I How can the stress tensor be non-zero where there is no matter?

The stress energy tensor is proportional to the Einstein tensor. So the Einstein tensor must be zero in vacuum, but the Riemann curvature tensor need not be zero in vacuum.
This is incorrect. The vacuum Einstein equations are given by ##R_{ab}=0##. The Reimann tensor is zero in vacuum. How could it be otherwise?
 

Ibix

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The vacuum Einstein equations are given by ##R_{ab}=0##.
This is true, but ##R_{ab}## here is the Ricci tensor, not the Riemann. The Einstein tensor, ##G_{ab}=R_{ab}-g_{ab}R/2##, is closely related to the Ricci tensor and is, as @Dale says, directly proportional to the stress-energy tensor. The field equations are, in fact ##G_{ab}\propto T_{ab}##. Both the Einstein and Ricci tensors are zero in vacuum.

But the Ricci tensor is not the same as the Riemann tensor.
The Reimann tensor is zero in vacuum. How could it be otherwise?
This is obviously incorrect. Simply derive the Riemann tensor for the Schwarzschild metric, for example. In Schwarzschild coordinates, twelve components are non-zero.
 
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Agreed. Yes I meant the Ricci tensor. My bad and apologies.
 
…and amazingly enough the two are one and the same! Yes you're right, the o
Yes, I should have mentioned that! Since the photon and graviton both being massless, they both have the same speed, c.
 
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Zero does not exist in the universe. It is only mathematically used as a limit.
Any equation can be rearranged to give a quantity that is zero, not in any limit, but over the entire domain of validity of the equation.
 
Are we trying to find out how the universe physically behaves? We must first understand our assumptions in solving our math. Sure we can always see that one term is approximately zero and can fall out of an equation we must solve. But an understanding of what happens (or must exist) just outside a singularity source is all important to understanding any modeled universal behavior. Is coherency lost, or did it ever exist?
 
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But an understanding of what happens (or must exist) just outside a singularity source is all important to understanding any modeled universal behavior
Not really. In fact, that is essentially the point of effective field theories. But you miss my point.

You incorrectly stated that zero is only mathematically used as a limit. This is false. Take any equation that is valid over some finite domain, such as F=ma. Rewrite it as F-ma=0. That quantity is 0, not as a limit, but over the entire domain of validity of the original formula.
 
Not really. In fact, that is essentially the point of effective field theories. But you miss my point.

You incorrectly stated that zero is only mathematically used as a limit. This is false. Take any equation that is valid over some finite domain, such as F=ma. Rewrite it as F-ma=0. That quantity is 0, not as a limit, but over the entire domain of validity of the original formula.
And, perhaps, you miss my philosophical point: that our mathematical model of the universe is not our universe. Though math is the best language to get at an explicit truth of our universe, we need to understand that it is NOT our universe, but only a language invented to predict universal behavior.

Just as the universe we measure is not the implicit universe, the potential universe, but the explicit universe (the one we describe with our math), so the word LOVE is not the feeling, but a description of it.
 
And, perhaps, you miss my philosophical point: that our mathematical model of the universe is not our universe. Though math is the best language to get at an explicit truth of our universe, we need to understand that it is NOT our universe, but only a language invented to predict universal behavior.

Just as the universe we measure is not the implicit universe, the potential universe, but the explicit universe (the one we describe with our math), so the word LOVE is not the feeling, but a description of it.
Which just shows what a load of bunk philosophy is.
 

Ibix

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And, perhaps, you miss my philosophical point: that our mathematical model of the universe is not our universe.
Nobody disputes that our models are models. This has nothing to do with whether or not "zero exists in the universe", which doesn't strike me as a particularly clear statement of anything let alone an answerable question.
 
Summary: Curvature comes from the stress tensor so how can there be curvature when there is no mass?

You're on Earth. You throw a ball and watch its trajectory. It's curved. That's because the Earth is curving space-time at every point along the trajectory. But the Earth itself is not present along the trajectory - there is no matter along the trajectory (let's ignore the air and any radiation that might be present) - so how is it curving the space there? There's not supposed to be action at a distance. Does it have something to do with gravitational waves? If so (and perhaps even if not because I'm still curious), what part of the field equations point to the existence of gravitational waves?
My journal-published experiments (Saffman-Taylor Instabilities In The Radial Domain http://link.springer.com/article/10.1007%2FBF00191691#page-1) suggest that sine-wave troughs are inertial fields analogous to gravity wells. As the universe expands, it may be that something (some near-zero-mass particles) flow around these wells (unable to push them outward) taking the path of least resistance to expansion.
 

Ibix

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My journal-published experiments (Saffman-Taylor Instabilities In The Radial Domain http://link.springer.com/article/10.1007%2FBF00191691#page-1) suggest that sine-wave troughs are inertial fields analogous to gravity wells. As the universe expands, it may be that something (some near-zero-mass particles) flow around these wells (unable to push them outward) taking the path of least resistance to expansion.
I can only see the first two pages of this paper since it's paywalled, but no such claim is made in the abstract and neither the references nor the papers citing this seem to suggest any work in that direction either.
 
PeterDonis said:
This is a Newtonian analysis, not a GR analysis.
Yes that's true. SamRoss was confused as to how curvature could exist at a point where there is no mass there, so I used the simplest example I could think of to show that a field extends beyond the immediate location of the field's source.
 
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And, perhaps, you miss my philosophical point: that our mathematical model of the universe is not our universe.
That is a completely non controversial point. I believe that only Max Tegemark might disagree, but as far as I know very few professional scientists take his idea seriously.

Nonetheless, this does not change the fact that your argument above was demonstrably wrong.

In any case, this forum is not for discussing philosophy, it is for discussing science as practiced by the professional scientific community.
 
I can only see the first two pages of this paper since it's paywalled, but no such claim is made in the abstract and neither the references nor the papers citing this seem to suggest any work in that direction either.
That is a completely non controversial point. I believe that only Max Tegemark might disagree, but as far as I know very few professional scientists take his idea seriously.

Nonetheless, this does not change the fact that your argument above was demonstrably wrong.

In any case, this forum is not for discussing philosophy, it is for discussing science as practiced by the professional scientific community.


The unsteady solution to the gravitational/density equation is found here (in a reference to my article): Chandrasekhar S (1961) Hydrodynamic and hydromagnetic stability. London: Oxford University Press (if anyone is interested in running the experiment. The paper I cited is mine and S.G. Advani's (work done at University of Delaware). It gives both the solutions to the analogous Instability on an expanding radial boundary from a source and the experimental setup through which the experimental space of the radial expansion was observed. The mathematical basis of the complex solution is The General Energy Equation across the expanding boundary.

[BOOK] Hydrodynamic and hydromagnetic stability
S Chandrasekhar - 2013 - books.google.com
Dr. Chandrasekhar's book received high praise when it first appeared in 1961 as part of
Oxford University Press' International Series of Monographs on Physics. Since then it has
been reprinted numerous times in its expensive hardcover format. This first lower-priced,
sturdy paperback edition will be welcomed by graduate physics students and scientists
familiar with Dr. Chandrasekhar's work, particularly in light of the resurgence of interest in
the Rayleigh-Bénard problem.

Gravitational Instability in an Expanding Universe. The expanding medium means that for any small density perturbation, there will be competition between its self-gravity which is attempting to increase the density, and the general expansion of the universe which decreases the density.
Gravitational Instability in an Expanding Universe

https://ned.ipac.caltech.edu/level5/Bothun2/Bothun5_1_4.html
 
That is a completely non controversial point. I believe that only Max Tegemark might disagree, but as far as I know very few professional scientists take his idea seriously.

Nonetheless, this does not change the fact that your argument above was demonstrably wrong.

In any case, this forum is not for discussing philosophy, it is for discussing science as practiced by the professional scientific community.
I am a member of the professional scientific community in that my research is published in juried journals. In case we forget before it was known as SCIENCE, it was called NATURAL PHILOSOPHY, which is the nature of the philosophy I refer to.
 
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my research is published in juried journals
And in any of your peer-reviewed reviewed publications or in any peer reviewed publications by other authors did you ever see the generic claim that zero “is only mathematically used as a limit”? If not then spare me the lecture on philosophy and the professional posturing.

When you make a mistake the best thing to do is to simply say “oops”, and learn from it.
 

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