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Conformal Transformation are those active coordinate transformations (diffeomorphism) $\sigma^a \longrightarrow \sigma'^a=\sigma'^a(\sigma)$ that change the metric in the following form: $$g'_{ab}(\sigma')\equiv\frac{\partial\sigma^c}{\partial\sigma'^a}\frac{\partial\sigma^d}{\partial\sigma'^b}g_{cd}(\sigma)=\Omega(\sigma)g_{ab}(\sigma)$$ So a conformal ...


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Suppose there is a flash event that we can represent as a light cone as the flash expands. There are three shutters with detectors around this flash event. The shutters open and close only once and this is almost instantaneous. One shutter opens in spacetime outside the light cone. One Shutter opens in spacetime inside the lightcone and the third opens ...


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Your question is as follows: Why centripetal force does not increase the value of tangential velocity? Answer: Assume you have circular motion as in the case where a person, with her hand, twirls a ball on a string. The string connects her fingers to the ball as the ball travels in a circle around her hand. In this case, the force in the string is ...


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I think the OP has made a mistake in applying second law of Newton. The law (about a particle) says: $$\Sigma \vec F=m\vec a$$ As it is seen, this is a vector equation. This means that corresponding components of both side of the equation must be equal. Although it is not said in the law's body, but it is obvious that we must write the equation above with ...


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Centripetal force is the name of the force that points towards the centre. This is in the radial direction. Tangential velocity is, as the name suggests, a velocity direction tangent to the circle. The radial and tangential directions are by definition always perpendicular - in the same way that the x and y axes are. You are of course right that if any ...


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I think confusion of the OP is related to the concept of the frame of reference. Frame of reference is a position in the space not a coordinate system. In other word, frame of reference is the position that measurements are evaluated with respect to it and for this evaluation, one can use any kind of coordinate system. So, the book is right when says: ...


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I think the best way to think of it is as follows.(It's not too different from what everyone has said, but may be put into better perspective). Choosing a frame of reference is a completely different job than setting up of coordinates. To observe an event in spacetime you must belong to some frame of reference(or equivalently, you create a frame of ...


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Sorry for the late answer, but I was kept quite busy these days. First, let me note that I find their notation, especially in respect to coordinate systems, a tad confusing too. I will still try my best to comment on these issues. Concerning your first point - I am quite positive that $ \left( \frac{\partial}{\partial t} \phi^a \right) e^a$ is a simple ...


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Actually the result is even stronger: Given a timelike geodesic $\gamma$ and a point $p \in \gamma$, there is a neighborhood $U \ni p$ equipped with coordinates, $x^0,x^1,x^2,x^3$ such that in the portion of $\gamma$ included in $U$, exactly along $\gamma$, the derivatives of the metric vanish in the said coordinates. Equivalently the Christoffel symbols $\...


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Here is what I understand: if you have a particle at state $|x \rangle$, active translating it by $a$ means moving the particle to state $ | x + a \rangle$. Passive transformation means you keep the particle in the same place, and change the coordinate by new variable $x = x' + a$ (note that the coordinate system is translated backwards $-a$). I am not very ...


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Yes, this is usually what is meant by a negative longitude, it's just a convenient way to express a range that happens to straddle the zero-point. And in the context of the HiGAL survey you were looking at, this is precisely what is meant.



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