ok, I am quoting from Schutz here:
"For instance, suppose we have a stationary gravitational field. Then a coordinate field can be found in which the metric components are time independent, and in that system ##p^0## is conserved [...]. The system in which the metric components are stationary is...
My problem is the way the thing is presented. The lab on Earth is given as an example of
$$ m \frac {dp_\beta} {d\tau} = \frac 1 2 g_{\nu\alpha,\beta } p^\nu p^\alpha $$
Which in turn was obtained from ## p^\alpha p^\beta_{;\alpha}=0 ##, i.e. a geodesic.
Maybe the result above is valid for any...
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...
It does not mean that. The pressure is acted upon the (imaginary) pressure detector of unit magnitude. The particles do not have to collide. Pressure in a gas does not come from particles colliding with one another. Statistically, there will be collisions in a real situation, but that is not the...
Why? Forget the cuboid. Just imagine a pressure sensor (which is placed perpendicular to the x axis at any point inside the gas) being hit by both particles.
I don't think that the idealization works at the edges of the volume of gas. In fact, a gas of particles with random velocities would...
I found this exercise from Schutz:
4.22 Many physical systems may be idealized as collections of *noncolliding particles* (for example, black-body radiation, rarified plasmas, galaxies and globular clusters). By assuming that such a system has a random distribution of velocities at every point...
If the net momentum is zero for my two particles, bouncing or not, how is there a momentum flux to account for the non-zero component of the tensor? Because it's obviously not zero. ##T^{11}## is by definition x-momentum flux. According to Pencilvester it should be zero.
But if the two particles in my example collide and bounce back, isn't that situation indistinguishable from the one where each particle goes on its own way? Both represent the exact same transfer of momentum between the two boxes you alluded to.
If the momentum cancels out, where does the pressure come from, if not from adding up the contribution from the particles traversing the x=constant surface?
I am probably wrong (that's why I am asking!). But I thought ##T^{11} = T^{22} = T^{33} = p## where p is the pressure in a perfect fluid in the MCRF. The ##2pv/A## would correspond to ##T^{11}##
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...
Do you mean that each inertial frame may have its spatial coordinates rotated in its own way, giving different coordinate values but the same magnitude to the observed acceleration vector?
What I mean is that ##T_{\alpha\beta}=0## away from the source. If the wave contains energy then ##T_{\alpha\beta}\neq0## at that point, leading to a contradiction.
No, not really. I am aware of the sticky bead thought experiment. I want to reconcile that in my mind with the fact that spacetime is not a physical field.
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...
Okay. Thank you very much. I think that my interpretation of the definition of the stress energy tensor was wrong. I assumed that the flow of α-momentum across a surface of constant ##x^β## meant the net flow of the α-momentum in an element of volume in the β direction. That was an unwarranted...