Why isnt there a theory of everything after decades of research by the best minds of the world. Is it because that general relativity and quantum mechanics are so different in their ideals
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What is currently stopping us from having a theory of everything? i.e. what mathematical barriers, or others, are stopping us from unifying GR and QM? I have read that string theory is a means to unify both, so in this case, is it a lack of evidence stopping us, but is the theory mathematically sound?
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The question I guess is pretty clear. I am a physics undergrad wishing to pursue research in quantum gravity(string theory?). What are the subjects I should learn other than the usual compulsory undergrad courses such as quantum mechanics, and GR, etc? Things that come to my mind are QFT, lie algebras, differential geometry, algebraic geometry etc? Other than this, could you suggest any topic that are not entirely essential but interesting in mathematical physics, gauge theory etc (e.g. solitons, monopoles etc). Actually I am looking for a mini project of sorts which would give me an idea of what these fields really are. I have done some lie algebras, classical field theory and am currently studying QM.
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I am new to general relativity but have a background in computer science.  Why is it so hard to do n-body simulations in GR?  For example, there could be a program which takes the properties (mass, position, velocity, etc.) of each particle as input and numerically integrates the evolution of the system using discrete but very small time steps.  If we throw enough memory and computing power at it, we can get arbitrarily close to the 'real' solution.  I assume the entire histories of the worldlines would need to be known, in order to calculate things like gravitational waves.  I have downloaded programs which show the evolution of the Schrodinger equation, so I am surprised they are so hard to find for GR.  Some responses I've heard elsewhere are 'the equations are nonlinear,' but from what I hear numerical algorithms are great for these types of problems.  What types of numerical algorithms are out there for GR?
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Can someone suggest a textbook that treats general relativity from a rigorous mathematical perspective? Ideally, such a book would   Prove all theorems used. Use modern "mathematical notation" as opposed to "physics notation", especially with respect to linear algebra and differential geometry. Have examples that illustrate both computational and theoretical aspects. Have a range of exercises with varying degrees of difficulty, with answers.   An ideal text would read a lot more like a math book than a physics book and would demand few prerequisites in physics. Bottom line is that I would like a book that provides an axiomatic development of general relativity clearly and with mathematical precision works out the details of the theory.  Addendum (1): I did not intend to start a war over notation. As I said in one of the comments below, I think indicial notation together with the summation convention is very useful. The coordinate-free approach has its uses as well and I see no reason why the two can't peacefully coexist. What I meant by "mathematics notation" vs. "physics notation" is the following: Consider, as an example, one of the leading texts on smooth manifolds, John Lee's Introduction to Smooth Manifolds. I am very accustomed to this notation and it very similar to the notation used by Tu's Introduction to Manifolds, for instance, and other popular texts on differential geometry. On the other hand, take Frankel's Geometry of Physics. Now, this is a nice book but it is very difficult for me to follow it because 1) Lack of proofs and 2)the notation does not agree with other math texts that I'm accustomed to. Of course, there are commonalities but enough is different that I find it really annoying to try to translate between the two...  Addendum (2):  For the benefit of future readers, In addition to suggestions below, I have found another text that also closely-aligns with the criteria I stated above. It is, Spacetime: Foundations of General Relativity and Differential Geometry by Marcus Kriele. The author begins by discussing affine geometry, analysis on manifolds, multilinear algebra and other underpinnings and leads into general relativity at roughly the midpoint of the text. The notation is also fairly consistent with the books on differential geometry I mentioned above.
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The GPS is a very handy example in explaining to a broad audience why it is useful for humanity to know the laws of general relativity.  It nicely bridges the abstract theory with daily life technologies!  I'd like to know an analogous example of a technology which could not have been developed by engineers who didn't understand the rules of quantum mechanics.  (I guess that I should say quantum mechanics, because asking for a particle physics application could be too early.)  To bound the question:   No future applications (e.g. teleportation). No uncommon ones (for, who has a quantum computer at home?). A less frequently-cited example than the laser, please. If possible, for sake of simplicity, we'll allow that the quantum theory appears in form of a small correction to the classical one (just like one doesn't need the full apparatus of general relativity to deduce the gravitational red-shift).
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Did Einstein completely prove Newton wrong? If so, why we apply Newtonian mechanics even today? Because Newton said that time is absolute and Einstein suggested it ?   So, if fundamentals are conflicting, how can both of them be true at a time?
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Even if many interesting similarities between the classical and the quantum mechanical framework have been worked out, e.g. in the subject of deformation quantization, in general, there are some mathematical problems. And in the conventional formulation, you don't want to make things like $\hbar\rightarrow 0$ for the expression $P=-\text i\hbar\tfrac{\partial}{\partial x}$.   In special relativity there are many formulas where one optains the non-relativistic formula by taking the naive limit $c\rightarrow \infty$, e.g.  $$\vec p=\frac{m\vec v}{\sqrt{1-|v|/c}}\ \rightarrow\ \frac{m\vec v}{\sqrt{1-0}}=m\vec v.$$  I wonder if it is know that you can always do that. Is there a formulation of special relativity (maybe it's the standard one already), where the starting assumptions/axioms/representations of objects of discourse involve the constant $c$, and as you take them with you to do all the standard derivations, you always end up with results which reduce to the Newtonian mechanics if you take that limit?
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Does String theory say that spacetime is not fundamental but should be considered an emergent phenomenon?  If so, can quantum mechanics describe the universe at high energies where there is no spacetime?  Schrodinger equation assumes the presence of time dimension. So how can such equations be valid at extremely high energy?  And is it meaningful to do physics if there is no time dimension?  Would the concepts of Hilbert space, wavefunction collapse etc be useful in that case?
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What is a good definition on Space,  and the most specific topic ""? Because someone told me that spacetime is the foundation of the entire universe?  And also, Is it possible to create spacetime?
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If the theory of everything indicates a singularity-free and finite universe, will  be relevant to the universe? If so, is there any interesting physical consequence, e.g. in superstring or quantum gravity?
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Can someone please give me the prerequisites and mathematics required for string theory? Are there some good references to study it, both online and in a book? Please consider I am a newbie in string theory, and only about up to the level explained in Hawking's Brief History of Time and Penrose's Road to Reality. I would prefer something which does not omit rigor, but builds it up.Also, are there good online lectures on this?  (I have covered most of GR, but I am not really comfortable with advanced Quantum mechanics. I have studied QM only from Griffith's book, and a few McGraw hill lecture notes.)
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I'm still self-learning general relativity. I have been a huge fan of  on GR, black holes and cosmology. He goes through GR in pretty much full tetrad formalism. The reason he wants to do that is because some of the physics becomes much easier to understand in local tetrad frame, where everything just follows special relativity. That's the reason people working with particle physics in GR are interested in working with tetrads, I suppose. While I think Andrew Hamilton's notes are the most amazing thing in this world, I find myself sometimes wanting to read more books on tetrads and particle physics. That's why I'd like to ask if anyone knows more fairly comprehensive sources to learn about tetrads and particle physics in GR. One thing I sometimes find is lacking in the above notes is exercises / example calculations (only in some chapters). This is why I'd especially appreciate resources with many exercises to master the basics. I'd like to think my background in GR is not too bad right now, but I still find myself struggling with quantum physics from time to time. Thanks a lot! Too Long Didnt Read I would like resources which have the following:  General Relativity Lots of tetrads Particle physics Exercises &amp; applications
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For years I have been very fascinated by the "mystery" aspect of gravity. Functionally, we understand it for our applications, but in my (limited to my 3 quarters of undergrad general physics) understanding, we don't know why it exists or how the "pull" works.   To provide an example of what I mean: there has to be some "measurable material" of gravity between, say, the earth and the moon -- some kind of 'graviton' (not that it needs to be a particle or anything, I have zero claims as to the nature of how gravity does what it does). The [butchered] saying of "pluck a flower and move the furthest moon" originates because gravity in theory, has an infinite range (right?) and moving a flower on earth could maybe move an atom on Jupiter ever so slightly.  Anyway, similar to how we had "very good" hypotheses about the existence of Higgs boson that the LHC 99%+ confirmed, do we have any "very good" hypotheses in regards to the fundamental way how gravity does what we have long-known it to do?
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I'm a bit confused. QFT is claimed to incorporate both Quantum Mechanics and Special Relativity. Therefore it should address the problem of non-locality caused by entanglement. However when I search for an answer on the Internet, I found nothing.   I'm not complaining. But it seems that most people only use QFT to do some fancy particle stuff and forgot we should care more about the more fundamental stuff.
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Is there a difference between a postulate and a principle in physics?  Both seem unproved statements taken as true. If thats correct, why the different names?
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When I read descriptions of the many-worlds interpretation of quantum mechanics, they say things like "every possible outcome of every event defines or exists in its own history or world", but is this really accurate? This seems to imply that the universe only split at particular moments when "events" happen. This also seems to imply that the universe only splits into a finite number of "every possible outcome".   I imagine things differently. Rather than splitting into a finite number of universes at discrete times, I imagine that at every moment the universe splits into an uncountably infinite number of universes, perhaps as described by the Schrödinger equation.  Which interpretation is right? (Or otherwise, what is the right interpretation?) If I'm right, how does one describe such a vast space mathematically? Is this a Hilbert space? If so, is it a particular subset of Hilbert space?
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I know classical physics, quantum mechanics, special relativity, and basic nuclear physics. I would like to get into some particle physics.   Where can I find a good introduction? It might be useful to segment books by whether they require quantum field theory or not.