Ok let us expand out your expression above :
(pμξμ),λ\frac{∂(p_{v}η^{v})}{∂p_{λ}}−(pvηv),λ\frac{∂(p_{v}ε^{v})}{∂p_{λ}} =
( pμ,λεμ + pμεμ,λ ) ( ηλ + pv\frac{∂(η^{v})}{∂p_{λ}} ) -
( pv,ληv + pvηv,λ ) ( ελ + pv\frac{∂(ε^{v})}{∂p_{λ}} )
Now we have : \frac{∂(ε^{v})}{∂p_{λ}} = 0 =...
hey Mate,
Thanks for your reply. Well it seems I got confused here. I had based my second commutation on the first one. In my very first attempt I had indeed written your expression but had forgotten that the other variable was x not either of the killing vectors ε or η...
Damn that got me...
hi there,
In this Ex ( see attached snapshot ), point b), the poisson bracket equation is not so straightforward to obtain.
Please correct my Poisson Bracket expansion here :
The first one which is provided is simpler :
[ε,η] = εμδμηρ - ημδμερ = ζη
and the monster one :
[pε,pη] =...
Indeed I saw it and also attempted to derive my own which yielded the correct result based on the continuity equation and on the assumption that the divergence of the density is negligible...
hello
In MTW excercise 22.6, given a fluid 4-velocity u, why the expression :
∇.u is called an expansion of the fluid world lines ?
Is the following reasoning correct ?
We know that the commutator : ∇BA - ∇AB is (see MTW box 9.2) is the failure of the quadrilateral formed by the vectors...
All N = 2 spaces are conformally flat.
This would mean that since the Weyl tensor vanishes for the conformal space whose Riemann tensor has the form [R], thus one can conclude that for N=2, the Weyl tensor is null.
This might make sense. But i do not know why the computation above did not...
hello,
The Weyl tensor is:
http://ars.els-cdn.com/content/image/1-s2.0-S0550321305002828-si53.gif
In 2 dimensions , the Riemann tensor is (see MTW ex 14.2):
Rabcd = K( gacgbd - gadgbc ) [R]
Now the Weyl tensor must vanish in 2 dimensions. However, working with the g
g =
[-1 0 0...
Thanks to all.
In wikipedia it is mentioned :
"Specifically, the scalar curvature represents the amount by which the volume of a geodesic ball in a curved Riemannian manifold deviates from that of the standard ball in Euclidean space"
Does this mean that the volume of the sphere in the curved...
hello
Can you perhaps explain what does the Riemann curvature scalar R measure? or is just an abstract entity ?
What does the Ricci tensor measure ?
I just want to grasp this and understand what they do.
cheers,
typo: What DO they measure in the title.
hello
For the same Friedmann metric, Landau (Classical theory of fields) finds a value for the Riemann curvature scalar which is given in section 107 :
R = 6/a3( a + d2(a)/dt2)
whereas in MTW , in box 14.5 , equation 6 , its value is :
R = 6(a-1 d2(a)/dt2 + a-2 (1 + (d(a)/dt)2 ) )
The...
Personally, I had to learn tensor calculus beforehand. Then Started with Schutz book, then Callaghan book on Space-time geometry, then the master of them all : Gravitation by MTW - but do not only read , you have to do every exercise in MTW. Do not spend one hour or two per Ex, if you can not do...
If you were in a lift in free fall on Earth (discard air drag ), and then you let go of an apple, it will follow a straight line. This is a local inertial frame, limited in time and space so that tidal effects are neglected...
There are also instantaneous inertial reference frames which can...
As always, I end up trying to answer my own question :) . So that others benefit .
We have for a skew symmetric tensor :
det(F) = Pf(F)2 , we would need to show that 1/8 εαβγδFαβFγδ = Pf(F)
Now since F is skew symmetric, we would find 2!2!2! = 8 terms which have similar (and...
hi there,
In this wikipedia article https://en.wikipedia.org/wiki/Electromagnetic_tensor
we have the following invariant :
FαβFμη εαβμη = 8 E*B
However the determinant is the square of this quantity divided by 8, i.e. ( E*B )2 .
Now from the definition of the determinant for a 4x4...
hi there,
In this wikipedia article https://en.wikipedia.org/wiki/Electromagnetic_tensor
we have the following invariant :
FαβFμη εαβμη = 8 E*B
However the determinant is the square of this quantity divided by 8, i.e. ( E*B )2 .
Now from the definition of the determinant for a 4x4...
I'm not saying this but rather this professor of Physics :
http://wuphys.wustl.edu/~katz/scientist.html [Broken]
Do you think he is right ? shouldn't he just shut up since many young people would get put off by this !
those lecturers, they spent their lives deciphering the universe's hidden secrets . They won nobel prices, field medals,..they have made advancement in science. Saying they do not know what they are talking about is utterly preposterous. These people deserve to be kings , deserve to lead the...
Indeed. It takes work to separate them : this solves the problem. They are gravitationally bound together. If this was not so they would have continued their paths unhindered...
I just wanted an explanation in terms of the spacetime curvature...not Newtonian physics.
this reminds me of to...
Yes then if these forces do not consume energy, then why the mass energy of the Earth moon system is less than the mass energy that the system would have if the 2 objects were at infinite separation (MTW Chapter 20 section 4)? this is non intuitive at all for me at least up to my very limited...
Because Earth is creating a spacetime curvature, which means there is energy in that location. Is this energy infinite ? does it get lost ? wasted ? somehow ?
If is exerting a gravitational force, then that means, there is energy being 'consumed' ? otherwise how would it maintain this force ...
I meant that it is subject only to gravitational forces...Imagine we tie a rope to an asteroid which is coming from infinity towards the sun. Gently, due to the curvature of the sun, the stiffness of the rope will change due to the Sun's gravitational forces...
Now if we attach a spring to the...
hey there
MTW mention that the mass energy of the Earth moon system is less than the mass energy that the system would have if the 2 objects were at infinite separation.
MTW say that it is due to gravitational forces.
Please correct me if I'm wrong : The fact that the moon follows a...
Remark : The product of the covariant g with its contravariant counterpart is assured to equal the unit tensor to first order In this case. See footnote of Landau & Lifgarbagez page 350.
Hi there,
in Linearized theory we know that :
gαβ = ηαβ + hαβ
If I multiply out both terms by ηθαηλβ, wouldn't one get :
gθλ = ηθλ + hθλ ? EQ1
But we already know gθλ = ηθλ - hθλ + O(h2) EQ2
How can we reconciliate EQ1 and EQ2 ? Was it an error to have raised the perturbation h with the η...
hello
In part 8) of this Ex, MTW mentions that the dominant non linear terms must be proportional to the square (M/r)2. The problem is that since I got the value :
h00 = A0/r + 6Qijninj/r3 (Qij is the quadrupole moment) and following the translation of coordinates suggested by MTW which is...
hello,
in MTW Ex 19.1, it was considered a Weakly gravitating body whereas in Ex 19.3, a Stationary relativistic source.
What relativistic means here ? does it mean the source is very dense and massive ? or does it mean it is rotating at near the speed of light ?
Can I specify the mass as the...
I see. this is related to my attempt to solving MTW Ex 19.2 where I need to find the ij0 component of the Christoffel Symbol with a zero time derivative of the gij.