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# Tag Info

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 ...

### 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 ...
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 ...
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 ...

### 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

### 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 ...
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 ...

### 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 ...
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 ...

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

### 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 ...
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 vote

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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