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    Independent components of the curvature tenso

    That's the first thing I thought when he specified GR. There was a discussion here a few months ago on vanishing Ricci tensors and what it says about "flat" spacetime in n-dimensions in comparison to a vanishing/non-vanishing Curvature tensor that (I think) led to some discussion on...
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    Einstein's Field equations & quaternions / octonions

    I've always wondered why quaternions aren't/haven't been taught very much. Perhaps there has been more investigation into their use than I'm aware of, but it seems like the choice of Gibbs' vector calc (IIRC developed from his study of quaternions) was the rather arbitrary result of Gibbs'...
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    Do photons create gravity?

    This thread is in the relativity section, and according to GR, "light" does produce gravity. If you looked at a completely "empty" Universe that was void of any form of energy, matter, radiation, etc., in a 4 dimensional spacetime where \mathbf{R(X,Y)Z}=R^{a}_{\ bcd}X^{c}Y^{d}Z^{b}=0=R_{bd}...
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    Gravitational lensing paradoxon!

    Agreed, unless I'm missing something?
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    Is Gravity a force, or not a force?

    It probably should bother you that the major theories of the world have definite areas where they can't agree. But don't let the word "force" be too much of a factor in that. Force is just the word given to describe a particular mathematical act. One theory has gravity being sent out...
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    Is there any reason for tensor indices being ordered one way or the other?

    Sorry...misread the OP's post Saying that a tensor is of type (r,s) means that it has r number of contravariant indices and s number of covariant indices. Contravariant and covariant vectors transform slightly different under coordinate transformations. A contravariant vector...
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    Is a function contravariant?

    My understanding of differential forms is that, given the map \phi : V \to V^{*} If f \in V^{*} is a 0-form (a real-valued function on V^{*} ), then \phi defines the function \phi^{*} f on V where \phi^{*}: F^{0}(V^{*}) \to F^{0}(V) as the function such that...
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    Deducing some GR from SR?

    Exactly. I wish there were some sort of standard "WARNING: HAND-WAIVING NON-MATHEMATICAL ANALOGY AHEAD" for situations like this. It seems that the vast majority of posts in this forum are clearing up the confusion that comes from misinterpreting "worded" analogies that were meant to...
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    General relativity flaws

    That sounds more like a personal philosophy. I don't need a worded definition for space in order for it to be complete. I'd actually prefer that it be defined by the mathematics so as to avoid the semantics and possible ambiguity of defining it with words as well. What do you mean by this...
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    Is Gravity a force, or not a force?

    It depends on what theory you're using. General Relativity doesn't consider gravity as a force. Massive objects create a space-time geometry with a "curvature" described by the curvature tensor. There is no actual "force" of gravity, just the curvature of space-time that is created by...
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    General relativity does not seem to address gravity

    It's the opposite of this. GR says that geometry of space is due to gravity, not that gravity is due to the geometry. If it were symmetric then geometry would create gravity as well, but GR (I believe) doesn't include this. It only allows for gravity to create geometry.
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    Tangent space vs. Vector space

    That's a good idea. I seem to like learning the differential geometry more than the actual physics of GR so far. I know I've heard Isham's book mentioned before too....I'll look for it at the library today. Thanks again for the help!
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    Tangent space vs. Vector space

    Thank you very much for the detailed post, it's very helpful. I think these posts have cleared up my confusion
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    Tangent space vs. Vector space

    Thanks for the replies. I know that the tangent space at the point P (from my first post) is a vector space, I wasn't sure when/how they differed. So, if I'm following this correctly......then the following statements are correct (at least in their usage of "vector space" and "tangent...
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    Tangent space vs. Vector space

    I'm not sure I fully understand the difference between these two terms when used in differential geometry/general relativity. If I were to describe covariant differentiation to someone, I would say something like this: "On a curved manifold (imagine a basketball), you could assume a tangent...
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    Spacetime Fabric

    There are some 256,000 words in the English language. Most of those are archaic forms that have fallen out of favor, so the "actual" number is much less. Giving worded analogies of what the mathematics is saying is always a slippery slope. Use the idea of a "fabric" as a vague basis to...
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    Question on wording for paper

    Yes, exactly. Thank you for organizing it so concisely. My original concern (assuming everything you listed is true), is can I make the generalizations that "A vanishing Ricci tensor in 3 OR LESS spacetime dimensions implies flat spacetime." ***The emphasis is on the generalization...
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    E=mc2 and Postulates to prove it

    If it is wrong, then Postulate II is no longer valid. I don't think it will affect the theory much though. There is more than enough experimental data to show that the equation is correct (as others have said). If experiment proves the equation, it doesn't really matter if the postulates that...
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    Question on wording for paper

    I am attempting to speak with "physics" terminology (which, unfortunately, may not be good for my case. lol). My understanding of GR and Differential Geometry is primarily from physics texts on GR. I have studied mathematical texts on the subjects, but the VAST majority of my understanding...
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    Question on wording for paper

    For the record, I am NOT saying that a flat space-time, which is known beforehand to have no disturbances of any kind, will have any value for the Ricci tensor other than R^{\mu \nu}=0 . In a Euclidean geometry, or a geometry that we know BEFOREHAND is flat, we can say that definitely, YES...
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    Question on wording for paper

    Question on wording for paper (re-explained in a second post) In a paper (undergrad thesis type paper) on GR, I have the statement: A vanishing Ricci Tensor, i.e., R_{\mu \nu}=0 is not enough to explicitly define a flat space-time. In the case where the manifold being investigated has...
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    Can you rip the space time fabric?

    No problem....I wanted to quote one of the OP's posts, but it was too many posts behind mine to show up in the "topic review" below the reply box. lol I was hoping you wouldn't take me up on the flowers anyway....that would get expensive. lol
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    Can you rip the space time fabric?

    My post, by definition, was not an ad hominem attack against you. I specifically stated that my post was NOT a specific attack against TCS, it just happened to be the first post I saw with the word "fabric" in it, a word I wanted to quote. If including the specific announcement that *This...
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    Can you rip the space time fabric?

    Yes....I believe you can actually help minimize the stiffness of this "fabric" by using fabric softener. I tend to use Bounty, unless the store brand is on sale. So, I think we should make sure that we mention that ripping this fabric is only possible if God hasn't done the laundry recently...
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    Why is energy-momentum tensor Lorentz invariant?

    If you look at the Bianchi identities with the Ricci tensor, you can get an expression like R ^{\alpha}_{\ \rho :\alpha}-R_{:\rho}+R^{\nu}_{\ \rho :\nu}=0 which, because of the symmetry of the Ricci tensor gives you 2R^{\alpha}_{\ \rho :\alpha}=R_{:\rho} Then, raise the suffix rho to...
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    SR and differential geometry

    Doesn't Weinberg have a book on gravitation where he tries to de-emphasize the geometric view? I think he says something like "I believe the geometric view has driven a wedge between GR and the theory of elementary particles." I don't know how relevant that is with what you're saying, but I...
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    Curvature (Ricci things)

    This. If you have a geometry that doesn't map to a Euclidean geometry and has a non-vanishing Ricci tensor, you have a non-euclidean geometry. Or, I believe you can say that if the curvature tensor vanishes, then covariant derivatives will commute, which indicates a coordinate system that...
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    Does Light Have Mass?

    Does this mean pain is analagous to mass? I never really felt them until they announced their presence in the unsavory way they're known to do. The other common usage of the word mass is "relativistic mass". This is the hornet's nest refered to earlier. The concept of "relativistic...
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    Does Light Have Mass?

    How do hornets have mass? I can't feel them. So, mass always has energy so that total energy is concerved, but kinetic energy doesn't have mass? Isn't that just arguing semantics? (Let it be known that I had no ill will when poking the hornet's next with a stick...just bored)
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    Does Light Have Mass?

    Wouldn't we lose the "feel" of air for a different reason? I can feel it when I drink from a straw. lol Light has enough mass to feel the effects of a gravitational field and also exerts its own gravitational attraction, so it must have mass. But, by definition, it can't have any rest...