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

In summary: Maxwell's equations?In summary, according to the field equations, the curvature of spacetime at any time and place is fully described by the Riemann tensor. The Einstein field equations are differential equations that restrict certain combinations of derivatives of the Riemann tensor to be equal to the stress-energy tensor. So in vacuum, they say that some combination of derivatives of curvature are zero, not that curvature is zero. Gravitational waves do not come into this.
  • #36
SamRoss said:
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.
 
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  • #37
DoctorSatori said:
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.
 
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  • #38
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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  • #39
DoctorSatori said:
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.
 
  • #40
Ibix said:
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.
Dale said:
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
 
  • #41
Dale said:
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.
 
  • #42
DoctorSatori said:
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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  • #43
Thread closed for moderation.
 
<h2>1. How can the stress tensor be non-zero in empty space?</h2><p>The stress tensor is a mathematical representation of the distribution of forces in a given region of space. It is not dependent on the presence of matter, but rather on the presence of energy and its associated fields. In empty space, there may still be energy and fields present, such as electromagnetic radiation or gravitational potential, which can contribute to a non-zero stress tensor.</p><h2>2. What is the physical significance of a non-zero stress tensor in empty space?</h2><p>A non-zero stress tensor in empty space indicates the presence of energy and fields, which can have important implications for the behavior of particles and objects in that region of space. It can also affect the curvature of spacetime, as described by Einstein's theory of general relativity.</p><h2>3. Is the stress tensor always symmetric, even in empty space?</h2><p>Yes, the stress tensor is always symmetric regardless of the presence of matter or energy. This is because the laws of physics are symmetric with respect to the direction of forces, and the stress tensor is a mathematical representation of those forces.</p><h2>4. Can the stress tensor be negative in empty space?</h2><p>Yes, the stress tensor can have negative components even in the absence of matter. This indicates the presence of attractive forces, such as gravity, in that region of space.</p><h2>5. How is the stress tensor related to the concept of stress in materials?</h2><p>The stress tensor is a generalization of the concept of stress in materials to the entire space. In materials, stress is a measure of the internal forces that act on a small element of the material. Similarly, the stress tensor represents the distribution of forces in a given region of space, which can affect the behavior of particles and objects within that region.</p>

1. How can the stress tensor be non-zero in empty space?

The stress tensor is a mathematical representation of the distribution of forces in a given region of space. It is not dependent on the presence of matter, but rather on the presence of energy and its associated fields. In empty space, there may still be energy and fields present, such as electromagnetic radiation or gravitational potential, which can contribute to a non-zero stress tensor.

2. What is the physical significance of a non-zero stress tensor in empty space?

A non-zero stress tensor in empty space indicates the presence of energy and fields, which can have important implications for the behavior of particles and objects in that region of space. It can also affect the curvature of spacetime, as described by Einstein's theory of general relativity.

3. Is the stress tensor always symmetric, even in empty space?

Yes, the stress tensor is always symmetric regardless of the presence of matter or energy. This is because the laws of physics are symmetric with respect to the direction of forces, and the stress tensor is a mathematical representation of those forces.

4. Can the stress tensor be negative in empty space?

Yes, the stress tensor can have negative components even in the absence of matter. This indicates the presence of attractive forces, such as gravity, in that region of space.

5. How is the stress tensor related to the concept of stress in materials?

The stress tensor is a generalization of the concept of stress in materials to the entire space. In materials, stress is a measure of the internal forces that act on a small element of the material. Similarly, the stress tensor represents the distribution of forces in a given region of space, which can affect the behavior of particles and objects within that region.

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