Tim Thompson
Muse
- Joined
- Dec 2, 2008
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Energy and the Accelerated Expansion II
I would like to revisit an interesting exchange from late January (back on page 22) ...
Now, I am not an expert in general relativity (GR); I took one survey class as a graduate student and everything else is self taught. My remark about energy and the expansion comes from my understanding of GR, limited as it is, and the fact that a visiting cosmologist (I can't remember who) made the same comment during a lecture at JPL some time before I retired. So at least I can suggest that the matter of energy or geometry in the accelerated expansion of the universe might not be too clear. I will come back to this shortly.
Meanwhile, let us once again look at Einstein's equations ...
[latex] R_\mu_\nu - \dfrac{1}{2}g_\mu_\nu R + g_\mu_\nu\Lambda= \dfrac{8\pi G}{c^4}T_\mu_\nu [/latex]
We see here that the cosmological constant appears as a multiplier of the metric tensor. If we look in the epic tome Gravitation (Misner, Thorne & Wheeler, 1970; page 410) we find their discussion of the introduction of the cosmological constant, and that its dimensions are cm-2. I don't want to assert that I am necessarily right in the first place, given my limited experience, but I do want to assert that if I am mistaken, it's an understandable mistake for someone in my circumstances to make.
So much for my limited experience. Let's look a little deeper. See Sean Carroll's book Spacetime and Geometry, an Introduction to General Relativity (Pearson Addison-Wesley, 2004), specifically section 4.5 "The Cosmological Constant" and page 172. Here we find ...
This all amounts to me "thinking out loud" as I try to understand a field of physics outside my own field. But there is also a point and I said I would come back to the apparently unclear nature of energy & geometry in the accelerated expansion of the universe ...
It is one of the great lessons of modern science that some things which it seems to the layman should be readily comprehensible are in fact really confusing. The trick is to know enough to realize you are confused.
I would like to revisit an interesting exchange from late January (back on page 22) ...
But doesn't acceleration inevitably involve energy?
Actually, from the point of view of classical physics (quantum mechanics not allowed), not necessarily (as I have already pointed out elsewhere). One need only look at Einstein's equations to see that the "cosmological constant" is pure geometry. So, if the cosmological constant is the simplest explanation for the accelerating expansion, then it does not involve "energy" at all in any classical sense, but only the pure geometry of space. Hence, "dark energy" may have been an unfortunate choice of words for the name, since energy need not be involved.
Of course, once we try to establish a quantum field theory for the cosmological constant, like vacuum energy or some such thing, then we have energy back in then game as one might expect. It's just not there as an entity of classical physics.
That's not correct. The cosmological constant appears in the Einstein-Hilbert Lagrange density as, well, a constant. If you carry out the integral it you get the 4-volume of spacetime times that constant. In other words including the CC is equivalent to including a constant energy density (note that what I've said has nothing to do with quantum mechanics, it's purely classical field theory).
In non-gravitational physics such an energy density wouldn't have any effect (since it's only energy differences that matter, and so a universal additive constant doesn't do anything), but gravity turns out to act on all forms of energy, even constants, and so the CC affects the geometry and can drive acceleration. Without such a term there is no way to achieve an accelerated expansion, because all ordinary forms of matter and energy act to decelerate it.
Now, I am not an expert in general relativity (GR); I took one survey class as a graduate student and everything else is self taught. My remark about energy and the expansion comes from my understanding of GR, limited as it is, and the fact that a visiting cosmologist (I can't remember who) made the same comment during a lecture at JPL some time before I retired. So at least I can suggest that the matter of energy or geometry in the accelerated expansion of the universe might not be too clear. I will come back to this shortly.
Meanwhile, let us once again look at Einstein's equations ...
[latex] R_\mu_\nu - \dfrac{1}{2}g_\mu_\nu R + g_\mu_\nu\Lambda= \dfrac{8\pi G}{c^4}T_\mu_\nu [/latex]
We see here that the cosmological constant appears as a multiplier of the metric tensor. If we look in the epic tome Gravitation (Misner, Thorne & Wheeler, 1970; page 410) we find their discussion of the introduction of the cosmological constant, and that its dimensions are cm-2. I don't want to assert that I am necessarily right in the first place, given my limited experience, but I do want to assert that if I am mistaken, it's an understandable mistake for someone in my circumstances to make.
So much for my limited experience. Let's look a little deeper. See Sean Carroll's book Spacetime and Geometry, an Introduction to General Relativity (Pearson Addison-Wesley, 2004), specifically section 4.5 "The Cosmological Constant" and page 172. Here we find ...
"... we see that the cosmological constant is precisely equivalent to introducing a vacuum energy density [latex]\rho_v_a_c = \dfrac{\Lambda}{8\pi G}[/latex]. The terms "cosmological constant" and "vacuum energy" are essentially interchangeable
Of course the vacuum energy density is a classical concept, so it does appear that no appeal to quantum mechanics is necessary. So it appears to me that one can see the cosmological constant both ways, either as geometry or energy, in Einstein's equations, without even looking as far as a Lagrange density.This all amounts to me "thinking out loud" as I try to understand a field of physics outside my own field. But there is also a point and I said I would come back to the apparently unclear nature of energy & geometry in the accelerated expansion of the universe ...
I will assert that the answer is that a layman is not supposed to understand it. That's why we have universities with classes & textbooks & degrees. As long as any individual remains a "layman" then that individual is necessarily limited in ability to understand. Some things are easier to understand than others, and somethings can be more deeply understood than others, even by the heroic efforts of the layman. But some things cannot be understood by anyone, layman or even expert, without deep effort. My own expertise is a general masters degree in physics and a 28 year career in astrophysics, all of which included one survey course in GR as a grad student, and no professional experience at all. If I have a hard time with it, one might excuse the layman for also having a hard time with it.So how's a layman supposed to comprehend this stuff, when "experts" like Tim and Sol can't even agree? FYI, Sol is correct.
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It is one of the great lessons of modern science that some things which it seems to the layman should be readily comprehensible are in fact really confusing. The trick is to know enough to realize you are confused.