneity, and thus a universal absolute time.


Imagine if, instead of being in a universe where it was incredibly difficult to accelerate, we lived in a universe where it were incredibly difficult to turn left and right.

Consider a long meandering line of people all holding hands, and facing the same direction. At large scales, this line can curve around like an S or even into a circle. All these people have a shared concept of left and right.

Let's imagine that the people are so unaccustomed to looking far away, that they have never given any thought to the idea that someone might be facing any different direction than forward. The Friedmann of this universe develops the Hypothetical Theory of Relativity.

He invents a metric that maps all of the people into a straight line, so that their concept of absolute left and right can be preserved. Now, whenever someone talks about the idea that distant people might have different concepts of left and right, they will be met with a chorus of "Hypothetical Friedmann's Relativity makes this sort of consideration is irrelevant." or that "HR makes the concept of left and right ambiguous and obsolete."

In fact the concepts of left and right are clear and unambiguous, but they represent different directions for different observers.
​

In the Friedmann General Relativity, the same thing is going on, except instead of relying on a shared concept of left and right, Friedmann GR relies on some shared concept of simultaneity, and invokes an absolute time.

 

[Chalnoth]

While General Relativity may well do what you say ("allows you to carry all of the effects of that choice through and get the right answer no matter what choice you make"), Friedmann GR is a mutant stepchild of General Relativity, which invokes a shared concept of simultaneity, and thus a universal absolute time.

Except the Friedmann solution to the Einstein field equations is just that, a particular solution. It does not break General Relativity in any way, shape, or form, and thus all of the effects of choosing a particular coordinate system are properly carried through in all of the calculations, provided, in this case, you don't try to calculate anything too close to t=0.

You can, if you wish, select a different set of coordinates, and you will get all of the exact same physics. You can choose, for instance, coordinates which have a completely different notion of simultaneity from the Friedmann coordinates, and all of the answers you'll get for observable quantities will be the same.

The math won't be as easy, of course, because the Friedmann coordinates follow the symmetries of the system, but it is still doable.

 

[JDoolin]

I think this conversation has dissolved into a game of "Yes it does" "No it doesn't."

I started another thread regarding the https://www.physicsforums.com/showthread.php?t=431843".