Here's an exact vacuum solution I finally found for the space around a thin infinite sheet of matter in the x-y plane whose pressure terms along x and y are equal and whose values are arbitrary being proportional to k. I've had other solutions corresponding to different pressure states, but this ones new. \(ds^{2}=\frac{kdct^{2}}{\left ( 1-\frac{\alpha \left | z \right |}{c^{2}} \right )}+2\left [ 1-k\left ( 1-\frac{\alpha \left | z \right |}{c^{2}} \right ) \right ]dctdz-\left ( 1-\frac{\alpha \left | z \right |}{c^{2}} \right )^{2}\left ( dx^{2}+dy^{2} \right )-\left [ 2-k\left ( 1-\frac{\alpha \left | z \right |}{c^{2}} \right ) \right ]\left ( 1-\frac{\alpha \left | z \right |}{c^{2}} \right )^{2}dz^{2}\)
I'm slightly suspicious of this because, if I'm not mistaken, the first derivative of the metric must be continuous due to the Israel junction conditions and your metric is not... A more general question is were exactly do you want this thread to go?
No, there is matter as a delta function there at z=0. Such a condition is only valid where the matter density is finite. As I said, it was for a thin sheet of matter. If I didn't take the limit as the matter went to a delta function at z=0 then intead of \(\alpha\left | z \right |\) terms I would have had just \( \Phi \left ( z \right )\) terms in the metric as the solution corresponding to an energy density and two pressure terms in the stress-energy tensor that would be proportional to \(\frac{d^{2}\Phi }{dz^{2}}\). If you don't take the limit that the matter goes to a delta function then you don't get the absolute value function. You just get \(\Phi \) terms in the metric corresponding to \(\frac{d^{2}\Phi }{dz^{2}}\) terms in the stress-energy tensor. This is the same kind of thing as having an r=0 singularity in collapsed matter, or lets say as having an electric field that is a step function different in value just outside a charged sphere than it is just inside. The electric potential just across the surface looks like an absolute value connection, the field just across looks like a step function and the charge density is a delta function. I didn't have to take the limit where \(\Phi \to \alpha\left | z \right |\). I just felt like it.
In what sense? New as in this is one you just found or it's new as in no one has ever found it before? Planes of matter, even charged matter, are extremely common in the literature, particularly given the string theoretic motivation of branes. The requirement of pressures in T to give static planar configurations is known in the literature. For example, a well known domain wall solution which has pressures and involves very similar metric coefficients to yours is \(ds^{2} = \frac{1}{\sqrt{1-2\alpha |z|}}(-dt^{2} + dz^{2}) + (1-2\alpha |z|)(dx^{2}+dy^{2})\) Much like your last thread about "What's the coordinate transform which turns this into the SC metric?" your formulation has coefficients which are extremely similar to the standard well known result but usually squared. I suspect you have some particular reformulation you're using in both instances. Even has the same singularity structure, allowing for some coordinate changes. Unless you're doing a convoluted physical setup I suspect, much like your Birkhoff's theorem thread, you're reaching a well known result by a non-direct manner. It's an occupational hazard, I've rederived a number of known results over the years, with it only becoming apparent when the conclusion is reached and you say to yourself "Oh, this is just an obfuscated version of that. Arse!". It's particularly common if you approach a problem indirectly. After all, it's easy to make a metric which is an exact solution to the EFEs with lots of parameters and which is not recognisable to anyone, you just pick a standard metric and do some unpleasant coordinate transform on it which has lots of free parameters. That's essentially what happened in your Birkhoff's theorem thread so even having a free parameter in your metric doesn't mean you've circumvented the theorem. Unfortunately there's no simple way to check whether two metrics are really the same up to coordinate transformations. In the case of your metric given it's such a bog standard physical configuration, you have the same singularity structure as a known result, the metrics look very similar and you haven't given any indication there's anything out of the ordinary about the system I'm pretty much certain you've just got an obfuscated version of something like Taub's result. Unfortunately it seems to be a recurring theme of yours, giving results already known.
New in the sense that the ones already known correspond to particular pressure states, whereas this one unlike the ones already known has the two pressure terms proportional to k so that the pressure can be anything. You again are just plain wrong. For example the stress energy tensor for the domain wall you wrote down given the sign conventions I use is \(T^{00} = \frac{\alpha \delta (z) c^{2}}{2 \pi G} = -4T^{xx} = -4T^{yy}\) All other \(T^{\mu \nu} = 0\) The stress-energy tensor for the domain wall I wrote is \(T^{00} = \frac{\alpha \delta (z) c^{2}}{2 \pi G}\) \(T^{xx} = T^{yy} = -\frac{k}{4}T^{00}\) All other \(T^{\mu \nu} = 0\) The k is what makes the pressure state and therefor the solutions different.
The k can be absorbed by a rescaling of spatial coordinates vs the time ones. This is why fixing certain scales at the start is useful, you don't generally end up thinking there's a free parameter in a system which isn't actually just a coordinate choice. Your result is part of the same 'equivalence class' of results, with the known result being a representative. That's part of the power of coordinate stuff, you need only consider a single element of an equivalence class. Your general attitude seems to be "I'm awesome at GR! Look, some new metric no one has ever done before!". Writing down a valid metric no one else has ever written down is easy. Your Birkhoff thread was just such an illustration because regardless of what your metric looks like it's undeniably just the SC metric in funny coordinates. I (and I think others) am somewhat unsure what you're trying to get out of posting these various metrics. If you think you honestly, really have a metric which is inequivalent to previously known ones then don't post it here, publish it. But before you bother a journal I think you need to brush up a bit on the known results because you don't want a repeat of your "I've united EM and GR!" fiasco.
Only on one side. if you transform one side the same transformation applied to the other gives you the behavior of the metric through the wall that retains the arbitrary pressure described by the corresponding stress-energy tensor. Yeah I know your sore that I understand general relativity better than you do to the point that I have found solutions you couldn't. Get over it. Its not the behavior of the metric on one side of the wall that is important here. It is how the metric behaves going through the wall itself that is what is unique about it and makes the metric "through the wall itself" MY metric. On one side yes the solution is a transformation of a known solution, but only on one or the other "side". What I figured out that makes this solution mine is how to choose the metric behavior through the wall itself so that the pressure becomes something different than the already known wall solution. Had you asked, I would have told you how I got my solution. Why do you suppose the stress-energy tensor bears such a resemblance to yours but with different pressure? What I did was take a known vacuum solution which happens to be equivalent to the one you wrote down and first did a transformation on it and then instead of patching the old solution to its mirror taking the abs(z), I instead patched the transformed solution to its mirror taking the abs(z). This sets up a unique behavior of the metric through z=0 which is what makes the metric MINE and what allows the pressure to be something different. The behavior of the metric through z=0 is unique to how I patched the solutions. Oh and I never said that I unified em and gravity. I was saying that because of a solution that I found, I have come to realize what a few others have already suggested which is that Einstein already classically unified them without realizing it. General relativity is an already classically unified theory of gravity and electromagnetism. For those interested, at the following link I describe what led me to search for this solution and gives other vacuum solutions which are independent of others known solutions. http://www.youtube.com/watch?v=qsqNWcjOVZ0
You don't seem to have understood what I said. I wasn't talking about sides of the domain wall, I was talking about sides of the equation. I would have thought that was pretty obvious. The s0lution Firstly I don't experience soreness when I come across people better at some part of maths or physics than I. If I did I would be unable to do my job because all of the people I work with have PhDs in maths or physics and the vast majority are in different areas than my PhD. As such every single person I work with is more competent than I in some area of maths or physics, just as I am more competent than them in some things. Such is the nature of life and people who don't learn to accept that go around with a constant chip on their shoulder assuming everyone else has the same insecurity that they do. The type of insecurity you're displaying. Heck, I don't mind even pointing to people on this forum who are better than me in things. Rpenner is much wider read than I in many areas of physics. Pryzk is better at electrodynamics and more practical areas of quantum mechanics. Guest could beat me blindfolded (him, not me) at functional analysis and more formal areas of mathematical physics. Prom, Trippy, the list goes on. Of course those are other people so secondly your comment assumes I think you more competent than I at general relativity. I would certainly say you're more competent than the average person and quite possibly better read in several areas of GR. Do I think you're more capable than me? I hadn't really thought about it, plus it's hard to say given we're not working on the same sorts of problems. Truth be told nothing you've posted here makes me think you're exceptional, none of the subject matter has been beyond stuff I did pre-PhD. For example, your EM+GR thread covered a subject I literally got lectured on in my 4th year. Actually I enjoy talking to people who know more than me, particularly if they are working in some area vaguely related to something I'm familiar with because then it gives me added motivation to ask them questions and try to gain some understanding on something I might otherwise not grasp. I've asked more than my share of daft or basic questions in front of groups of people before. I would rather gain some understanding and people know I didn't understand before than remain ignorant but hope everyone thinks I understand. Hence I don't have any issue with appearing to be less knowledgeable than someone else. You obviously spend your time playing with metrics and physical setups. I've done some of that myself, I even have a paper published on a particular string theoretic metric I proved a uniqueness result for which gives insight into mass gaps in gauge theory. So if you are more capable than me in any significant way you have yet to demonstrate it to me and thus I don't feel any insecurity towards you. The tone of my previous posts have been as they are because it's clear you think yourself superior, not just to me but to a significant chunk of physicists, hence your attempts to 'subtly' say "Look, a new exact solution! I'm **** hot at this!". While you might be able to impress yourself or the non-physicists around you nothing you've done so far jumps out at me as something ground breaking. Being competent converting physical setups into metrics and doing coordinate manipulation is a means to an end, not the end itself, when it comes to GR. To be perfectly honest I'm not terribly interested. Like I've said, if you think you have something major and new than send it to a journal. Anything actually new is going to get very little constructive discussion on a forum like this because of the very low number of people capable of research level science and those of us who are are particular areas we're interested in or minimal time to spend crunching through other people's work. Of course if you have something specific to say beyond "Here's a metric" then discussion can follow but just posting a metric will not get much response. Most people don't know what it means and even less are going to say anything other than "Okay....?". That's why there's little point in any of us researchers posting our work, very little constructive comment or input will be given from others. Of course we could do it just to show off but that's hardly a good reason. It's my experience that those who post what they deem to be publishable work on forums without being asked are either cranks whose posts are nowhere near publishable or egotistical. Your posts essentially advertise your own work, despite it not having first been put through peer review. Is there some reason you aren't sending your work to journals? If you're so sure you're coming up with new stuff and you can demonstrate it clearly and coherently then a journal wouldn't need much time to look over your calculations to confirm it. Posting on a forum isn't going to reach many of the right people, yet it takes just as much, if not more, effort to write your In the thread in question you tried to claim that the fact you could find very constrained special systems where knowing the EM field told you the metric and vice versa was somehow a unification. That wasn't justified. It's possible to put classical EM and gravity into a single framework, which is just using EM to contribute to the energy-momentum tensor. Alternatively you can use Kaluza-Klein models to view EM as just a difference face of some gravitational degrees of freedom, though it costs you a scalar field. Your approach of saying "It works when everything is extremal and not moving therefore it'll just require magnetic fields to account for motion" is not a justified argument and ignores how the degrees of freedom are removed by assumptions of static configurations and extremality (which isn't actually a word but never mind). The fact you kept calling it 'my solution' when it was well known even to students (hence why I knew about it) shows that you've got gaps in your knowledge about what is or isn't a new result. And can you clarify something. In your Birkhoff thread were you trying to take the position the metric contradicted Birkhoff's theorem or were you just wanting an excuse to post another metric?
If you don't think you're going to get anything from this forum then you're welcome to leave. Like I said, I don't have any insecurity about you or anyone else. I don't need to start posting buckets of algebra in an attempt to show you I'm a competent mathematician/physicist who can read. I get plenty of job satisfaction in real life (even got a promotion last week!). If all you want to do is post "Look, I claim I've done some new exact solution which is something no one else has ever done!" thread after thread then don't bother. Find a journal and submit your work to them, which is what you should have already done if you're serious. You are welcome to be a constructive member but I suggest you wind in the attitude.
Well, at least he can spell. You called me a stocker, when I feel that it is perfectly normal for someone to be curious about whether or not, you’re right or wrong. You obviously want people directed towards your website, which is showing up on page two now. I assumed that something was not quite right because according to Wikipedia. “Although new "classical" unified field theories continue to be proposed from time to time, often involving non-traditional elements such as spinors, none has been generally accepted by physicists.” You said that it is not safe to assume anything, not about this. Not about anything in science, ever. If I wanted to know whether you or anyone else is ever right or wrong, you don't assume. You learn the material and figure it out for yourself. You also said that it was all there in black and white and if it’s wrong, and as far as the already unified material in chapter 7 goes, then it’s NOT Me that is wrong, but general relativity. That is how AlphaNumeric and you are being cranks. :bugeye: Sorry, I can’t figure it out but that doesn’t mean I’m a crank. It means that I'm only a homemaker. “As far as naming a peer in the field of general relativity goes you tell me; do you know of anyone in the forum or even still living anywhere on the planet besides myself that has been able to find multiple independent nontrivial exact solutions to Einstein's field equations. Anyone? Frankly, the biggest names in the field today don't even seem to be aware of Vaidya's solution. They're still arguing over implications of Hawking radiation in despite of it. I can count on my fingers without counting a finger twice the number of people anywhere on the planet that have ever sufficiently understood general relativity since Einstein to be able to do this.” It looks like they're aware of Vaidya’s contributions to me. :shrug: http://en.wikipedia.org/wiki/P._C._Vaidya Explain it then, Bruce. What exactly was said in the unification thread? He said that general relativity is a classical theory in which electromagnetism is an already unified field with gravity. He was not the first to realize this, as others have argued this before. He just finally accepted it for himself when he figured out how to write the electromagnetic field tensor in terms of a covariant gradient of a Killing vector for the exact solution to Einstein's field equations he discussed, which means that the electromagnetic field is a function of space-time geometry. YouTube - Unification He is extremely knowledgeable in GR, and I did appreciate his contributions to the forum, but I couldn't help being curious about the whole unification thingy. When his paper is published and accepted, let me know, and I’ll be happy to eat crow.
That axe you keep grinding must be plenty sharp by now. If all you're going to do is snipe you're hardly setting a stellar example, are you? Like I said, if you or anyone else a problem with me you're welcome not to read threads you see I'm the last poster in. You're welcome to read other forums. You're welcome to submit any work you think deserves serious discussion by learned people to journals, where it's their job to evaluate work rather than here, where the few of us who do research don't come here to be at your beck and call. I know part of the reason you have issue with me is I ignored some of your questions in the past. Tough. I either didn't read them or didn't care to reply. People here are free not to reply if they don't want to or can't be bothered to. I'm sorry you feel slighted by that..... actually no, I'm not sorry, I really don't care but it's one of those vapid platitudes people tend to say out of vague social habit. Notice how when I gave waitedavid137 non-supportive comments I explained myself. If all you're going to do in reply to things I've said is 'do an AlexG' then you'll end up getting much the same response from moderators as Alex tends to get, suspensions for adding nothing. Of course if you're so up to speed on general relativity and electromagnetism you're free to bump that thread of waitedavid137's so you and he can talk all about it, you don't need to just complain how others fall short of your head spinningly stellar standards. As Trooper's quotes of waitedavid137 shows, he clearly has a very high view of himself. He might even meet your high standards. Maybe you and he can get together and write up all his world beating work and get it published in a journal? After all, if a lesser mortal you two consider to be a troll and a crank can manage it surely it's within your capabilities, right? Right? Like I said to waitedavid137, I don't need complements from you or waitedavid137 to give myself existential motivation, I get plenty in real life. Unlike waitedavid137 and plenty of other people here I don't feel the need to post page after page of my work to try to bamboozle them with algebra. That's the strategy of people who can't get their work recognised by the research community. After all, why is waitedavid137 posting his work here if he views his capabilities at pretty much top in the world? Why isn't he going straight to journals? If he thinks my comments aren't worth jack and no one else with any GR experience is replying to him then why is he posting it here? It's not like posting work here is how you get yourself a research position and your work accepted by the community. But hey, I imagine you've stopped reading by now, thinking to yourself "He's on a rant, the idiot". I'm sure you want to get back to grinding that axe a bit more.
I already know what you do but thanks for the update. I wished I had a 'smidgeon' of your training so I generally enjoy your posts. Being you're a real scientist with experience. You know what I don't like about your posting style but who really cares. You trolled him and you know it. He claimed to have found a static solution not a general solution. When you discuss physics I appreciate what you share. The rest is always easy on the Internet.
Hi Lady I know AN understands but you know me. The basic claim was it was a static solution not a general solution. So it's and indication for David. I know David knows GR. What's going on is David is really strong with the mathematical analysis and does it. Don't know if he has anybody to bounce it off. Certainly not me . What gets me is the adversarial bs even between really smart folks. BTW I'm going to push the post button when I really don't know what was said over the entire thread. I better get my hard hat on. One thing is for sure nobody is going to wind up claiming something is right when it isn't. So nobody in this conversation is a crackpot.
