EDIT later that night: Fixed some stuff about the strong energy condition
So Nic, who has been writing his
Physics for Mathematicians series, recently put up his article on
general relativity. One thing he didn't go into a whole lot of detail is, just what is the stress-energy tensor?
Like he introduces it as, let's consider charge, if you want to do charge density as a scalar it doesn't transform properly, but you turn it into a 4-vector by combining it with current density, and, aha, now you'v got a perfectly good 4-vector. So for energy density, well, we know that in relativity you never consider energy by itself (it doesn't transform properly!), you consider it as a component of 4-momentum; but then if we make that into a density, 4-momentum density, that won't transform properly by itself, but you turn it into the energy-momentum tensor and now it does. Yay!
OK, but, like, what is this tensor? What are its coordinates, when we write in coordinates? We've got the energy density (time-time); we've got the energy flux or momentum density (time-space); but what are the space-space coordinates?
Well, they're exactly what they sound like -- things like, the density of x-momentum flowing in the z-direction. Now, these are often called "pressure" (for the diagonal coordinates) and "shear stress" (for the off-diagonal ones), but I don't like that description. Why? Because pressure is about
force, that is to say,
acceleration. But the energy-momentum tensor is supposed to be about, well,
momentum,
motion,
velocity.
Sure, the x-pressure coordinate is equal to the pressure density that a plane oriented perpendicular to the x-axis would experience if we placed it there and held it in place and that plane stopped everything hitting it; but I mean we don't have to think of it that way, that's not to my mind fundamentally what it is, and like again this is assuming that it stops everything hitting it (as opposed to say, deflecting it, or letting it pass through, or whatever...). (Although I'm still going to say "pressure" as a convenient shorthand, even though I don't like thinking of it that way.)
But, there's something kind of weird here. A standard example in GR is the "perfect fluid", with energy density ρ and pressure P. How is this possible for nonzero P?? Like -- there's no momentum density, so nothing is moving. So how can there be flux of momentum?? (Note I deliberately don't say "pressure" here -- focus on what's moving, which is what the tensor is acutally about, not your physical intuition of pressure!)
Well, I realized -- imagine that at our point, energy is moving at a certain speed in a certain direction; but, simultaneously, it's also moving at the same speed in the opposite direction; that is to say, energy is *dispersing* in two opposite directions. Then the net momentum density at that point will be zero -- but there is still motion, and the space-space coordinates in T
ab capture that! (This doesn't cancel out due to the quadratic dependence of the diagonal terms on velocity; if momentum flows in the +x direction, then it's positive x-momentum flowing in the x-direction, while if flows in the -x direction, it's negative x-momentum flowing against the x-direction, so either way it contributes positively.) They capture some component of the motion which is hidden but still relevant (and which will show up directly in the net momentum after a change of coordinates, so it's very real).
So in the perfect fluid, then, each point is constantly dispersing energy in all directions; so at each point there's no net momentum, but there is nonetheless motion, and the diagonal terms capture that. I much prefer this way of thinking about it -- this hidden component of motion -- to saying that it's pressure! For the reasons stated above.
I mean I'm wary of reifying too much, right? Turning too many derived notions into primitive notions. Treating them as things that must make sense and can vary freely, instead of thinking about what they actually mean and how this constrains them.
And this brings me to the main thing I wanted to talk about --
energy conditions. Like, T
ab shouldn't be able to be just anything, right? It should be something that physically makes sense. (Not in terms of
actual physics, mind you, where things are made of atoms; but, like, in terms of the sort of continuum mechanics that GR is based on.) At first I didn't think even a perfect fluid made physical sense in this sense, but now I accept it does. So what stress-energy tensors are possible?
Well, energy conditions are conditions on T
ab that attempt to answer this, or at least give conditions that are necessary if not sufficient. Though he doesn't use the term, in his article Nic discusses the
weak energy condition, which is that one should have v
av
bT
ab≥0 for timelike (and null) v
a; i.e., a timelike observer should always observe nonnegative energy density. There's also the
dominant energy condition, which is stronger (and which Nic uses as motivation for the weak energy condition but then kind of discards?), which is that for forward-pointing timelike or null v
a, -v
aT
ab is also forward-pointing timelike or null; that is to say, observed 4-momentum should never be superluminal. So, both those seem pretty reasonable!
(There's also the weaker
null energy condition, where you only require v
av
bT
ab≥0 for null v
a; I'm not clear on what the use of this condition is.)
What do these conditions imply in the perfect fluid case? The null energy condition implies ρ+P≥0, quite a weak statement. The weak energy condition implies ρ≥0 and ρ+P≥0. So now ρ≥0 at least -- energy is nonnegative -- but pressure is still allowed to be negative, as long as it's not too negative. And the dominant energy condition implies ρ≥|P|. So again, negative pressure still allowed, but now also we've got an upper bound on pressure in addition to the lower bound.
What about the strong energy condition? (Which apparently is not actually stronger than the weak energy condition or the dominant energy condition, but which could be applied alongside them.) I don't really understand the strong energy condition, and some people apparently think it's too strong, but it's a thing people talk about. In the perfect fluid case it apparently implies ρ+P≥0 and also ρ+3P≥0, although, maybe I'm doing someting wrong but I haven't been able to derive the latter. But negative pressure is still allowed!
But I want to suggest a way stronger energy condition, based in what I was saying above about how T
ab should represent actual motion. I want to suggest that T
ab should lie in the closure of the
convex cone on tensors of the form v
av
b. And like -- I'm not suggesting this as merely a necessary condition like the other energy conditions, but also a sufficient one.
This condition implies both the dominant energy condition and the strong energy condition. Like, to use coordinates, it implies that all the diagonal coordinates are nonnegative, so in the perfect fluid case, we finally get P≥0. But it also implies T≤0 (where T denotes T
aa), which means that, in the perfect fluid case we also get ρ≥3P, a stronger upper bound on P than we got from the dominant energy condition.
Is all this reasonable? I dunno! But I feel like it ought to be, because dammit, T
ab should represent motion, rather than being some reified free-floating thing!
Worth noting that what WP says about why the strong energy condition is unreasonable is because of the cosmological constant -- which I assume means it's counting the cosmological constant as part of the stress-energy tensor. But you don't have to do that! You don't have to interpret it that way, you can keep it separate. Or if you do want to interpret it as part of the stress-energy tensor, I'm fine saying that fine then, that part doesn't represent actual motion, my condition only applies to the rest. :P
So, uh, yeah... how reasonable is this condition? I sure think it's reasonable, but yeah. Also, can it be expressed in terms of coordinate-free inequalities? I hope so! My formulation is coordinate-free, obviously, but it's not in terms of inequalities; some of the inequalities I derived from it I only know how to express in coordinates.
Questions I should ask more people about, I suppose!
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joshuazelinsky, I submitted the Snake problem as a second-year research problem to PROMYS. And the students (unfortunately I don't know their names, aside from one Rakesh Rivera) found that I made some incorrect claims! Here's a big one -- I said that Θ2,2,k was winnable, but in fact it's only winnable for k=1 and k=2!
