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6

Parametrize the particle's worldline w.r.t. $t$: $$~x^\mu (t) = (t,1)$$ Its four-velocity is: $$u^\mu =\frac{d x^\mu}{d \tau}$$ To evaluate this we use the fact that: $$d \tau^2 = coshx~dt^2-dx^2$$ Also use $x=1$ and $dx=0$: $$d \tau^2 = cosh(1) dt^2$$ Therefore: $$u^\mu = \frac{d x^\mu}{d \tau} = \frac{d t}{d \tau} \frac{d x^\mu}{d t} = ... 6 Comments to the post (v2): Note that \sqrt{|g|} transforms as a density rather than a scalar under general coordinate transformations. In particular, the covariant derivative of \sqrt{|g|} does not necessarily coincide with the partial derivative of \sqrt{|g|}. Here is a heuristic explanation using local coordinates. The Levi-Civita connection is ... 6 Your intuition that the Einstein equations are equations for the metric tensor, not for the manifold is mostly on the right track, but the details are wrong. That core bit of intuition is best phrased, I think, as saying that the Einstein equations are local equations for the geometry of the manifold. That is, they tell you that, whatever manifold your ... 6 Conserved quantities in GR In GR, energy (or mass) is typically an ill-defined concept. In flat spacetime, we define energy as the conserved quantity corresponding to time translational symmetry. Extending this to GR is quite tricky mainly because, what one is calling time is already observer dependent (this is of course also true in flat spacetime, but at ... 5 Is there another way to conclude the Schwarzschild solution has a mass M It's not so much a conclusion as a definition. From Schutz in "A first course in general relativity", section 8.4 "Newtonian gravitational fields", pages 207 - 208: Any small body, for example a planet, that falls freely in the relativistic source's gravitational field ... 4 The proper time of a time-like curve is its length. 4 Let us for simplicity work in units where the speed of light c=1 is equal to one, and assume that there is no cosmological constant \Lambda=0. A spherically symmetric vacuum solution to the EFE of the form$$\tag{1} ds^2~=~g_{tt}(r)dt^2 + g_{rr}(r)dr^2 +r^2 d\Omega^2,$$and such that it asymtotically becomes Minkowski space$$\tag{2} ...

3

I read a few lines about general relativity and [... an equation for] the eigentime of a time-like curve. I suppose that this is referring to an equation similar to $$\tau A_J^Q := \int_0^1~dt~\sqrt{g[~\dot\gamma, \dot\gamma~]},$$ where $A$ denotes a particular participant ("material point", "principal identifiable individual"), the quantity being ...

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Here's a heuristic calculation: Let $\{E_i\}$ be an orthonormal frame ($g(E_i,E_j)=\epsilon_i\delta_{ij}, \epsilon_i=\pm 1$). Then $\mu$ is the canonical volume form $\sqrt{g}\,\mathrm{d}x^1\wedge\cdots\wedge\mathrm{d}x^n$ iff $\mu(E_1,\dotsc, E_n)=1$. Then $$(\nabla_X\mu)(E_1,\dotsc,E_n)=\nabla_X(\mu(E_1,\dotsc,E_n))-\sum \mu(E_1,\dotsc,\nabla_X ... 2 A general diffeomorphism does not map geodesics to geodesics. Some simple counter examples You can a build diffeomorphism on the Euclidean plane by imagining putting one finger on a tablecloth at point x and dragging it. This map is clearly smooth, a smooth inverse is constructed by dragging your finger back. Any geodesic on the plane (a line) passing ... 2 There are lots of ways of approaching special relativity. My own preferred approach is the invariance of the line element. Suppose you move a small distance in spacetime (dt, dx, dy, dz) then the length of the line element ds is defined by:$$ ds^2 = -c^2dt^2 + dx^2 + dy^2 + dz^2 \tag{1} $$This equation is known as the metric equation and is derived ... 1 The set of transformations that leaves the speed of light unchanged is the Lorentz group. Representation theory enables us to investigate the irreducible representations of the Lorentz group. The lowest-dimensional representations act on scalars four-vectors However, take note that usually we consider representations of the corresponding Lie algebra ... 1 In special relativity there are two major assumptions: -the laws of physics are the same in all inertial frames -the speed of light that you observe is always the same, (thus independent of the relative motion between the light source and the observer). From this two assumptions follows the famous Lorentz transformations. In these Lorentz transformations ... 1 The concept of 'straight' is a bit ill defined in GR and has no real definition. In fact in a sense the geodesics themselves be seen as 'straight' lines; they are the shortest paths connecting 2 points (this is what in normal Euclidean space would be a 'straight line') In the LC connection they are the integral curves of some vector field V with  ... 1 The square root of the determinant of the metric can be understood as a particular function of the components of the metric g_{ab}$$\sqrt{-g} =f(g_{ab})$$By the chain rule we of course have$$\nabla_a \sqrt{-g} = \nabla_a f(g_{bc}) = \frac{df}{d g_{bc}} \nabla_a g_{bc} But we know that $\nabla_a g_{bc}=0$ so that of course $\nabla_a \sqrt{-g} =0$. This ...

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Exactly as an ideal clock at rest with the observer (here pictured as a timelike curve) measures the proper time of the observer, ideal rulers at rest with the observer measure the distances in the rest space of the observer. Mathematically these rulers are pictured as an orthonormal basis made of $3$ vectors normal to the unit tangent vector to the ...

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