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In this nice reference the autor assumes the relativity principle + homogeneity + isotropy and deduce the general coordinate transformations which contains both Lorentz and Galileo transformations. Further he imposes the postulate of the constancy of the speed of light, restricting the transformations to be the Lorentz type.


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Diffeomorphism Invariance Let $M$ be a smooth manifold. Let $\phi: M \to M$ be a diffeomorphism. A simple property of the Einstein equations is $$ g \in \otimes^2 TM \text{ is solution to vacuum Einstein equation} \implies \text{ so is } \phi^*g $$ To see that this is true, simply pull back both sides of the Einstein equation by $\phi$, and use the ...


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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 ...


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How do we formally define vectors in physics? An excerpt from chapter one, page 12 of "Mathematics of Classical and Quantum Physics" Originally, we introduced a vector as an ordered triple of numbers. The rule for expressing the components of a vector in one coordinate system in terms of its components in another system tells us that if we ...


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A physical quantity is a vector if it transforms in the same way as a position vector when the coordinate system undergoes a transformation.


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The continuity equation is described by: $$ \frac{\partial \rho}{\partial t} + \nabla\cdot(\rho{\vec u}) = 0 $$ For incompressible and steady flow, it reduces to: $$ \nabla\cdot{\vec u} = 0 $$ The incompressible continuity equation in spherical coordinates is: $$ \nabla\cdot{\vec u} = \frac{1}{r^2}\frac{\partial}{\partial ...


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From those we can get $f(x(t),t) = x + Ct + D\,$ after integration. How can I interpret this $f(x(t),t)$? The constants of integration $D$ and $C$ are, respectively, the displacement between the origins of the two frames at time $t=0$ and the constant velocity with which the origin of the primed frame moves with respect to the origin of the unprimed ...


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Let two orthonormal systems $Oxyz$, $O'x'y'z'$ with a general motion (translational plus rotational) between each other and a point particle $\rm P$, see Figure. Symbol Conventions : 1.The vectors for position $\mathbf{R}$, velocity $\mathbf{U}$ and acceleration $\mathbf{A}$ of a particle with respect to $Oxyz$ expressed by coordinates of this same ...


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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 $$\tag{1} ds^2~=~g_{tt}(r)dt^2 + g_{rr}(r)dr^2 +r^2 d\Omega^2,$$ such that it asymtotically becomes Minkowski space $$\tag{2} ...


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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 ...


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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 ...


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Your mistake is that you left out the summation over double indices (which would give another delta). Since you have to sum over all possiblities of e.g. $c$, one of these possibilities is when c=d and only this term remains. I'll give a simplified example: $$\frac{\partial}{\partial{x^d}}({Q_{bc}}x^c) = \frac{\partial}{\partial{x^d}}({Q_{b0}}x^0) + ...


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$du,dv$ are light-like, i.e. they could, in principle, be viewed as some affine parameters of some light-rays. However, we will focus (in the spirit of the usual coordinate-nature analysis) on what is the nature of either $u,v$ constant. I.e., we want to know what is the nature of $u,v$ constant hypersurfaces and derive the nomenclature from this. We will ...


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The first figure shows the parallel and normal components of a vector $\mathbf{r}$ relatively to a direction $\mathbf{n}$. Based of this, the second figure shows the centripetal acceleration. In case of plane circular motion $\omega R = v$.


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Assuming that $\theta$ is the polar angle (angle between $\vec r$ and $\hat z$) and $\phi$ the azimuthal angle then the following relationships can be used. $x = r \;\sin \theta \;\cos \phi$ $y = r \;\sin \theta\;\sin \phi$ $z= r\; \cos \theta$


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You said this is the $r$-component, then you've missed the $\hat{\bf r}$. Use $$r=\sqrt{x^2+y^2+z^2}$$ $$\cos\theta=z/r=\frac{z}{\sqrt{x^2+y^2+z^2}}$$ $$\hat{\bf r}=\frac{x\hat{\bf x}+y\hat{\bf y}+z\hat{\bf z}}{r}$$


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Since there is no time dependence in the given equation (in other words, you have only the trajectory in terms of some parameter -- $\theta$ -- with no given connection to the time), you cannot arrive at the velocity by differentiating. Of course $k = \cot\alpha$ for a well-chosen $\alpha$ ... ($\alpha = \arctan(k^{-1})$) :) But why does this prove ...



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