Einstein Field Equations??? I have not been able to comprehend the Einstein Field Equation, the Stress Energy Tensor, the Ricci Tensor, the Einstein Tensor, and Christoffel Symbols. Though I am reasonably proficient at working with nested loops in programming, and I have a rudimentary knowledge of partial differential equations, there seems to be no source for spanning the gap between diff-eq, and Tensor calculus, and there also seems to be no source that carefully defines the units and terms of the Tensor calculus. Am I correct in thinking that there is some analogy with the kinematic equations: v=dx/dt a=dv/dt Can one make similar equations with 4-displacements, 4-velocities, and 4-accelerations? What I'd like to do is get from where I am now, to where I can sort of comprehend the articles on wikipeda on these topics. Like for instance, I know the chain rule in partial differential equations [tex]d f(x,y,z)=\frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy+\frac{\partial f}{\partial z}dz[/tex] Is this something that could be expressed in tensor notation?
Re: Einstein Field Equations??? I think you can benefit from watching the Leonard Susskind Lectures on General Relativity, available on youtube. He is someone who has the knack for teaching effectively. Although, these lectures aren't the best source for the basic Tensor calculus information, they give the important intuitive explanations that will make it much easier to consult other more formal sources. I also recommend the lectures by Alex Maloney (audio and notes available). Here you will get into more details and more formality to get everything rigourously correct. However, without the Susskind Lectures, this may be too big a jump for you, given the background you mentioned. http://www.physics.mcgill.ca/~maloney/514/
Re: Einstein Field Equations??? Have you had vector calculus? Here is my own shot at introducing tensors from scratch. You might also find it helpful to read Exploring Black Holes, by Taylor and Wheeler. It discusses a lot of GR *without* tensors, and if you can understand all of that physics, you should have a very good foundation for understanding the same material using tensors. Yes, these have direct analogies in terms of 4-vectors: http://www.lightandmatter.com/html_books/genrel/ch04/ch04.html#Section4.2 But there is no real analogy with the Einstein field equations. (Don't know if that's what you were asking.) To understand the physical content of the field equations, try the Feynman lectures and/or Penrose's The Road to Reality. Yes. For an example, see http://www.lightandmatter.com/html_books/genrel/ch05/ch05.html#Section5.7 , subsection 5.7.2.
Re: Einstein Field Equations??? If you want to really know this stuff, you will at some point have to study differential geometry. I like Lee's books. "Introduction to smooth manifolds" covers the basics (and a lot more). The stuff about connections, covariant derivatives, parallel transport and curvature is in "Riemannian manifolds: an introduction to curvature". If C:[a,b]→M is a smooth curve, then its tangent vector at C(t) is denoted by [tex]\dot C(t)[/tex] and defined by [tex]\dot C(t)f=(f\circ C)'(t)[/tex] for all smooth functions f from an open neighborhood of C(t) into the real numbers. So the tangent vector of a curve, at a specific point on the curve, is an operator that takes functions to numbers. When C is the mathematical representation of the motion of a particle in a 4-dimensional spacetime, its tangent vector field [tex]\dot C[/tex] is called its 4-velocity. (In special relativity, these definitions can be simplified). There's this calculation: [tex]df|_p (v)=v(f)=v(x^i)\frac{\partial}{\partial x^i}\bigg|_p f =dx^i|_p(v)\frac{\partial}{\partial x^i}\bigg|_p f[/tex] (where v is an arbitrary tangent vector at p), which implies [tex]df=\frac{\partial f}{\partial x^i} dx^i[/tex] but to really explain this requires a lot more time than I'm willing to spend tonight. Some of my earlier posts may help, e.g. this one and this one. Note that [tex]\{dx^i|_p\}[/tex] is a basis for the cotangent space at p. Specifically, it's the dual basis of [tex]\bigg\{\frac{\partial}{\partial x^i}\bigg|_p\bigg\}[/tex]
Re: Einstein Field Equations??? Hey, I just wanted to say, there's a lot of really good stuff here. It will keep me busy for a while. Thanks to Fredrik, bcrowell, and stevenb
Re: Einstein Field Equations??? The Leonard Susskind lectures have been really helpful. Let's see if I've got any of this right: The covariant tensors (indices downstairs) are like gradients; vector quantities that are functions of position. The contravariant tensors (indices upstairs) are like differential elements; quantities that represent the positions themselves.
Re: Einstein Field Equations??? Sounds like you're off to a good start. Just a word of caution about the lectures. They are very streamlined to give you the most critical concepts so that you have a sense of the landscape. From there, it will be much easier to get into the nitty gritty details. As an example, if I remember correctly, he presents the formula for covariant derivative of a covariant tensor with a sign error. There are some other minor mistakes too. Keep in mind that even the best physicists gets signs and factors of two wrong, ... perhaps even more often than the average Joe, for whatever reason. Consider this lecture series like that "thin book" on a library shelf. One of my favorite professors back in school had a useful piece of advice for his students. He gave it in the form of a question by saying, "If you go to the library to get a book to learn a new subject, which book should you take"? Students would then throw out a bunch of answers until someone finally said, "Pick the thickest book!", to which he would reply, "No, pick the thinnest book." His point was simply that it's better to start with a brief overview of a subject, hitting the major concepts, before getting into the details. Anyway, the Susskind lectures (even the ones on other topics), are those "thin books" you can read as introductions to new topics. Of course, the book needs to be good as well as thin, but Susskind succeeds there as well, in my opinion.
Re: Einstein Field Equations??? I haven't had a chance to look at this very much in the last couple of weeks, but maybe something that I was teaching in algebra might apply to rank 0 tensors. I want to take a function f(x) and stretch it along the y-axis by a factor of V, stretch it along the x-axis by a factor of H, then transpose it along the x and y axis, by x_{0} and y_{0}. The end product looks like [tex]\[g(x)=V f(\frac {1}{H}(x-x_0))+y_0\][/tex] Are there contravariant and covariant 0 degree tensors involved here?
Re: Einstein Field Equations??? What follows may be dumbing down the subject, but it helped me when I was first learning about curved spaces and tensor calculus. Imagine two axes, representing say, x and y directions. One usually thinks of two straight lines at right angles crossing at the origin x=0, y=0. In this case defining a position vector from the origin to a point x_{0}, y_{0} is trivial i.e. drop a perpendicular from the point to each axis and read off the value. Now imagine that the axes are not at right angles, and curve smoothly. One can still drop a perpendicular to the axis and read off a value. But there's a nother set of values available - by drawing a line through the point parallel to each axis and reading off the value where this line cuts the other axis. The two sets of values obtained thus are called the perpendicular projected components, and the parallel projected components. The motivation for this is that the quantity defined as the inner product of these two 'vectors' is a geometric invariant. It will not change under coordinate transformations. Writing the perpendicular vector as X_{a}, and the parallel as X^{a}, the quantity X_{a}X^{a}=X_{1}X^{1}+X_{2}X^{2} = L^{2} is invariant. This means that if we can write our physical observables in curved space as scalars formed by contracting tensors, they will be covariant under general coordinate transformation. To summarize : to define an invariant length in curved space, we must allow two ways of expressing positions, the covariant and contravariant vector. And, the metric of the space, a rank-2 tensor can be used to transform covariant -> contravariant and vice-versa, viz. X^{a}=g^{ab}X_{b}
Re: Einstein Field Equations??? Okay. Let's see if I can envision an example. If we've got the idea we can curve space, we could turn an arc-lengths of several concentric circles into a parallel straight lines, and we could take the radial lines rom the center, and also make parallel straight lines from them. Or, we could take hyperbolas and lines from the origin (see attached) and convert them to parallel straight lines. If we can come up with precise mathematical transformations that do these things, then we could discuss which elements are covariant, contravariant, and invariant, based on how the transformation is accomplished, right?
Re: Einstein Field Equations??? Yes, it's basically partial differential equations. Like Maxwell's equations written with the 4-vector potential - the vector potential is not observable, and there are many vector potentials corresponding to the same physical situation or E and B fields. So in addition to the differential equations, there is a rule saying which potentials correspond to the same physical situation. This identification of different solutions as the same is what we call geometry - the true object that remains unchanged by different descriptions of it. In practical calculations, one can gauge fix.
Re: Einstein Field Equations??? Sorry, I can't see the relevance of the above to what I wrote. A (crude) diagram is attached to illustrate what I'm saying. The key is that the axes are curved. Notice that if the curved axes deform back to the usual Euclidean axes, the parallel (contravariant) and perpendicular ( covariant) components of the vector coincide. So in flat space you can put tensor indexes high or low.
Re: Einstein Field Equations??? It's the concepts of invariant, covariant, contravariant that throw me. From Special Relativity I know that distances, times, simultaneity are all observer-dependent quantities. But proper time and proper distance between events are invariant. However, when you look at the proper time between events in a gravitational field, this is no longer invariant. If you are looking down,you see things moving in slow-motion. Looking up, you see things moving in fast motion. If you told me we were working on a problem to find the work, and said [tex]W= \vec{F}\cdot \vec{d}[/tex]. I will take a guess and say Work is invariant, Force is covariant, and distance is contravariant. Is that right? What are some examples of invariant quantities, and do these remain invariant in gravitational fields?
Re: Einstein Field Equations??? If F and d were contravariant and covariant then W would be invariant under coordinate transformations. The points you raise are about physics in curved space-time, which is not GR. GR is the theory of how matter curves space-time, and it turns out that pseudo-Riemannian geometry is the correct way to formulate it, and the field equations are the 'final' result. I realize that just seeing how covariant and contravariant vectors arise in curved spacetime is not enough to understand tensors. I may be overplaying the invariance card. You will have to think abstractly to understand the Riemannian manifold, which is about worldlines and the vectors and vector fields associated with them. The kernel of thing is 'curvature', and you've been looking at that in earlier posts. Two points about physics in curved spacetime. One obvious adjustment is that the distance^{2} between points (a,b) and (c,d) is no longer (a-c)^2 + (b-d)^2 ( i.e. dx^2+dy^2) but is now given by integrating ds^{2}=Sum g_{ab}dx^{a}dx^{b}. The most important is that the usual derivative is replaced by the covariant derivative. A good book on GR is Stephani's "General Relativity : An Introduction to the Theory of the Gravitational Field" (1986) . It really does start easy with 'Force-free motion ... in Newtonian mechanics'. [edit] cut manifest falsehoods.
Re: Einstein Field Equations??? Let me ask some basic questions that I'm getting out of Misner/Thorne/Wheeler; MTW. If you watch an ant walking around an apple, do you really believe that ant thinks its walking in a straight line? Do you think the apple is flat and geometry is somehow curved, or do you think the apple is actually meaningfully round in some global reference frame? If you see light bending around the sun in a solar eclipse, do you think that the light traveled in a straight line, but space was somehow bent, or do you think that the light actually meaningfully changed directions in a global refence frame? If you see two satellites travel along geodesic paths and somehow meet at the same point twice, do you think they both traveled straight lines in some warped space-time, or do you think that the satellites actually meaningfully changed directions in a global reference frame? From the writing, I have this sense that the authors MTW really truly believe there is no global reference frame, and that in each instance, the ant, the light, and the satellites really are moving in straight lines. They manage to convince themselves of this by saying that the light and the satellite feel no forces; thus they must be traveling in straight lines. These are the arguments they are using to entice me down the rabbit hole, as Einstein said "Why were another seven years required for the construction of the general theory of relativity? The main reason lies in the fact that it is not so easy to free oneself from the idea that coordinates must have an immediate metrical meaning." I have no such desire to free myself from this idea. To me, any legitimate coordinate system should have immediate metrical meaning. And all coordinate transformations simply change the metrical meanings. When I think of the distance between two points, I think of the shortest distance between two points. Integrating along the path to find the distance between two points represents the distance traveled along the path between the two points. I don't see how this is a change in space-time. It is a change in what you mean by distance.
Re: Einstein Field Equations??? Don't take analogies too seriously. You have to imagine a two-dimensional ant and a mathematical apple for this to be really accurate. It's a "straight line" (technically a geodesic) in spacetime, not space. You need a coordinate system just to define which slice of spacetime to call space, and in the coordinate systems that would be convenient to use in this situation, the path through space is curved. It sounds like you're asking us to ignore what the theory is saying and tell you what we think actually happens. Since I don't think my intuition is more accurate than general relativity, I can't even tell you what it would mean to do that. A coordinate system is just a function that assigns a 4-tuple of real numbers to each event. Yes, and the result is the length of the curve. "Length" is the appropriate word here, not "distance". In spacetime, the function we're integrating involves a square root of something that can be positive or negative. This means that we can define the "proper length" of a curve only for curves such that the quantity under the square root is positive everywhere on the curve. But we can also define a similar quantity, "proper time", for curves such that the quantity under the square root is negative everywhere on the curve. We just flip the sign of that quantity in the definition, so that we have something positive under the square root.
Re: Einstein Field Equations??? With some requirements, I hope. You can't just take arbitrary sets of 4 numbers and assign them randomly to the events and call it a coordinate system. The coordinates have to be somehow meaningful. If you can establish a one-to-one and onto relationship between the coordinate systems then you have a global coordinate system that can be viewed many different and valid ways. But how do you determine an appropriate way to find the proper length, using an integral of a curve? If you have spacelike separation, there is no valid "path" between the events. In any case, the coordinates of the events are well defined in any given coordinate system. You don't need to do an integral to determine the distance or the time between them. The space-time interval between events tells you properties only of a hypothethical object, that would have certain properties if it were there. The vast majority of event pairs don't have an object in between them. And the events themselves exist and have locations and times in a global reference frame, describable without doing any integration at all.
Re: Einstein Field Equations??? If you have a curve y=f(x), the the length along the curve between two points is (as you know) [tex] s=\int_{p_0}^{p_1} \sqrt{1+(dy/dx)^2} dx [/tex] or something similar. We have to think in terms of curves because worldlines are curves. In Lorentzian space ( t,x) we define the proper length as [tex] s=\int_{p_0}^{p_1} \sqrt{1-(dy/dt)^2} dt [/tex] Most of your remarks are based on flat-space. If the 'axes' of a space are not Euclidean straight lines then there is no canonical distance and you need to define lengths of curve sections.
Re: Einstein Field Equations??? In your equation you've got events p0, and p1. What if you don't need s? What if you're more interested in what is happening in your own frame of reference, and you're not really concerned about some nonexistent ruler or particle traveling between these events? For instance, what if p0 and p1 are two supernova explosions, that happen in two different parts of the sky several thousand years apart. Certainly we could calculate the space-time interval between those two events, but what possible use would it have? It would be an answer to this question: (What is the distance between those events in a reference fram where the two events appear to be simultaneous?) but that question is completely irrelevant unless you happen to be traveling in just the right velocity where the two events were simultaneous. With the Euclidian distance [tex]r^2=x^2+y^2+z^2[/itex], there are all kinds of things you can do with it: Work = Force dot distance, Torque =Force Cross Distance, velocity = distance over time, etc, etc. But with this s distance [itex]s^2=t^2-x^2-y^2-z^2[/itex], what good physics comes from this? I think none. I think that t, x, y, z are contravariants. s is invariant. But if you want to talk about velocity, momentum, these are covariant quantities, based on properties of matter within the space. Perhaps it is here where general relativity can shine. But it is not because somehow you warped the space; up down, left, right, forward, backward, future and past, are still defined in a global inertial reference frame. You KNOW you're spinning, so what was once left is now forward. But that's a global change! EVERYTHING that was once to your left is now forward. You can't make the claim that the universe is somehow only "locally Euclidian," when everyday experience tells you that turning your head allows you to see what's over to your left or right--no matter how far away it is. Extreme distances do not free events from the laws of Euclidian rotation. Why should extreme distance free events from the laws of Lorentz Transformation? Yet that's what the argument seems to be in Chapter 1 of Misner Thorne Wheeler. Well, technically, they argue that "the geometry of a sufficiently limited region of spacetime in the real physical world is Lorentzian" and "the geometry of a tiny thumbprint on the apple is Euclidian." (I may be making the fallacy of the inverse). However, with Chapter 7 entitled "Incompatibilty of Gravity and Special Relativity" I suspect they also meant the inverse: that Lorentzian spacetime can only exist in sufficiently limited regions.
Re: Einstein Field Equations??? MTW Chapter 7 Section 3: "... Schhild's [argument uses] a global Lorentz frame tied to the earth's center. It makes no demand that free particles initially at rest remain at rest in this global Lorentz frame" Now I see at least part of the problem. You can't tie a global inertial frame to anything unless it never accelerates. The title of the chapter is "Incompatibility of Gravity and Special Relativity" and it is "Track-2" material, meaning that except for the obvious mistake of trying to tie a global inertial reference frame to an accelerating object, it mostly uses notation that I don't know.