Firstly, I am not a mathematician. So as I fumble through a likely ignorant question, please show pity. A part of my teaching and research frequently leads me to consult theories of astrophysics, in particular, EFE's. e.g. G mu nu = 8 pi T mu nu. The terms mu nu must be so fundamentally a part of the furniture that none of the discussions address these values within the elaboration. I would greatly appreciate any insight into the use and meaning of these values in the context of field equations. Sincerely, Oyate ps don't worry about being too technical in replying.

They are indices. Something like \(G_{ab}\) is an array of expressions, not just a single one. For instance, in 1+1 dimensional space-time a,b will take 1+1 = 2 values each and each entry in G will be a different expression, ie \(G_{ab} \sim \left( \begin{array}{cc}G_{11} & G_{12} \\ G_{21} & G_{22} \end{array} \right)\) This is an extension of the vector component notation, \(v_{a} \sim \left( \begin{array}{c}v_{1} \\ v_{2} \\ v_{3} \end{array} \right)\). If you are unfamiliar with index notation then you're over stretching yourself by looking at the Einstein field equations. The reason no book on GR ever explains the index notation is that its assumed knowledge, you won't get to a course in GR without courses in SR, electromagnetism, linear algebra and vector calculus, all of whom develop and build on the notion of index notation. GR makes extensive use of not just vector and matrix indices but to much more complex ones like a Bianchi identity \(R^{a}_{b[cd;e]} = 0\), where you get definitions like \({R^\rho}_{\sigma\mu\nu} = \partial_\mu\Gamma^\rho_{\nu\sigma} - \partial_\nu\Gamma^\rho_{\mu\sigma} + \Gamma^\rho_{\mu\lambda}\Gamma^\lambda_{\nu\sigma} - \Gamma^\rho_{\nu\lambda}\Gamma^\lambda_{\mu\sigma}\). If none of these expressions make any sense to you then better to leave the astrophysics for a year or two while you get up to speed on the requirements for such things. Otherwise you'll spend 10 times longer constantly looking up every single notation which you should already know and that'll only waste your time.

That was exactly what I was looking for and precisely what I expected. So much the furniture, and thus, a given. Indices to an array is a very concise and useful definition. We’re currently working on diagnosis, prediction and treatment of epilepsy and seizure disorders in children (petit mal or absence seizures in particular). The behaviors described by the nonlinear dynamics of Lorenz (not to be confused with Lorentz) and, surprisingly, the physical interactions described by Einstein: mass effects on space-time geometry, with Maxwell’s description of the relationship of electricity to magnetism, are providing analogs and models in which to frame EEG and MEG findings. That is, electrical and magnetic activity in the brain. Even though the energy is electrochemical (from sodium and potassium ions) at levels of only -70 to +30 mV, making the resultant EM fields quite weak; chaos theory, whether it’s modeling climatic storms in the atmosphere or electromagnetic storms in the brain, and oddly enough, the effects of gravitation on the trajectory of photons and the effects of overlapping biological EM fields on electron flow in nerve conduction velocity, are providing useful models as means of coherently organizing these data. This is a forum is remarkably well done and the level of scientific stringency being pursued is to be applauded. I only just found it in pursuit of an answer to my simple question. Science always progresses in greater strides when the disciplines cooperatively interact. Don’t you find? I really don’t need to look up every single notation to derive the value GR has to offer in this endeavor. And at 62, I probably won't be getting to any more GR courses, although I'd love to. So thank you, your reply was at least as useful as your pomposity was entertaining.

Forgive me if I'm sceptical that someone doing medical work on epilepsy firstly has any need for the Einstein field equations and Maxwell equations, much less has sufficient mathematical background to understand them. And how did astrophysics come into it? First you're researching astrophysics, now you're doing medicine? Would you care to demonstrate an example of what you speak? My previous post wasn't pompous, just because you didn't want to hear everything I said. You talked about how your research and teaching regularly leads you to astrophysics, yet you're unfamiliar with index notation. If you're doing research which leads you to the Einstein field equations you should already be familiar with index notation, else how can you be lead to something you can't understand? What guides your 'research' to astrophysics? Your second post only makes me wonder even more if you're just taking the piss. How many buzzwords and unrelated things could you throw into a single post? You're researching medicine, astrophysics, chaos theory, electromagnetism all under a single heading? Sure you are.

An EEG machine that is powerful enough to map the muon frequencies at a time by time basis would be a potentially useful and noninvasive tool in the enhancement of neuroscience. As in quite clearly showing the parts of the brain that are more active during epileptic fits/ normal brain function to a very precise degree. I think you have a fine understanding Oyate.

I think you might understand something from here. Please Register or Log in to view the hidden image! W transverse mass in W -> mu nu (eps)

That's W bosons decaying to muons and neutrinos, which makes sense if you understand the electroweak sector of the Standard Model.

His question has nothing to do with muons or neuroscience. Nor does the muon has anything to do with neuroscience. And you have no understanding at all.

Damn...AlphaNumeric is harsh...I am now afraid to ask questions for fear of the backlash. Please Register or Log in to view the hidden image!

I'll be nice if people are rational and make an attempt to understand. Moose is just bat shit crazy and Oyate questionable. How does someone do research which leads you to the Einstein field equations yet not cover the basic algebra said equations are framed in? Astrophysicists have to do the same basic mathematical methods courses as any other physicists, including vector calculus. And how such research leads into medical research on epilepsy I have no idea and clearly Oyate doesn't want to say. People who really want to honestly learn and discuss things rationally I have all the time in the world for and I'll be so pleasant it'll make the Care Bears seem like antisocial dicks. Now, who wants a hug? Please Register or Log in to view the hidden image!

Good lord, I get back from holiday and find a Magna Carta here replete with cartoons. Your post wasn't pompous, you are. No I really didn’t hear everything you said, however I did read everything you wrote. Demonstrate an example? All one has to do is a keyword, or I should say buzzword search with all those unrelated “things”. A quick search yielded: Chaos Theory and Epilepsy. LEONIDAS D. IASEMIDIS and J. CHRIS SACKELLARES. The Neuroscientist Even this is interesting: A MULTIRESOLUTION FRAMEWORK TO MEG/EEG SOURCE IMAGING. [posting links not yet permitted, pdf's to large to attach, you'll have to go look] There are thousands of us out there. And all the research is being done by a mixture of very unlikely bedfellows. Lorenz is a “weatherman” what on earth would he want with analog computers, logistic difference equations and fractal mathematics? As it happens, seizure disorders, like atmospheric turbulence, are chaotic events in the electromagnetic environment of the brain, that are constrained by “attractors” that confine these fields within highly predictable and reproducible parameters... As it is with all chaotic phenomena. Cardiologists have found that a heart in fibrillation is electrically, in a high dimensional state of chaos. This is old stuff. Electromagnetic fields and their effect on the movement of electrons, Gravitational fields and their effect on the movement of photons are both field behaviors. Whether these EM fields are at microcurrent levels and we read them with electroencephalographic or magnetoencephalographic instruments or they are the result of colliding black holes generating fields as gravity waves and we have no instruments to detect them. They are all field effects and these fields exhibit consonant phenomena. These consonances have lessons to teach. Whether they occur in the brain, in the atmosphere or in space. My end of the work is at the biological, electrochemical level, interfacing with patients, employing the fruits of the technological research of my colleagues who wallow in these equations like they’re in their mothers arms. It’s a wonderful thing to see. So I feel compelled to better understand these mathematical hieroglyphics as a part of my job so we can better communicate. Communicate, now is that a new word for you? Some of my colleagues have about the same level of understanding of biologically produced EM fields as I do about quadratic equations. So they do their best to understand (at a sufficient level) the 'languages' we speak. I’ve been practicing medicine for 27 years, does that number ring a bell for anyone? They say if you haven’t rung the bell in physics by the time you’re 26 then it just ain’t gonna happen for you. Well, so much for that Nöbel prize, at least you’ll have your hubris to keep you company. kids these days...

I asked about the Einstein field equations and Maxwell's equations. I don't deny that some medical technologies use advanced physics, MRIs, PET scanners etc are all products of such physics. The people, the engineers, who design and build them understand the relevant physics but the doctors which use they don't need to. They need to know not to bring metal into an MRI room in case its magnetic but do they need to understand how to do a discrete inverse Radon transform? Nope (and believe me, they are a bitch to do for some things), the technology is 'under the hood' just like I don't need to know how to build an engine to drive a car. You first talked about astrophysics. How does your medical research lead you to that? If you don't know tensors how do you know your research leads to the Einstein field equations, which are written in terms of tensors? If you don't know calculus how do you know your work leads to Maxwell's equations? Your example wasn't something specific to your work, it was something Google threw out for you. Buzzword Bingo is an easy game to play if you avoid answering people's direct questions. No one who knows the models 'weathermen' use would ask such a thing. The worlds largest supercomputers regularly get put to task on global weather phenomena. I don't deny that seizures can be examined in terms of the electrochemical patterns in the brain but its a long way from a doctor using an EEG or whatever they are to see if a patient is okay to a doctor needing to know about chaos theory and Maxwell's equations. That's the interface between a doctor saying "I need a machine which can do X" and an engineer/physicist saying "I know how to build a machine which makes use of A, B and C which will do X for you, you just press this button.". Now you're just stretching analogies. Yes, electromagnetic and gravitational effects often are described using waves. Yes, some are hard to detect. That doesn't mean a doctor needs to know general relativity. It doesn't mean someone researching the effects of seizures needs to know how EM signals are detected by their diagnostic equipment, never mind analogous models within physics (not biology). You've mentioned the EFEs, Maxwells equations, gravitational waves etc. I have yet to see you present any reason why your research requires you to know them on a working level. Then you've just agreed with what I've been saying, the doctor is at the other end of the technology/science, where they use the outputs of machines, machines whose underlying principles they don't understand, but the engineer who built them do.