I'm looking at Schutz 7.4 where first he obtains the following expression for a geodesic:
$$ m \frac {dp_\beta} {d\tau} = \frac 1 2 g_{\nu\alpha,\beta } p^\nu p^\alpha $$
This means that if all the components of ##g_{\nu\alpha }## are constant for a given ##\beta##, then ##p_\beta## is also...
I have worked out (and then verified against some sources) that ##R^\theta_{\phi\theta\phi} = sin^2(\theta)##. The rest of the components are either zero or the same as ##R^\theta_{\phi\theta\phi} ## some with the sign flipped.
I was surprised at this, because it implies that the curvature...
I am trying to understand the stress-energy tensor. Say you have a gas of particles moving with random direction, but all with mass ##m## and velocity ##v## in a frame where the centre of mass is at rest. The contribution to the flux of x-momentum of a particle moving with ##\vec p = p \vec...
I understand that any source of gravitational waves loses energy, which is carried away by the waves. But since the waves are perturbations in spacetime rather than a physical field, they cannot carry energy the way photons do. I have read that this used to be a source of considerable...
Hi all -
I am trying to follow a derivation of the above. At some point I need to find gαβ for
gαβ = ηαβ + hαβ
with |hαβ|<<1
I am stuck. The text says
gαβ = ηαβ - hαβ
but I cannot figure out why. Can anybody help?
Dear all,
I am self-studying GR using A First Course in General Relativity by Bernard F Schutz. I am halfway through the course, trying to solve all the exercises. But I worry that I can solve maybe 80% of them, the remaining 20% I find them just too hard.
I know I am no genius, and I don't have...
If a large cloud of dust of constant ρ is moving with a given ##\vec v ## in some frame, then at any given time and position inside the cloud there should not be no net energy or i-momentum flow on any surface of constant ##x^i ## (i=1,2,3) because the particles coming in cancels those going out...
I understand gravitational waves as ripples in spacetime. As PeterDonis said in an earlier post,
" The 4-d spacetime geometry does not have to "propagate" anything; it just is."
On the other hand, this is block-universe speak. In this language, verbs that imply change are forbidden. We might...
I have a problem with static/non-static spacetime. The problem is that the notion of spacetime includes time itself, so how can it change with time?
Imagine an asteroid approaching the Earth-Moon system. The Earth-Moon system is a non-static spacetime, so presumably is giving off gravitational...
Hi all,
I can't seem to find any historic account of when those two theories got their (quite strange) names. Can someone point me to a source?
One of the reasons I am interested in this is the persistent idea in popularizing texts, especially from the 1970's, that Special Relativity could...
Hi,
I am having trouble understanding why Tij can be non-zero for i≠j. Tij is the flux of the i-th component of momentum across a surface of constant xj. Isn't the i-th component of momentum tangent to the surface of constant xj and therefore its flux across that surface zero? What am I...
This is taken from B. Schutz's First Course in General Relativity.
Am I alone in considering this an abuse of language? After all, energy and spatial momentum are NOT the same concept, although they may be both expressed in kg in the units used (in which c=1).
I would rather say that a...
I'd like to understand why i cannot seem to be able to define unit polar basis vectors. Let me explain:
We have our usual polar coordinates relation to Cartesian:
x = r cosθ ; y = r sinθ
if I define \hat{e_{r}}, \hat{e_{\vartheta}} as the polar basis vectors, then they should be...
Trying to understand the conservation of relativistic momentum, I thought of this problem.
It's very simple, and possibly my mistake is embarrassing so I apologize in advance :blushing:
Two particles of identical mass m and speeds v and -v in some frame S collide inelastically and they stop...
This is probably a stupid mistake I am making, but I can't figure it out. My apologies in advance...
I am familiar with the text-book derivation of the Lorentz transformation (I don't have any problem with it). It starts out stating:
x2+y2+z2-c2t2 = x'2 + y'2+z'2-c2t'2
meaning that a...
I have been puzzled by this for years, so I would welcome some enlightenment.
It seems that Einstein was enamored with Mach's principle while searching for GR, but in the end GR does not seem compatible with it - or rather has nothing to say about it.
What I mean is that the proverbial...
I keep on reading that cosmologists contemplate two possibilities: either the universe is closed, unbounded and therefore finite, or else it is open (possibly flat) and infinite. I hear that a flat, finite universe "introduces many problems" and is discarded. My question would be "what...
I'd like someone to confirm whether I am on the right track here.
Most formulations of the twin paradox involve a sharp turn-around with infinite acceleration. I suppose that there is an SR-only description of a non-infinite acceleration - a kind of 'smooth' version of the twin paradox. But my...