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1

We have the frame $\{e_\mu\}_{\mu=0,\dotsc,3}$ in terms of which the velocity vector is $v=v^\mu e_\mu$. There are a few properties of the affine connection which I would like to summarize: $$\nabla_{fX}Y=f\nabla_XY$$ $$\nabla_X(fY)=f\nabla_XY+X(f)Y$$ $$\nabla_{e_\mu}e_\nu=\Gamma^\lambda{}_{\mu\nu}e_\lambda$$ Using this, let's get to work. We have ...


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In a sense, parallel transport, covariant derivative and connection are all synonym for you can recover one from the other. So given a manifold one usually starts by giving one notion (e.g. how a vector field is transported parallel to itself along a family of curves) and then, if needed, the other objects are derived. In physics, when dealing with a ...


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Good terminology question. Let's work in some differentiable manifold $M$, our transformation is a smooth map $T: M \to N$. In the case of a rotation $M = N$. Our $\phi$ is a smooth function $\phi: M \to \mathbb{R}$. In classical field theory the fact that $\phi$ maps to $\mathbb{R}$ is often expressed by the statement "$\phi$ is a scalar field". Now ...


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As usual when dealing with transformations one has to be careful whether they are active or passive. If I understand your question you are implying a passive transformation, which is a mere change of coordinates. In this case all you are doing is changing the way you assign a scalar value to a "vector" of coordinates. Therefore $\phi(\mathbf x) = ...


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Comments to the question (v3): I) The notions of vectors, tensors, scalars, etc, depend on contexts in physics, cf. e.g. this and this Phys.SE posts and links therein. II) In OP's context, these notions refer to representations $\rho$ of the Lie group $SO(3)$ [and the corresponding Lie algebra $so(3)$] of 3D rotations, cf. e.g. Ref. 1. Let $\mathrm{i}L_k$, ...


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The Poisson bracket you wrote only works for position (which is not a vector in general, as coordinates do not transform as vectors or any other tensor quantity). For general vectors the correct Poisson bracket is $$\{L_i,A_m\} = - (\mathbf p\times\nabla_{\mathbf p} + \mathbf r\times\nabla)_i A_m,$$ which reduces to the relation you wrote if you take $A_m = ...


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I think one can enter a dispute regarding the notion of "accepted" but the idea is that General Relativity is successfully described by a Pseudo-Riemannian Manifold, subject to Einstein Equations, with free-falling objects following geodesics. Now you look for a set of axioms that give you this structure. One such set, although not entirely rigorous, is ...


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General relativity can be constructed from the following principles: The Principle of Equivalence Vanishing torsion assumption ($\nabla_XY-\nabla_YX=[X,Y]$) The Poisson equation (or any other equivalent Newtonian mechanics equation) Explanations: The Equivalence Principle can be used to show that spacetime is locally Minkowskian, i.e. the laws of ...


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In layman's terms, it just means that the laws of physics are the same everywhere. This means that we are talking about one common set of laws. The fun part is figuring out how one common set of laws can behave the same, while they are taking place within different frames of reference. Thus we have a one, that is shared by a many. How can this be, when each ...


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The approximate equality is intended to remind you that Nakahara considers transformations with an infinitesimal parameter $\epsilon$ here, but you could as well take it as a full equality, it doesn't matter. The phase space is a cotangent bundle, where the coordinate $x$ are coordinates of the underlying manifold and the momenta $p$ lie in the cotangent ...


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this phrase doesn't specify what laws are invariant It doesn't need to since it is a guiding principle, a razor. It is a statement about the nature of physical law. Put another way, on this principle, an alleged 'physical law' that isn't invariant under inertial coordinate transformations is not a genuine physical law. or even what it means to be ...


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The laws of physics are invariant means slightly different, but (almost) equivalent things depending on what formulation you are working with. Given a collection of transformations (a symmetry/transformation group) and a Lagrangian formulation, you can check whether the Lagrangian changes when you apply the transformation. If it does not change (or ...


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In leyman's terms, it just means that the laws of physics are the same everywhere. Here, on the Moon, even in another galaxy, or in a spaceship travelling at near light speed to another galaxy.


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When constructing equations of motion which are the reflection of laws of nature so to speak, we must make them Lorentz invariant and invariant to spacial rotations. This means that they must have the same form under these transformations. One example is construction of a field theory, in which you begin by forming an action which is Lorentz invariant making ...


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The laws of physics are the same in every inertial frame of reference. If the laws differed, that difference could distinguish one inertial frame from the others or make one frame somehow more correct than another. Here are two examples: Suppose you watch two children playing catch with a ball while the three of you are aboard a train moving with ...


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According to Einstein: All our well-substantiated space-time propositions [and consequently, all of our statements concerning facts and findings in physics] amount to the determination of space-time coincidences. If, for example, the [course of events] consisted in the motion of material points, then [...] nothing else are really observable except the ...



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