# Do Einstein's Field Equations Show Up Anywhere Else?

• cuallito
In summary, Yang Mills theory is a theory that tries to explain the behavior of matter as a result of the forces that act between particles. The equation of motion looks kind of similar, and it's formulated in similar language (what with the covariant derivatives, etc).

#### cuallito

I was wondering if equations similar to Einstein's field equations show up anywhere else?

I have a degree in physics but a guilty pleasure of mine is watching those popular physics documentaries, and I invariably grit my teeth when they use the rubber sheet analogy to explain gravity, because it uses gravity to explain gravity!

So, do analogous equations pop up in another area?

I'm not sure how "real world" this is, but the issue is discussed in
http://arxiv.org/abs/0909.0518:
Bold Assertion:
(a) Some ordinary quantum field theories (QFTs) are secretly quantum theories of gravity.
(b) Sometimes the gravity theory is classical, and therefore we can use it to compute interesting observables of the QFT.

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cuallito said:
I have a degree in physics but a guilty pleasure of mine is watching those popular physics documentaries, and I invariably grit my teeth when they use the rubber sheet analogy to explain gravity, because it uses gravity to explain gravity

I don't mean to go off topic, but I just want to say that this is one of my favorite analogies! There are very few, if any, topics in either the pure or applied sciences (physicians take note!) that cannot be understood by the average layperson. It is simply up to the expert to find a simplified way of explaining the topic. Analogies to simplified models are a fantastic way to show a layperson (or even a student learning the material for the first time) how the natural world operates. If you must, feel free to throw in the caveat that you're using only a crude model to make an analogy because doing so is more beneficial than saying, "it's too complicated for you to understand."

In closing, I'll leave you with this quote from the preface of the Feynman Lectures:
Feynman was once asked by a Caltech faculty member to explain why spin-1/2 particles obey Fermi-Dirac statistics. Rising to the challenge, he said, "I'll prepare a freshman lecture on it." But a few days later he told the faculty member, "You know, I couldn't do it. I couldn't reduce it to the freshman level. That means we really don't understand it."

cmos said:
I don't mean to go off topic, but I just want to say that this is one of my favorite analogies! There are very few, if any, topics in either the pure or applied sciences (physicians take note!) that cannot be understood by the average layperson. It is simply up to the expert to find a simplified way of explaining the topic. Analogies to simplified models are a fantastic way to show a layperson (or even a student learning the material for the first time) how the natural world operates. If you must, feel free to throw in the caveat that you're using only a crude model to make an analogy because doing so is more beneficial than saying, "it's too complicated for you to understand."

In closing, I'll leave you with this quote from the preface of the Feynman Lectures:
Feynman was once asked by a Caltech faculty member to explain why spin-1/2 particles obey Fermi-Dirac statistics. Rising to the challenge, he said, "I'll prepare a freshman lecture on it." But a few days later he told the faculty member, "You know, I couldn't do it. I couldn't reduce it to the freshman level. That means we really don't understand it."

I agree with you on everything you said. I don't have a problem with trying to explain things to laypeople. It's just that this particular example uses the thing to explain the thing! So it's more of an analogy than explanation. Actually that was the reason I was asking, I was hoping that there'd be another place more in "common experience" that equations similar to Einstein's field equations pop up, so you could say, "Gravity works a lot like..."

I guess it depends on what exactly you mean by "analogous" equations. Do you mean equations that "look" similar? Or equations expressed in the same geometrical fashion? Or equations expressed as tensors, or what.

Yang Mills theory http://en.wikipedia.org/wiki/Yang–Mills_theory

The equation of motion looks kind of similar, and it's formulated in similar language (what with the covariant derivatives, etc). Here the "curavture" terms are like the F's (actually field strengths), and the stress-energy terms are like the current J.

Of course, if your goal is to be able to make a statement like "Gravity works a lot like..." aimed towards laypersons, then this analogy would be completely useless. (lol).

Equations that are like Field Strength = constant*Source strength are all over the place in physics. Maxwell's equations, for example, have a nice analogy with GR, replacing the gravitational "field" (the metrics) with E and B fields, and the stress-energy-tensor with the currents.

In fact, if you write down Maxwell's equations in covariant form, they look kind of similar to the EFE's. Maxwell's equations, are, however, linear.

Good question. I suppose I'm looking for something else of the form curvature = stress/energy. Seems like there should be something in the fields of fluid dynamics or materials.

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In the space of all gauge fields, over which you perform the path integral, the space of Instanton solutions (the moduli space) also satisfies the Einstein equations.

cuallito said:
. . .and I invariably grit my teeth when they use the rubber sheet analogy to explain gravity, because it uses gravity to explain gravity!

I've always gritted my teeth at that one, though I wasn't completely sure why until reading this. It just *seemed* wrong to me, if this makes sense. Of course, having a gut feeling doesn't necessarily mean anything in physics, but you never know when it will. :P

I don't think anything ought to rub us the wrong way about that analogy.

In the analogy, the objects on the rubber sheet are attracted to one another because of deformations of the sheet due to the objects being attracted to earth, not because of their real-life gravitational attraction to one another. You can think of the objects' attraction to Earth as standing in for the property of objects having mass (yes the objects do have mass, but keep analogy-land and real life straight in your head! in the rubber sheet analogy, the deformation of sheet causes an attraction >> than actual gravitational attraction). In both cases, the object has a quality that deforms space-time/rubber sheet.

So while gravity does play a role in the analogy, it is NOT directly the reason for the why the objects are "attracted" to one another on the rubber sheet. It is the deformation CAUSED BY, but which IS NOT, real life gravity.

There is no circular reasoning anywhere in the argument, and the argument is sound.

ThereIam said:
There is no circular reasoning anywhere in the argument, and the argument is sound.
I think it would work better with the sheet presented "upside-down" (ie. as a hill), but the apple on the front cover of MTW is probably the best analogy I've encountered.

I don't think anything ought to rub us the wrong way about that analogy.
There is everything wrong about that analogy. It teaches something wrong that requires lengthy explanation and patience to unteach. And if you've been on PF any length of time, you've seen the completely plausible but totally left-field questions that it raises in beginners' minds.

What's the fundamentally wrong idea? Namely, that the curvature of space is what makes planets go about the sun. Is that what you believe too? Well join the crowd then!

The curvature of space has a higher order effect, of course, but is not the cause of Newtonian attraction, as this analogy certainly makes it seem. What causes a planetary orbit is the metric component, g00 = 1 - 2M/r. Curvature in time, if you will. In the linearized approximation this is the only term that survives.

A "curved" orbit is produced solely by the planet's attempt to extremize its proper time in this nonuniform field of time dilation. Space curvature does not enter.

## 1. What are Einstein's Field Equations?

Einstein's Field Equations are a set of equations proposed by Albert Einstein in his theory of General Relativity. They describe how matter and energy interact with the fabric of spacetime to produce gravitational effects.

## 2. Where do Einstein's Field Equations show up?

Einstein's Field Equations are used in a variety of fields, including cosmology, astrophysics, and engineering. They are also used to study black holes, gravitational waves, and the large-scale structure of the universe.

## 3. Can Einstein's Field Equations be applied to everyday situations?

While Einstein's Field Equations are primarily used for studying the behavior of massive objects in extreme conditions, they can also be applied to everyday situations. For example, GPS systems rely on the equations to account for the effects of gravity on time and space.

## 4. Are there any limitations to Einstein's Field Equations?

While Einstein's Field Equations have been incredibly successful in explaining gravitational phenomena, they are not able to fully explain the behavior of the universe on a quantum level. This has led to ongoing efforts to develop a theory of quantum gravity.

## 5. How have Einstein's Field Equations impacted our understanding of the universe?

Einstein's Field Equations have greatly expanded our understanding of the universe, from the behavior of stars and galaxies to the nature of space and time. They have also been the basis for the development of technologies such as GPS and gravitational wave detectors, and continue to be an important area of research in modern physics.