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If the observer is not in free-fall, the metric-tensor $g_{\mu,\nu}(s)$ at the observer's position, expressed in local coordinates around the observer, will not be $\eta_{\mu,\nu}$. Your first assumption about the path $(\gamma)$ is wrong. I guess what you are aiming at is the notion of the space of coordinates around a point, which is indeed a flat space ...

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Let $M$ be your spacetime, a smooth manifold equipped with (pseudo) Riemannian metric (for example $\mathbb{R}^{(1,3)}$ for special relativity). The set of reference frames is the frame bundle over $M$, usually denoted $FM$. Explicitly a frame at point $p$ in $M$ can be viewed as an ordered orthonormal basis (with respect to the the inner product defined ...

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you can either evalute the integral numerically or search for a good coordinate transformation to evaluate it "by hand". This transformation is the one to spherical coordinates, though. Also you could try this: \begin{align} Q & = 2 \pi A \int_{-\infty}^{\infty}\int_0^{\infty} r \cdot H(R^2-r^2-z^2)~dr dz \\ & = 2 \pi A ...

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I) Many of OP's questions on how the Lagrangian formalism works is already addressed in e.g. this Phys.SE post and links therein. For instance the question about the total time derivative in the EL equations is discussed in my answer. II) In this answer, we would like to explain mathematically the various definitions in the Lagrangian formalism (of ...

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You were pretty close already. There is a handy table on Wikipedia for a variety of coordinate systems. But for the polar system: $$\vec{\nabla} \cdot \vec{U} = \frac{\partial U_r}{\partial r} + \frac{1}{r} \frac{\partial U_\theta}{\partial \theta}$$ and you can look up the curl in the same table. These can be derived from the Cartesian definitions by ...

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The columns of a 3×3 rotation matrix contain the coordinates of the local xyz axes (expressed in world coordinates). With Euler angles (321) you apply the elementary rotations $R_Z$, $R_Y$, $R_X$ in sequence to form the local → world rotation matrix. That is $$E = R_Z(\varphi) R_Y(\psi) R_X(\theta)$$ This is interpreted as each rotation occuring about ...

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Let's say you do the calculation $C_{final}=R_xR_yR_zC_{o}$, where the $C$s are coordinates. The first rotation is about the $z$ axis of $C_{o}$ and will produce a new coordinate system in which the new $z$ is the same as the old, but the $x$ and $y$ axes are different. The next rotation will be about the new $y$ axis and will produce a newer, new $x$ and a ...

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Based on Einstein's assertion: All our well-substantiated space-time propositions amount to the determination of space-time coincidences {such as} encounters between two or more material points. the notion "reference frame" should likewise be expressed in terms of (requirements on) "material points" and "space-time coincidences" in which they did, ...

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So based on this, what's really a reference frame? In pre-relativistic mechanics, reference frame is a system of points whose mutual distances are assumed constant - a rigid body. For measurements of position on Earth, the reference frame is often Earth's body, assumed to be rigid. For measurements of position in space, Earth's body could be used. ...

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It would help to read this section of Galilean Invariance as this reconciles it nicely with more intuitive notions of relative frames of reference. Two observers moving at different speeds (or help us all accelerations), would not agree on the simultaneity of some events. This represents a shift in relative time due to motion, which is why you see a $vt$ ...

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The most direct and beautiful answer to this important question has been provided by Taylor and Wheeler in their famous book "Spacetime Physics". If you google "spacetime physics wheeler frame of reference" for images, you will be led to a picture of space divided regularly into a 3-dimensional grid pattern. The crucial thing though is that at each grid ...

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You can also do the following, which may not be as general as you want it, but the idea might be usefull for other problems. You already know that the given metric is Minkowski metric in different coordinates, so look at the null geodesics. In the usual coordinates $(t',x')$ they are given by $x'\pm t'=const$. Then find the null geodesics in the given ...

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In the general case you want the Cartan-Karlhede algorithm. It is an algorithm for producing a complete set of classifying invariants for a metric, expressed as functions of the coordinates. Given the components of the metric $g$ in the coordinates $x_1, x_2, \ldots$, the algorithm produces a list \begin{align} \Lambda & = \Lambda(x_i) \\ \Psi_k & = ...

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If you were to Wick rotate $t \rightarrow i \theta$, the metric would be $ds^2 = dr^2 + r^2 d\theta^2$, which is just flat space in polar coordinates. The standard cartesian coordinates can be obtained by $x=r\cos\theta$, $y=r\sin\theta$. The same procedure works in the original Lorentzian signature metric, but with hyperbolic trig functions instead of sines ...

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