I am trying to answer exercise 5 but I am not sure I understand what the hint is implying, differentiate with respect to ##p_\alpha## and ##p_\beta##, I have done this but nothing is clicking. Also, what is the relevance of the hint "the constraint ##p^\alpha p_\alpha = m^2c^2## can be ignored...
Since ##\nu## is contracted, we form the scalar product of the metric and inverse metric,
##g_{\mu\nu}g^{\nu\lambda} = (\vec{e_\mu} \cdot \vec{e_\nu}) \cdot (\vec{e^\nu} \cdot \vec{e^\lambda}) = \vec{e_\mu} \cdot (\vec{e_\nu} \cdot \vec{e^\nu}) \cdot \vec{e^\lambda} = \delta^\lambda_\mu##
I...
I have read many GR books and many posts regarding the title of this post, but despite that, I still feel the need to clarify some things.
Based on my understanding, the contravariant component of a vector transforms as,
##A'^\mu = [L]^\mu~ _\nu A^\nu##
the covariant component of a vector...
Since it is stated that ##E'_x = E_x##, I am going to set a special case where ##z' = z = 0##, ##E_x## in (5.10) reduces to,
##E_x = \frac{1}{4 \pi \epsilon_0}\frac{Q}{x^2}##
However, ##E'_x## in (5.13) reduces to,
##E'_x = \frac{1}{4 \pi \epsilon_0}\frac{Q}{\gamma^2 x'^2}##
There is an...
So, here's an attempted solution:
With ##r_{min}##,
$$r_{min} = \frac{1}{B + \frac{\beta}{\alpha^2}}$$
With ##r_{max}##,
I get:
$$r_{max} = \frac{1}{B - \frac{\beta}{\alpha^2}}$$
or
$$r_{max} = \frac{1}{\frac{\beta}{\alpha^2}}$$
Other than this, I and the team have absolutely no idea on how...
Show that, according to relativistic physics, the final velocity ##v## of a rocket accelerated by its rocket motor in empty space is given by
##\frac{M_i}{M} = \Big ( \frac{c+v}{c-v} \Big) ^ \frac{c}{2 v_{ex}}##
where ##M_i## is the initial mass of the rocket at launch (including the fuel)...
Below, I have already solved - I assume - correctly for question 1. Question 2, I am nearing to what I believe is the solution. Question 3, I simply have no idea where I should begin considering that it is interconnected with question 2.
With that said, below is the lengthy and somewhat tedious...
I'm struggling to get the hang of killing vectors. I ran across a statement that said energy in special relativity with respect to a time translation Killing field ##\xi^{a}## is: $$E = -P_a\xi^{a}$$ What exactly does that mean? Can someone clarify to me?
To begin with, I posted this thread ahead of time simply because I thought it may provide me some insight on how to solve for another problem that I have previously posted here: https://www.physicsforums.com/threads/special-relativity-test-particle-inside-suns-gravitational-field.983171/unread...
Below is an attempted solution based off of another user's work on StackExchange:
Source: [https://physics.stackexchange.com/questions/525169/special-relativity-test-particle-inside-the-suns-gravitational-field/525212#525212]
To begin with, I will be using the following equation mentioned in...
Two four-vectors have the property that ##A^\mu B_\mu = 0##
(a) Suppose ##A^\mu A_\mu > 0##. Show that ##B^\mu B_\mu \leq 0##
(b) Suppose ##A^\mu A_\mu = 0##. Show that ##B^\mu## is either proportional to ##A^\mu## (that is, ##B^\mu = k A^\mu##) or else ##B^\mu B_\mu < 0##.
Part (a) is...
Right, so I thought I'd done this correctly but clearly not because my velocity is greater than the speed of light, where have I gone wrong?
P = (M, 0, 0, 0)
p1 = (E1, p1x, p1y, p1z)
p2 = (E2, p2x, p2y, p2z)
P = p1 + p2
p2 = P - p1
square each side
to get (p2)2 = P2 - 2Pp1 + p12
therefore
(m2)2...
I've tried using gammamc^{2} = E1 + E2 but how do i find gamma?? If i try to use the kinetic energy then I just get gammamv^2 = 1gev but i dont know v? very confused
In A.P. French's Special Relativity, the author said the following,
As I understand, photons are massless, so I don't think the last equation above applies to photons, but then, when deriving it, he used an equation proper to photons (##E=pc##).
So in which context is ##m=p/c## valid?
According to the 2nd postulate of Special Relativity, speed of light in vacuum is the same in all inertial reference frames.
If I take a beam of photons and see the other photons in the beam from a frame of reference of a single photon, do they look stationary or moving at the speed of light...
This is my first thread. I hope I do it right. I just started reading the book Special Relativity by W.Rindler. And as I was reading it, I stumbled upon a pickle. So in Lorentz theory, it says, supposedly we could measure the original to-and-fro time T2 directly with a clock, and suppose we...
Recently, I've been studying about Lorentz boosts and found out that two perpendicular Lorentz boosts equal to a rotation after a boost. Below is an example matrix multiplication of this happening:
$$
\left(
\begin{array}{cccc}
\frac{2}{\sqrt{3}} & 0 & -\frac{1}{\sqrt{3}} & 0 \\
0 & 1 & 0 & 0...
A problem of an isolated system's velocity in different inertial systems, in special relativity
##\ \ \ \ \ ## In the inertial reference frame ##K##, the velocity of each component of an isolated system in ## x## direction is zero at a certain time.
##\ \ \ \ \ ## The inertial reference frame...
[BEGINNGING NOTICE]
Before I begin showing my attempted solution, I would just like to quickly mention that this is a "repost" of the same question I had around a week ago. While I would usually use the "reply" function on the same thread, I believe that thread is getting pretty messy (sometimes...
Below is the attempted solution after researching the contents available on Introduction to Electrodynamics by Griffith.
To begin with, I defined the rod as having a length of ##l'## at rest in frame ##S'##. Thus, in frame ##S'##, the height of the rod is ##l' sin(\theta ')## and its horizontal...
"My" Attempted Solution
To begin, please note that a lot - if not all - of the "solution" is largely based off of @eranreches's solution from the following website: https://physics.stackexchange.com/questions/369352/scalar-invariance-under-lorentz-transformation.
With that said, below is my...
Summary: The problem is to generalize the Lorentz transformation to two dimensions.
Relevant Equations
Lorentz Transformation along the positive x-axis:
$$ \begin{pmatrix}
\bar{x^0} \\
\bar{x^1} \\
\bar{x^2} \\
\bar{x^3} \\
\end{pmatrix} =
\begin{pmatrix}
\gamma & -\gamma \beta & 0 & 0 \\...
Unfortunately, I am not entirely confident of the above equations being able to do the trick and ultimately solve for the question. However, my guess is that using the equation written above for "boost", I could perhaps use ##v## and insert it into the ##x##-direction part of the matrix...
Homework Statement: Problem:
The planet X is far 48 light-years from Earth. Suppose that we want to travel from Earth to planet X in a time no more than 23 years, as reckoned by clocks aboard our spaceship. At what constant speed would we have to travel? How long would the trip take as reckoned...
I have been getting back to studying physics after a long break and decided to go through the problems in Rindler. But there is something I don't quite understand in this problem.
To first answer the second part, Exercise II(12), I wrote $$\frac{du_2}{dt} = \frac{du_2}{du_2^\prime}...
Clarification:
The statement in the title is actually from the solution to the homework question, as given by the textbook (you can see the whole thing below under "Textbook solution"). The solution doesn't explain everything, which is where my confusion comes from. Usually in my classes we...
I'm a bit lost at how to exactly start this exercise... As far as I understand we need to first determine ##d\tau_E## and ##d\tau_S##.
First question: Since we can neglect the earths movement, can I also neglect the movement of the satellite with respect to the far away observer? If so, I...
I'm struggling in the details of this exercise. Let ##S'## be the reference frame where the acceleration of the spaceship is constant, in which case we have ##u'(t')= a' t'## (since we assume no acceleration at the beginning). The rest frame of the rocket ##S## is connected to ##S'## via a...
Homework Statement: A spaceship has two clocks: one in the front and one at the back. The clocks are synchronized in the spaceship's frame of reference.
The spaceship zooms past the earth's surface at a relativistic speed. Prove that in the earth's frame of reference, the clock at the front...
Hello, I have a couple of questions related to reference frames in Special Relativity.
Let's consider a rocket that is inertially moving towards a star with a relative velocity 0.9c.
I'd like to look at this example from both the rocket's and the star's perspectives.
In the reference frame of...