Tendex-Vortex visualization of curvature tensor, and
Brane-world scenario
Tarique Monsoor
several facts in hand…
• The Ranque-Hilsch Effect3,4 Ranque reported the phenomenon
of noticeable temperature distribution (fig.3) in confined steady
rotating gas flows in 1933.
• Superfluid Vortex: Experiments performed by Hall and Vinen
established the existence of quantized vortex lines in superfluid
helium. Experiments performed by Rayfield and Reif established the
existence of quantized vortex rings. Packard has observed the
intersection of vortex lines with the free surface of the fluid.
several facts in hand…
• Violation of the zeroth law of thermodynamics in Black
Hole, which is one the most fundamental assumptions concerning
macroscopic systems in equilibrium. A. Ramirez-Hernandez, H.
Larralde, and F. Leyvraz showed1 that systems with negative
specific heat can violate the zeroth law of thermodynamics.
• Negative Specific Heat of Black-hole: from the works of W.
Thirring2 we are informed of the fact that if radiation energy is
extracted from a star whose nuclear fuel is exhausted, the star will
contract and heat up. Thus a star acts like a system with negative
specific heat.
Several recent paradigms…
• The paradigm of foliating spacetime into spacelike hypersurfaces;
introduction of tools for visualizing curvature tensor e.g. frame-drag
vortex lines, tidal tendex lines, vorticity, tendicity, vortexes and
tendexes are several recent developments.5
• Brane-world scenario, the other recent developments, the spacetime
itself is a foliate/ brane. 6,7
• Holographic Duality
Splitting of Weyl curvature tensor
C


 R


 2
[
[
R
]
]
 
1
3
[
[

]
]
R

Here,   represents Kronecker delta and the square bracket
represents antisymmetrization.
Denoting the 4-veocity of the observers who move orthogonal to

foliation’s space-slices by u
and the induced spatial three
metric by    g   u  u .

Using the projection operator,   , we can split the Weyl
tensor covariantly into two irreducible parts…



E       C  u u




B        C  u u

which are symmetric and trace-free. The even parity field is
called “electric” part and the odd parity field is called “magnetic”

part of C   .
Fig.1:
the ‘red’ lines have negative tendicity
the ‘blue’ lines have positive tendicity
Fig.2:
the ‘red’ lines have negative vorticity
the ‘blue’ lines have positive vorticity
Black-hole: 5-D vortex or not?
From this Ranque-Hilsch effect we find that vortex and turbulent fluid
motion generates cooling effect. Since it is a 4-dimensional
phenomena, it also has manifested opposite effect, the heating
effect. This nature of vortex motion persuades me to think black hole
as vortex and which is the probable source of negative specific heat
of black hole. Now, there are two options; first one is that black hole
and white hole exists in pair; the second and the most imaginative
one is that black-hole actually is a higher dimensional object whose
4-dimensional cross-section or projection is our astronomical blackholes and resides in the brane, which is our universe.
References
[1] A. Ramirez-Hernandez, H. Larralde, and F. Leyvraz
arXiv:0802.1748v2 [cond-mat.stat-mech] 26 Apr 2008
[2] W. Thirring, Z. Physik 235 339 (1970); “Systems with negative specific heat”
[3] Vlad Bezprozvannykh and Hank Mottl; “The Ranque-Hilsch Effect: CFD Modeling”
[4] G J Ranque. .Experiments on Expansion in a Vortex with Simultaneous Exhaust of Hot and Cold Air.. Le Journal De
Physique, et le Radium (Paris), vol 4, June 1933, pp 1125-1130.
[5] David A. Nichols, Robert Owen, Fan Zhang, Aaron Zimmerman, Jeandrew Brink, Yanbei Chen, Jeffrey D. Kaplan,
Geoffrey Lovelace, Keith D. Matthews, Mark A. Scheel, and Kip S. Thorne
“Visualizing Spacetime Curvature via Frame-Drag Vortexes and Tidal Tendexes I. General Theory and WeakGravity Applications”
[6] Zurab Kakushadze and S.H. Henry tye; “Brane world” hep-th/9809147 v3
[7] A. Chamblin, S.W. Hawking and H.S. Reall; “Brane-World Black Holes” hep-th/9909205 v2
• Fig.3: the temperature distribution in a vortex tube
Thank you