All Questions
Tagged with differentiation inertial-frames
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Differential form of Lorentz equations
A Lorentz transformation for a boost in the $x$ direction ($S'$ moves in $+x$, $v>0$) is given by:
$$ t'=\gamma\left(t-v\frac{x}{c^2}\right),~x'=\gamma(x-vt)$$
In the derivation of the addition of ...
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Taylor expansion of scalar function for a coordinate infinitesimal transformation (Poincaré group)
For a coordinate infinitesimal transformation of the form $x^{\prime \mu} = x^{\mu} + a^{\mu} + \omega^{\mu}_{ \ \nu}x^{\nu}$, we want to derive its effect on a space of scalar functions $f(x)$. This ...
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Solving a PDE using $x-vt$ as a variable
So I was reading this Landau and Lifshitz paper:
https://doi.org/10.1016/B978-0-08-036364-6.50008-9
The article can also be found without a paywall by just searching its title, "On the Theory of ...
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Time derivative of a "general" vector $\vec A$ in an accelerating frame: what about e.g. velocity $\vec v$?
According to Morin "Classical Mechanics" (Section 10.1, page 459), the derivative of a general vector $\vec A$ in an accelerating frame may be given as
$$\frac{d\vec A}{dt}=\frac{\delta \vec ...
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Notation confusion about time derivative of a vector in a rotating frame
As far as I can tell, this question, or similar ones, have been asked a number of times:
Derivation of the time-derivative in a rotating frame of refrence
Time derivatives in a rotating frame of ...
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Derivation of a partial derivative equation by Albert Einstein in Special Theory of Relativity
I was reading Albert Einstein's "On the Electrodynamics of Moving Bodies". In section "Kinematical Part", on $3 (Theory of the transformation of coordinates and times from a ...
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Why can I write $\frac{d}{dt}=\frac{d}{dt'}\frac{dt'}{dt}+\frac{d}{dx'}\frac{dx'}{dt}$?
I’m dealing with a Lorentz invariance problem, and in one of the solutions I’ve seen to prove the wave equation the term above was used. However I don’t really understand why it can be written that ...
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3
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Derivative as a fraction in deriving the Lorentz transformation for velocity
Consider a frame $S$ and $S'$ which is coincides at $t=0$ and then $S'$ starts moving with velocity $v$ in $+x$ direction.
By Lorentz transformation equation,
\begin{align}
x'&=\gamma(x-vt) \\
...
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Special relativity - Einstein's transformations
I am reading "On the electrodynamics of moving bodies" and have got to page 6 and become stuck. Is anyone able to please help explain how:
Einstein went from the first line of workings to ...
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Differential form of the velocity equation in non-standard configuration
I'm reading a text on special relativity ($^{\prime\prime}$Core Principles of Special and General Relativity$^{\prime\prime}$, by James H. Luscombe, Edition 2019), in which we start with the equation ...
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Partial derivative of a 4-velocity
Trying to do some basic manipulations with 4-vectors and I have a question about the proper (no pun intended) approach. It's probably easiest if we look at a simple example. So let's define a 4-...
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Galilean transformation and differentiation
Given $x=x’-vt$ and $t=t’$, why is $\frac{\partial t}{\partial x’}=0$ instead of $1/v$? $t$ seems to depend on $x’$ because if $t$ changes, $x’$ changes. Also, in this problem, $dx=dx’$ as well, but I ...
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