
#37
Jan407, 03:38 AM

P: 2,043

Complexify each dimension, then are you still convinced that the loop is closed? Wick rotations generally do not work in curved spacetime. 



#38
Jan407, 12:18 PM

Emeritus
Sci Advisor
P: 7,434

In fact, if you reread my remarks, you will see that I suggest that the ict formalism, which is what I assume you are referring to by "Wick rotations", is not generally used in GR. One says "goodbye to ict" and deals with only real numbers. So I'm feeling like there is straw all over the floor. (Straw all over the floor? What does that mean? It means that a strawman argument has been totally demolished.) Since it appears to be your strawman, MeJennifer, (it certainly isn't mine!) I suggest you sweep up his poor remains ... 



#39
Jan507, 03:09 AM

P: 2,043

You are effectively producing some kind of rotation over some plane, that causes the complex axis of the complex space coordinate plane to map over the real axis of the complex time coordinate plane. Perhaps my usage of the term Wick rotation is a bit to wide for this kind of rotation but nevertheless it is a rotation.
Hopefully it is uncontested that rotations over a plane consisting of the real axis from one complex plane and the imaginary axis of another plane in a curved manifold with a Lorentzian signature is at least "fishy". As soon as the space coordinate becomes imaginary in the van Stockum dust solution you have to ask in which direction the curve is heading. Obviously it is heading in the orthogonal direction of the space coordinate axis. But is the orthogonal direction of the space coordinate axis the time coordinate axis? If you think the answer is yes you should ask yourself why you reason as such, since there is nothing in the theory of relativity that either implies or excludes that. 



#40
Jan507, 10:57 AM

Sci Advisor
P: 2,341

Hi all,
This has been an amazingly confusing thread, but I sense that at least some readers with less experience working with gtr might be clearing up some misconceptions from reading some of the comments by those with more experience, so forging ahead, I have some comments on points I haven't yet addressed (I plead exhaustion): At the same time, though (no pun intended), it is important to recognize that some charts feature coordinates which do have a notable geometric or coordinatefree interpretation! In particular, consider the Schwarzschild time coordinate used in the interior or exterior Schwarzschild charts in the Schwarzschild vacuum (which cover an interior or an exterior region respectively). Here, the coordinate vector field [itex]\partial_t[/itex] is also a Killing vector field. In fact, it is uniquely determined by the stipulation that we choose the Killing vector field on the full Schwarzschild vacuum which is timelike in the asymptotically flat regions and also tends to unit length near "spatial infinity". This condition is independent of which coordinate chart we use. One of the many important advances which occured in gtr during the Golden Age (say c. 19591979, covering Bondi radiation theory through the positive energy theorem) was that it became common for researchers to use this kind of coordinatefree thinking to construct charts "adapted" to the symmetries they were assuming in constructing some solution. See Stephani et al., Exact Solutions of Einstein's Field Equations, 2nd. Ed., Cambridge University Press, 2001, for some fine expository chapters on coordinatefree methods and for many examples of this kind of construction of solutions. cesiumfrog added: "all the existing EFE solutions which don't yet have much physical interpretation"; I am probably just misreading this, but I wouldn't want anyone to get the impression that "no known exact solutions possess a reasonable physical interpretation", for this is certainly not true! To the contrary, there are important classes of solutions, such as the static spherically symmetric stellar models, which not only have an unobjectionable interpretation but are also useful in modeling real astrophysical objects. Examples of exact solutions with clear physical interpretations (including a clear understanding of the limits on their applications to realistic physical scenarios) include plane wave solutions, some null dust solutions such as the Vaidya null dust, many cosmological models such as the FRW models and various generalizations, colliding plane wave (CPW) models, etc. Then there are solutions which have clear interpretations in that it is clear what one is trying to describe, but which on closer inspection have physically objectionable features; these include Weyl vacuum solutions with "struts", the Van Stockum "rotating" cylindrically symmetric dust, RobinsonTrautman vacuums with "pipes", and so on. The Kerr vacuum is unobjectionable and realistic (for black hole models) in the exterior regions, and unobjectionable but perhaps unrealistic (for black hole models) in the "shallow interior" regions, but as several commentators have mentioned, it is objectionable in the "deep interior" regions, since it there admits closed timelike curves (CTCs), as does the Goedel lambdadust. These CTCs are problematical. The properties you are referring to can be read off the components [itex]g_{ab}, \; g^{ab}[/itex] of the metric tensor as expressed in the given chart. If w is the one of the coordinates, it is associated with a coordinate vector field [itex]\partial_w[/itex] and a "dual" coordinate covector field [itex]dw[/itex], which is a simple or Darboux rank zero) oneform. Think of a covector as a "hyperplane element". Assuming without loss of generality +++ signature, at some event E, we can partially classify the geometric nature of the coordinate w as follows: 1. [itex] g_{ww} < 0, \; g^{ww} > 0 [/itex] means that [itex]dw[/itex] cuts the light cone at E ("++ signature hyperplane element") and [itex]\partial_w[/itex] is timelike at E. 2. [itex] g_{ww} = 0, \; g^{ww} = 0 [/itex] means that [itex]dw[/itex] is tangent to the light cone at E ("null hyperplane element") and [itex]\partial_w[/itex] is null is E. 3. [itex] g_{ww} < 0, \; g^{ww} > 0 [/itex] means that [itex]dw[/itex] is transverse to the light cone ("+++ signature hyperplane element") and [itex]\partial_w[/itex] is spacelike. It is quite possible to give charts for Minkowski vacuum which exhibit various combinations of these alternative behaviors for the four coordinates (not entirely independently, of course). The above "classification" is incomplete, and a complete classification of the local causal properties of all possible coordinate charts (local in sense of "local neighborhood") involves considerations I haven't mentioned. It turns out there are 199 types in the CollMorales scheme, which uses 4+6+4=14 parameters, including the signs noted above. See http://www.arxiv.org/abs/grqc/0507121 MTW has a good overview of the other wellknown systems, including the Eddington chart (rediscovered by Finkelstein) and the KruskalSzekeres chart (discovered independently by Martin Kruskal and Peter Szekeres). Synge is sometimes mentioned as the first to have understood the global geometry of the maximal analytic extension of the Kerr vacuum solution, but his writings were not entirely clear, so many tend to credit this to Kruskal and Szekeres. One can discuss complex coordinates, but these don't mean what you probably think, and this thread is already confused enough! Write out the sourcefree Maxwell field equations in flat spacetime in their full partial differential glory; the independent variables are t,x,y,z and the dependent variables are the components of the electric and magnetic fields. Now compute the Lie algebra of the point symmetry group of this system according to the method given by Sophus Lie. You obtain a large Lie algebra which includes the 15 dimensional algebra so(2,4) (the Lie algebra of the conformal group on Minkowski spacetime), plus the generator of scalar multiplications of the dependent variables, [tex] E^x \, \partial_{E^x} + E^y \, \partial_{E^y} + E^z \, \partial_{E^z} + B^x \, \partial_{B^x} + B^y \, \partial_{B^y} + B^z \, \partial_{B^z} [/tex] plus generators arising from the freedom to add any solution (so as with any linear system, the point symmetry group is technically infinite dimensional), plus the generator [tex] B^x \, \partial_{E^x} + B^y \, \partial_{E^y} + B^z \, \partial_{E^z}  E^x \, \partial_{B^x}  E^y \, \partial_{B^y}  E^z \, \partial_{B^z} [/tex] which generates the one parameter subgroup of "Wick rotations". These effect only the dependent variables and as you can see can be considered a "rotation" in a six dimensional space (the space of components of the two vector fields). 


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