Recent content by Thales Castro

Yeah, R is an arbitrary fixed radial coordinate (not necessarily the Schwarzschild radius 2M) in which the circular orbit takes place. r is the radial coordinate (in the same sense I would have an x coordinate and a fixed point x_{0})

Thanks for the reply. That makes sense. One thing that I discussed with my colleagues later is that the relation between Omega and R is not necessarily the one I gave in the text. The rocket could be using his propulsion to keep himself at the "wrong" speed in his orbit, and in this his...

In a circular orbit, the 4-velocity is given by (I have already normalized it) $$ u^{\mu} = \left(1-\frac{3M}{r}\right)^{-\frac{1}{2}} (1,0,0,\Omega) $$Now, taking the covariant derivative, the only non vanishing term will be $$ a^{1} = \Gamma^{1}_{00}u^{0}u^{0} + \Gamma^{1}_{33}u^{3}u^{3} $$...

I did the first part using the transfer matrix method: $$ Z = Tr(T^{N}) $$ In this case, the transfer matrix will be $$ T(i,i') = \begin{pmatrix} e^{\beta J} & 1 & e^{-\beta J}\\ 1 &1 &1 \\ e^{-\beta J} & 1 & e^{\beta J} \end{pmatrix} $$ To get the trace of $T^N$, you find the...

So, for the end of this semester's introductory couse in General Relativity (undergrad) I have to do a project on "The analogy between the mechanical laws on a black hole and the laws of thermodynamics". I couldn't find much (at least on my own) about this particular topic in my university's...

Ok, I just realized that the Hamilton-Jacobi Equation was already giving me the derivative of ## \bar{S}(t) ## :rolleyes:

Homework Statement A particle moves on the ##xy## plane having it's trajectory described by the Hamiltonian $$ H = p_{x}p_{y}cos(\omega t) + \frac{1}{2}(p_{x}^{2}+p_{y}^{2})sin(\omega t) $$ a) Find a complete integral for the Hamilton-Jacobi Equation b) Solve for ##x(t)## and ##y(t)## with...

Thanks for the reply! I understand that it is much easier in this case to just find the inverse map. But, for the sake of the exercise, I still want to find a proof that the maps are surjective, I just don't know how to formally write it.

Homework Statement The n-dimensional Euclidean group ## E^{n} ## is made of an n-dimensional translation ## a: x \mapsto x+a ## (##x,a \in \mathbb{R}^{n}## ) and a ## O(n) ## rotation ## R: x \mapsto Rx ##, ##R \in O(n) ##. A general element ## (R,a) ## of ## E^{n} ## acts on ## x ## by ##...

Thanks for replying. So, from your equation, I have (Since my textbook doesn't assume ## c = 1 ## for this exercise, I'll just keep it there): $$ mc^{2} = dE + (m+dm) \gamma(dv') c^{2} \implies dE = mc^{2} - (m+dm) \gamma (dv') c^{2} $$ Assuming ## p = \frac{u E}{c^{2}} ## and using...

Problem statement: A rocket propels itself rectilinearly by giving portions of its mass a constant (backward) velocity ## u ## relative to its instantaneous rest frame. It continues to do so until it attains a velocity ## v ## relative to its initial rest frame. Prove that the ratio of the...

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