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2. ### MTW Exercise 25.5 b) - killing vectors

Yes there is a much shorter version using Lie derivatives indeed...
3. ### MTW Exercise 25.5 b) - killing vectors

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 =...

5. ### MTW Exercise 25.5 b) - killing vectors

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...
6. ### MTW Exercise 25.5 b) - killing vectors

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] =...
7. ### 'Expansion' of fluid world lines

Great . Thanks ! I have Wheeler and Ciufolini's "Gravitation and Inertia". I will check that out. Thanks for mentioning that.
8. ### 'Expansion' of fluid world lines

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...
9. ### Riemann Tensor Derivation

You may also see the derivation in Dirac's book : General theory of Relativity under equation 11.1 if I'm not wrong
10. ### 'Expansion' of fluid world lines

Indeed this is what part b) mentions. Thanks !
11. ### 'Expansion' of fluid world lines

Oh I see Bill. I did not get to part b) yet .
12. ### 'Expansion' of fluid world lines

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...
13. ### Weyl tensor in 2 dimensions- confused

This is related to the last question of MTW ex 21.21
14. ### Weyl tensor in 2 dimensions- confused

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...
15. ### Weyl tensor in 2 dimensions- confused

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...
16. ### Riemann curvature scalar, Ricci Scalar.What does they measure ?

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...
17. ### Riemann curvature scalar, Ricci Scalar.What does they measure ?

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.
18. ### Riemann curvature scalar value different in Landau and MTW ?

I see Bill Thanks. Not so obvious though .
19. ### Riemann curvature scalar value different in Landau and MTW ?

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...
20. ### Good books on both Relativities

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...
21. ### Magnetic field of an electron and general relativity

Perhaps this important paper will shed some light on the subject ...

23. ### Question about inertial frames

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...
24. ### Correct GR publishing date?

You will need to consult the bible on GR history : https://www.amazon.com/dp/1402039999/?tag=pfamazon01-20
25. ### Determinant of the electromagnetic matrix

Typo : that should have been (2!*2!)*2 and not 2!*2!*2! - sorry
26. ### Determinant of the electromagnetic matrix

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...
27. ### Determinant of a 4x4 tensor

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...
28. ### Determinant of the electromagnetic matrix

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...
29. ### Do not become a scientist

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 !
30. ### Mass bending space makes me think there's another side to space

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...
31. ### Mass bending space makes me think there's another side to space

watch this :It may clarify the mathematics... http://www.youtube.com/watch?v=LoaOHvy5AcA
32. ### Can we extract energy from a gravitational field ?

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...
33. ### Can we extract energy from a gravitational field ?

indeed Bill, it has been a while...
34. ### Can we extract energy from a gravitational field ?

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...
35. ### Can we extract energy from a gravitational field ?

indeed, f=-kx
36. ### Can we extract energy from a gravitational field ?

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 ...
37. ### Can we extract energy from a gravitational field ?

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...
38. ### Can we extract energy from a gravitational field ?

Thanks for your reply. This is section 20.4, I mean chapter 20, section 4
39. ### Can we extract energy from a gravitational field ?

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...
40. ### MTW Exercise 19.3 last part

Full solution could be found in Landau and Lifgarbagez page 359 for those interested...
41. ### Raising and lowering indices in Linearized theory

Thanks Bill . I'm just trying to convince the little mathematician in my head ...
42. ### Raising and lowering indices in Linearized theory

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.
43. ### Raising and lowering indices in Linearized theory

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...
44. ### MTW Exercise 19.3 last part

I may have found it after a long struggle : we have according to Equation 7.4 of T'hooft GR lectures (...
45. ### MTW Exercise 19.3 last part

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...
46. ### MTW, Chap19, Stationary relativistic Vs Weakly gravitating source

thanks very much to you all. It is now clear.
47. ### MTW, Chap19, Stationary relativistic Vs Weakly gravitating source

But what does it mean a relativistic source - this is the term that got me confused...
48. ### MTW, Chap19, Stationary relativistic Vs Weakly gravitating source

If you look at the following document ...
49. ### MTW, Chap19, Stationary relativistic Vs Weakly gravitating source

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...
50. ### Metric tensor - index manipulation

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.