# Tag Info

1 vote
Accepted

### How to obtain orthonormal tetrad basis for an infalling observer?

Let me give you the general idea. When constructing an orthonormal frame of an observer, you want to start with the zeroth leg, which is the four-velocity of the observer. In this case you are looking ...

### What is the difference between a local and global coordinate transformation?

$x \rightarrow x' = (1+\sigma)x$, acts identically on each point. Similarly, if $\epsilon(x) = kx$, then scaling transformation is global as well.

### Does the singularity travel along the time axis?

In a curved spacetime there is no unique coordinate chart that covers all of spacetime, so the whole notion of "the time axis" is ill-defined. Moreover, the singularity itself is not part of ...

### Does the singularity travel along the time axis?

We can't observe the singularity but we can observe the event horizon since we are attracted towards it. This comment reflects a Newtonian approach and neglects GR where the interpretation is that ...
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 ...
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 ...
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.

### 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 ...

### 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 ...
Accepted

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 $\vec{s}$ ...
Accepted

### Direction of Area Vector of a simple 2D figure

For a open surface, it is usually in that direction which make acute angle with electric field.
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 ...

### Time dilation and understanding which is $\Delta t$ and which is proper time $\tau$

Superman's wrist watch is at rest with respect to superman. So the time range measured by superman's wrist watch is smaller (i.e. the time runs slower for superman and for his watch) than the ...
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 ...

### Time dilation and understanding which is $\Delta t$ and which is proper time $\tau$

Whatever Superman does, he does right where Superman is. Superman kisses Lois goodbye right where Superman is. Twenty hours later in Superman's proper time $\tau$, Superman fixes the space probe right ...

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 vote
Accepted

### 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. ...

What I was missing is the notion of analytical continuation. While the coordinate is $r$ is always positive geodesics should be continued to the antipodal direction $\theta \rightarrow \pi-\theta,~~~\... 0 votes ### Find canonical transformation for$P$given canonical transformation for$Q\$

TL;DR: The answer depends on the context. Examples: If we assume that a canonical transformation here means a symplectomorphism, then indeed OP's method is correct: the determinant of the Jacobian is ...
Accepted

### Components of velocity 3-vector of light in spherical coordinates

$$v_r = c\\v_\vartheta=\vartheta_0\\v_\varphi=\varphi_0$$ Basically by definition, or visual inspection of how the relations are meant to work.
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} ...