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How can I call a transformation where only the $z$-axis is modified by a function in each point?

In general, a map is conformal if the Jacobian is some positive scalar times an orthonormal matrix, so that it locally preserves angles between curves. Your suggested map is $$\begin{pmatrix} x' \\ y'\...
mike1994's user avatar
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Generalized momentum

The total momentum $P=\sum \partial L/\partial x_i$ in flat Euclidean space in Cartesian coordinates is conserved, if the Lagrangian $L=T-V(x_1,x_2,\dots x_n)=T-V'(R,x_1-x_2,\dots ),$ rewritten in ...
Roland F's user avatar
2 votes

Generalized momentum

Generalized momentum $p^i$ is conserved if the Lagrangian function $\mathscr{L}$ doesn't depend on the generalized coordinate $q^i$. It's readily proved from the Lagrange equation, $$\frac{d}{dt} \...
basics's user avatar
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Differential form of Lorentz equations

How do these differentials follow from the Lorentz transformations mathematically? And is there a chain rule involved? You simply take the total differentials of the Lorentz transform formula you ...
Dale's user avatar
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2 votes

Change of variables from FRW metric to Newtonian gauge

I'll show how you can do this with Mathematica very quickly. If you don't have access to Mathematica through your institution I believe WolframCloud is free. We're starting with the FRW metric $$ds^2 =...
delon's user avatar
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How to transform coordinates in the Action equation?

This could be the Lagrangian from Lenny's point of view, if: the transformation of coordinates is $$x = X + f$$ and you wrote the wrong sign in your question the potential is written with some abuse ...
basics's user avatar
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2 votes
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Radial reparametrization ansatz in Schwarzschild metric derivation

If we start from the Ansatz $$ds^2 = -e^{\alpha(\rho)} dt^2 + e^{\beta(\rho)} d\rho^2 + e^{\gamma(\rho)} ( d\phi^2 + \sin^2(\phi) d\theta^2) , $$ (This is equivalent with the OP's original Ansatz as ...
TimRias's user avatar
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Confusing Goldstein Statement about Magnitude of the Lagrangian

Briefly speaking, Goldstein is trying to convey that the Lagrangian $L(q,\dot{q},t)$ is invariant under passive change of generalized coordinates, cf. e.g. this Phys.SE post; while the Hamiltonian is ...
Qmechanic's user avatar
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1 vote

Change of variables from FRW metric to Newtonian gauge

$\def \b {\mathbf}$ from $$ ds^2=-dt_c^2+a(t)\,dx_c^2 $$ $$ds^2=\underbrace{\begin{bmatrix} dt_c & dx_c \\ \end{bmatrix}}_{\mathbf q_p^T} \underbrace{\begin{bmatrix} -1 & 0 \\ 0 & a(...
Eli's user avatar
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6 votes
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Change of variables from FRW metric to Newtonian gauge

Firstly as you suggested, $$ d t_{c} = \frac{\partial t_{c}}{\partial t} d t+\frac{\partial t_{c}}{\partial \vec{x}} d \vec{x}$$ $$d t_{c} = d t-\frac{1}{2} \dot{H}(t) \vec{x}^{2} d t-H(t) \vec{x} \,d ...
S.G's user avatar
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A worldline from the perspective of another in special relativity

I can give you a very simple physical description of how the rotation comes about and hopefully it will give you some insight for your investigations. My description is based on Wigner Rotation. This ...
KDP's user avatar
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1 vote

Removing the cosmic horizon in the de Sitter metric

You can remove the singularity by the same trick that removes the singularity of Schwarzschild coordinates: substitute $t = u + f(r)$ for some $f$. Taking $f(r) = \ell \tanh^{-1} (r/\ell) - r$ gives ...
benrg's user avatar
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2 votes

Birkhoff's theorem and Schwarzschild vacuum solution

JanG wrote: »Schwarzschild solution satisfies these conditions only for $r>r_S$« That's not true, the Schwarzschild metric is spherically symmetric all the way down to the singularity. Even if you ...
Yukterez's user avatar
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5 votes

Physical Quantities Sign Convention

For a vector quantity $u\,\hat u$, if the component $u$ is negative it means that the vector is in the $\hat u'= -\hat u$ direction, ie in the opposite direction to $\hat u$. Does negative velocity ...
Farcher's user avatar
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3 votes

Physical Quantities Sign Convention

One thing is a sign. Another thing is an operator. The negative sign means the value/vector/quantity in "the opposite direction", as you suggest. The subtraction operator causes reduction ...
Steeven's user avatar
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6 votes

Physical Quantities Sign Convention

Other answers have answered the direct questions in the OP. Here, I'd like to expand on the general question a little bit and talk about the fact that negative signs mean different things in ...
march's user avatar
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3 votes

Physical Quantities Sign Convention

If quantity $\vec q$ is a vector (like velocity, acceleration, force), then quantity $- \vec q$ is simply same vector $\vec q$ scaled by $-1$, so that now $$ -\vec q = \left[ \begin{matrix} -q_x\\ -...
Agnius Vasiliauskas's user avatar
1 vote

Physical Quantities Sign Convention

Any change in either the magnitude or direction of velocity is acceleration. Since velocity is a vector $\vec v$ its negative is a vector pointing in the opposite direction. They are often ...
Bob D's user avatar
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1 vote

Physical Quantities Sign Convention

Negative velocity means it is moving in the opposite direction, however negative acceleration just signifies that its velocity is decreasing at the given rate, to give some examples: If something is ...
The Burger King's user avatar
1 vote

Physical Quantities Sign Convention

Velocity is a vector, so minus means head of the vector arrow in the direction of decreasing axis values. Similarly for acceleration and force. But for something to accelerate oppositely still having ...
Radek D's user avatar
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0 votes

Interior Solution for Black Hole in Particular

This is a weird way of phrasing, but it seems about right. The interior metric for a Schwarzschild black hole can be written as $$\mathrm{d}s^2=-\left(1-\frac{2M}{r}\right)\mathrm{d}t^2+\left(1-\frac{...
Níckolas Alves's user avatar
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How a scalar component can be negative?

A scalar can totally be signed, in which case the minus sign indicates some sort of state rather than spatial direction. For example, an object at $-4^\circ C$ whose rate of change of speed is $-0.5$ ...
ryang's user avatar
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Can you tell who is moving through time?

By definition every observer is moving through time. In fact, in his own reference frame, am observer only moves through time, and not space. This time is called his Proper Time. If you are asking the ...
RC_23's user avatar
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Can you tell who is moving through time?

Special relativity has two main postulates which are true for any relativistic problem In any reference frame, light travels at the velocity $c$. Every reference frame is equally valid and no ...
Ronny's user avatar
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2 votes

Can you tell who is moving through time?

However, can you know who is moving through time? In the sense that this phrase has meaning, every massive object is moving through time at $c$. Could you just turn on a flashlight while you are ...
Dale's user avatar
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Can electric field be negative?

An electric field can be negative. The definition of the electric field is the force experienced by a positive charge in the field. the source particle is negatively charged, then the positive test ...
KDP's user avatar
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2 votes
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Number of independent reparametrization gauge invariances of the 'world $(n+1)$-manifold action' of $n$-dimensional objects

GSW is merely stating the fact that an $(n+1)$-dimensional world volume has $(n+1)$-dimensional local coordinate systems, or equivalently, is parametrized by $n+1$ independent parameters $\sigma^{\...
Qmechanic's user avatar
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3 votes

Event horizon is a null surface - what about the angular coordinates?

Two wrongs apparently do make a right. First, you’re using bad coordinates for the event horizon, and second, as a result, you divided by $0$ by plugging in $r=2m$. But, somehow you got that the ...
peek-a-boo's user avatar
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