12
votes
Accepted
Why is relativity of simultaneity so special?
The crucial mistake you are making is to focus on the reception of the signals, not the events that generated them. Clearly Sally receives the blue and red signals at different times, and that would ...
- 22.9k
9
votes
Why is relativity of simultaneity so special?
For one thing, in the sound example the pressure waves are moving through the air or some fluid medium. You could choose a reference frame where the bulk motion of the fluid with respect to the ...
- 6,332
7
votes
Accepted
Vector addition for differentials in the context of electric potential
I think that your professor is showing the differential vector for infinestimal change in each coordinate component.
The diagrams correspond to cartesian, cylindrical and spherical coordinate systems ...
7
votes
Accepted
Why is $dt/d\tau=\gamma$? What is $dt/d\tau$ supposed to mean exactly?
$$
d\tau= \sqrt{dt^2-dx^2-dy^2-dz^2}
$$
is the infinitesimal increment of proper time $\tau$ along a timelike trajectory
$(x(t),y(t),z(t))$ parametrized by the coordinate time $t$. This is standard ...
- 48.1k
5
votes
Partial derivatives vs Covariant derivatives in polar coordinates
Covariant derivatives take into account for both component and basis changes, thereby applicable for curved spaces - where partial derivatives only take component changes into account - is this ...
4
votes
Accepted
General Lorentz Transformation in Special Relativity
In units where $c=1$ a Lorentz boost in a general direction is: $$\left(
\begin{array}{cccc}
\gamma & -\gamma v_x & -\gamma v_y & -\gamma v_z \\
-\gamma v_x & \frac{(\gamma -1) ...
- 86k
4
votes
Accepted
Is my understanding of canonical transformations flawed?
Hint: In symplectic notation
$$z^I~=~(q^i,p_i)\qquad\text{and}\qquad Z^I~=~(Q^i,P_i),$$
and assuming no explicit time dependence in the transformation $Z^I=f^I(z)$,
OP has shown that
$$\begin{align}\...
- 188k
3
votes
Partial derivatives vs Covariant derivatives in polar coordinates
As OP correctly points out connections introduce a concept of differentiation of tensor fields or more in general of sections of vector bundles that takes into account how the bases of the fibers ...
- 1,607
3
votes
Accepted
Why can the dot product of two vectors be expressed as a differential?
I would imagine that the simplest way to show this is to note that the position vector $\mathbf x$ can be expressed in either basis:
$$x'^j \hat e_j' = \mathbf x = x^i \hat e_i$$
A given set of ...
- 63.3k
3
votes
Difference and meaning of index the derivative operator
You can eventually (if you need to) learn a more rigorous treatment later, so let me instead provide a cookbook approach:
An object with an open index means that its value changes when the observer ...
- 769
3
votes
Accepted
How does the wavefunction transform under an arbitrary change of variables?
TL;DR: As the overall phase of the wavefunction is not physical, OP's question has a non-unique answer that ultimately comes down to a choice of convention. Within a given class of situations we often ...
- 188k
3
votes
Vector addition for differentials in the context of electric potential
The infinitesimal displacement $d\vec{s}$ is derived from $\vec{s}$.
In cartesian coordinates :
$\vec{s}=x\vec{i}+y\vec{j}+z\vec{k}$
$$d\vec{s}=dx\vec{i}+xd\vec{i}+dy\vec{j}+yd\vec{j}+dz\vec{k}+zd\vec{...
- 184
3
votes
The center of the Schwarzschild black hole
The metric in isotropic coordinates (1) and (2) has a (coordinate) singularity at $r=a$. Consequently, there is no a priori relationship between the metric for $r>a$ and $r<a$. They are both ...
- 9,741
2
votes
Accepted
Kinetic equation material derivative
Using the coordinate system in Fig 5.1, the electric field can be written as:
\begin{equation}
\mathbf{E} = E\left(\cos \vartheta \hat{e}_v - \sin \vartheta \hat{e}_\vartheta \right).
\end{equation}
...
2
votes
The center of the Schwarzschild black hole
Equation for $r$ in terms of $r'$ (in the question as originally posed) has them the wrong way round. In notation $r$=isotropic, $r'$=Schwarzschild coordinate it should be
$$r' = r (1 + a/r)^2 .$$
...
- 53.1k
1
vote
The angular momentum of zero mass limit of Kerr metric
The Kerr metric describes a rotating body. $a = J/Mc$ is the angular momentum per unit mass of the body. $a$ characterizes the rate of rotation.
In the limit $M \rightarrow 0$, the metric becomes the ...
- 33k
1
vote
Equating 2 sides of EFE
First, note that for scalar functions, the covariant derivative reduces to the partial derivative. So for scalar functions, it is true that if the covariant derivative is zero at a point, then the ...
- 42.9k
1
vote
Why can the dot product of two vectors be expressed as a differential?
First observe, that for any matrix, we can pick up its $i,j$ entry by applying it first to a column vector that is zero apart from $1$ at its $j$'th position and then dotting it into a similar vector ...
- 1,116
1
vote
Difference and meaning of index the derivative operator
Here is a brief summary:
In (relativistic) physics, it is standard to adorn a local coordinate $x^{\mu}$ with a superindex.
We define a shorthand notation for the partial derivative $\partial_{\mu}:=...
- 188k
1
vote
Why is $dt/d\tau=\gamma$? What is $dt/d\tau$ supposed to mean exactly?
It might be helpful to think in terms of geometry and trigonometry using "rapidity" (the Minkowski analogue of the angle between vectors), which is defined between two future-timelike ...
- 10.7k
1
vote
Is there an intuitive picture for quasi-isodynamic versus quasi-axisymmetric versus quasi-helically symmetric flow in stellarators?
So the basic idea of the stellarator is to produce a magnetic field that results in the "lines of force" being shaped in such a way that the total path around the long axis is roughly equal ...
- 8,023
1
vote
Why if the metric tensor components are constant then SR applies?
Let me summarize some ideas in the comments and the other answer and add an important point regarding frames that I think is interesting to keep in mind.
In Lorentzian signature, if you go to an ...
- 547
1
vote
Why if the metric tensor components are constant then SR applies?
The metric is diagonalizable, yes, and then with further coordinate transformations it can be converted to Minkowski; see, for example, Schutz exercise 6.3 where he guides you through the steps.
- 33
1
vote
Vector addition for differentials in the context of electric potential
The diagrams are slightly misleading, because the infinitesimal changes in angular quantities are shown as being quite large. It may be more helpful to draw separate diagrams showing how $\vec{s}$ ...
- 1,077
1
vote
Time dilation and understanding which is $\Delta t$ and which is proper time $\tau$
superman is not at rest
There is no preferential frame in relativity. The problem supposes both Louis frame and Superman frame as inertial, with a relative velocity of $0.7$c.
The two events (say ...
1
vote
Time dilation and understanding which is $\Delta t$ and which is proper time $\tau$
I've always found talk of an observer confusing in Special Relativity. Much clearer, in my opinion, to use the idea of an inertial frame of reference (frame, for short).
The proper time between two ...
- 32.7k
1
vote
General Relativity manipulating tensors, tensor indices meaning
About the first part, you nearly answered your own question: it is indeed exactly because the definition of tensors arises naturally out of multilinear maps such as $t(e^a,e^b)$ that the order of ...
- 1,116
1
vote
Direct conversion of cartesian velocity to spherical velocity and vice-versa
Notice that $\frac{d\theta}{dt}$ and $\frac{d\phi}{dt}$ do not have units of velocity (m/s), so right there you have a problem.
Velocity in the $\hat \theta$ direction is "how much distance $ds$ ...
- 6,332
1
vote
Direct conversion of cartesian velocity to spherical velocity and vice-versa
This is one of those things that will, upon closer inspection, turn out to be horrible and confusing. Let us compute some stuff, with suggestive notation chosen to make the understanding easier.
...
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