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We usually write [math]T_{\mu\nu}[/math] in terms of energy density [math]\rho[/math] (where c = 1 and [math]\rho [/math] has units of pressure) and pressures p. Rev Prez

It has units of inverse force. Now you know the left side of the equation is in units of distance. What does that tell you about the right side of the equation? (ignore the volumetric components of [math]T_{\mu\nu}[/math]. Rev Prez

Thank you. That matches my intuition. But my problem remains. I can't even prove that a positive mass-energy distribution implies a positive trace; I just don't know how. I've never seen the proof and I am just a little lazy. On the other hand, I do have a question as to what happens to geodesics when flat space-time is introduced to a negative mass-energy distribution. Follow me so far? Rev Prez

Yes, I am familiar with the energy momentum tensor. My question is with a negative mass-energy distribution, what is the proof that [math]T_{\mu\nu}[/math] (or [math]T^{\alpha\beta}[/math] is positive definite, negative definite, semidefinite either way, or indefinite? Bear in mind I'm not a mathematician. I intuitively grasp that a negative mass-energy distribution is either indefinite, negative semidefinite, or negative definite, and in the case of the Schwarzchild metric I can see clearly that the energy momentum tensor is positive definite. I want to understand this generally in terms of the trace.

Any News Corps stuff--best consistent open source intelligence anywhere. Pretty much all the major newspapers in a week, CNBC, occasionally MSNBC and CNN, followed by the news analysis and opinion periodicals. Rev Prez

Wouldn't that mean that any tax-empt entity is barred from political activity? Say, the political parties or MoveOn.org or American Atheists? Rev Prez

This is wrong and of no consequence. Yes, the components of [math]T_{\mu\nu}[/math] are [math]\rho[/math] and [math]p[/math]. I am asking if geodesics are always lengthened by positive definite [math]T_{\mu\nu}[/math]? This is a conservation question, nothing more, nothing less. I have no idea where you're getting squared terms in the tensor. The components are [math]\rho[/math] and [math]p[/math] for a perfect fluid. The trace is simply [math]\rho + p_x + p_y + p_z[/math]. Rev Prez

Because you haven't taken the time to pick it up? I don't know. N. Any matrix A where Tr(A) > 0. Rev Prez

They haven't here. Rev Prez

Only in a twisted world where "tax" means something else entirely. How about turning education over to the market and let parents determine what they will and won't support? Then, Lord willing, secularists can pool together their own resources and leave the rest of the country in peace. Rev Prez

No it doesn't. Hell, I bet you don't even go through the day spending, excuse me, an inordinate amount of time on the subject. On topic, there's more reason to believe in God than time travel. At least the possibility of the divine doesn't require a revision of our perception of entropic progression along the time axis. That said, I checked uncertain. Rev Prez

I'm looking for the shortest path through a region of space-time containing some distribution of matter described dynamically by [math]T_{\mu\nu}[/math]. Specifically, if [math]T_{\mu\nu}[/math] is a positive definite matrix, is a geodesic through it always longer than one through empty space-time? Rev Prez

[math]T_{\mu \nu} [/math] is the energy momentum tensor in [math]R_{\mu\nu} - \frac{1}{2}Rg_{\mu\nu} = 8 \pi G T_{\mu\nu}[/math]' date=' where [math']R_{\mu\nu}[/math] is the Ricci tensor, R is the Ricci scalar, G is the gravitational constant, and [math]g_{\mu\nu}[/math] is the metric we're solving for. Rev Prez

Uh, that's not the trace. Rev Prez

Um, what? Rev Prez

My question has to do with the energy momentum tensor. If [math]T_{\mu\nu}[/math] is positive definite, is a geodesic across the corresponding topology always longer than its projection across a Minkowski spacetime? This is obvious for M>>0 in the Schwarzchild metric--the interval increases timelike. I'm asking if it is a general condition of all positive definite [math]T_{\mu\nu}[/math]. Rev Prez

The distance between two distant points may open up at a rate greater than c as measured by some observer at a third point, but this has nothing to do with comparing reference frames. Maybe this misunderstanding about expansion is due to how easy it is to replace "two points apart from mine" with "one distant point and myself." I don't know. Rev Prez

Quick question. Does positive definite energy momentum always imply a lengthened geodesic (compared to the vacuum state)? Also, is there an example of macroscopic indefinite energy momentum state in nature? Rev Prez

The reason I asked is because cosmology, as a whole, not an intuitive subject. If you're genuinely excited by the subject, challenge it and yourself by gaining competence in the prereqs. For me at least, the whole thing is far more beautiful when your intuition is informed by even a minimally rigorous foundation and focused on specific cosmological questions and problems. Rev Prez

Because it's a conservative system. Rev Prez

Yes, two really large ones. It explains by analogy and predicts nothing. There is no mechanism, either internal to the theory or applied, others might use to evaluate for internal coherence and against meaningful experiments/observations. That is in its explanatory mode, we're left with these unoperationalized (effectively "undefined") notions of "sheet (of paper)," "top," "bottom." Second, it draws on two subjects well understood in present physics--stars and blackholes--but offers nothing in the way to cohere this vague hypothetical relationship between the two you proposed with what we know about their statics and dynamics. IMO, the beauty and fun of scientific systems lies in discovering models that describe the empirical world around us. If this stuff really excites you, take the time to prepare yourself to approach, penetrate and master the arguments and tools used to derive and explain these models. It's not too overwhelming, a decent amount of exciting stuff is accessible to someone who's at least been exposed to freshman year mechanics and calculus. Some more coursework in linear algebra would just about put you on track to teaching yourself some of the necessary maths to get into the really interesting stuff. Rev Prez

Do you know why? Rev Prez

Just some background. I've got multivariate calc and diff eq under my belt, along with linear algebra, abstract algebra and real analysis. I'm trying to pick up differential geometry on my own, and while I can now laboriously understand the algebra, I'm still very iffy with the actual derivations and calculations. Also, I'm not anywhere near the point where I can take an interesting problem like this one and simply write down and follow through mathematically. I'll probably have some more questions after I digest this, but thanks for the help. My bad. I logged in as "Rev Prez" instead of "revprez." "Rev Prez" was never validated (the activation email never got to me). Rev Prez