First, I review the generalization to higher dimensions of the algebraic classification of the Weyl tensor and the Newman-Penrose formalism. Generalization of the Geroch-Hld-Penrose formalism to higher dimensions. with Vojtˇech Pravda, Alena Pravdov´a and Harvey S. Reall) Class.
Historical context
This observation stimulated the development of the general theory of relativity (GR), which was first fully written down by Einstein in 1916 [8]. Therefore, the study of general relativity in higher dimensions is an essential part of a better understanding of string theory.
Review of general relativity
We will see many more examples of how special the four dimensions are later in the thesis. BLACK HOLE IN FOUR DIMENSIONS 5 This thesis will focus on Einstein's times, where the only subject allowed is a.
Black holes in four dimensions
For the Kerr geometry there are two independent Killing vector fields 𝑘 = ∂/∂𝑡 and 𝜁 = ∂/∂𝜙, which leads to two conserved quantities 𝐸 = −𝑢.𝑘 and ℎ = 𝑢.𝑚 which we interpret as the energy and angular momentum of the particle (per unit mass). The Kerr black hole is thought to be the stationary end state of the collapse of sufficiently massive stars, under fairly common conditions.
Black hole uniqueness theorems
BLACK HOLE UNIQUENESS THEOREMS 9that one starts with (consistent) initial data on some Cauchy surface that is in some. In an asymptotically flat spacetime ℳ, a black hole region is defined as the subset of ℳ that lies outside the causal past of future null infinity.
Black holes in higher dimensions
THE BLACK HOLE IN A HIGHER DIMENSION 11 In four dimensions, the topology of (time slices) of the event horizon is finite. These are (globally) flat black hole asymptotic solutions of Einstein's vacuum equations, with an event horizon of the spatial topology 𝑆1×𝑆2.
Near-horizon geometries
In fact, the near-horizon (NH) geometries of all known extremal vacuum black hole solutions have more symmetry than manifests in the above metric, with the symmetry generated by 𝑘 and 𝑋 improving to 𝑆𝑂. Classification of near-horizon geometries proved significantly easier than classification of full black hole solutions in higher dimensions.
Stability of black holes
It is known that the area of the black hole horizon(s) in any spacetime always does not decrease [19]. This negative mode appears for all black holes with an angular momentum parameter 𝑎 greater than some critical value 𝑎0.
New results of this thesis
In particular, we show that the equations for linearized perturbations of the near-horizon geometry can be reduced to the equation of motion for a charged, massive scalar field in 𝐴𝑑𝑆2. In four dimensions, perhaps surprisingly, all these methods can be used to give different descriptions of the same classification.
Algebraic classification in four dimensions
In this language, a spacetime is algebraically special if and only if at least two of the PNDs coincide. On this basis, one can decompose the Weyl tensor with respect to the complex scalars.
Algebraic classification in higher dimensions
It is most useful to use this classification to do a boost-weight decomposition of the Weyl tensor. So Φ=0 is the statement that all boost weight 0 components of the Weyl tensor vanish on that basis.
Aside: Alternative methods of classification
In fact, it is quite simple to show that a space-time is algebraically singular on any null killer horizon, with a multiple WAND tangent to the null generators of the horizon.8 However, it is well known that null generators do not extend the horizon , and thus the conditions of Theorem 2.11 fail there. A completely different approach to a higher-dimensional generalization of the Petrov classification was given by De Smet [120].
The Newman-Penrose Formalism
This approach to the > 4 generalization of the 4D Newman-Penrose formalism was developed in Refs. In the case of asymptotic flat spaces, possible higher-dimensional generalizations of the peeling theorem are discussed in Refs.
The Geroch-Held-Penrose Formalism
Note that the action' on the weight gain components 0 of the Weyl tensor has one subtlety:. 2.52). These commutators depend on the spin 𝑠 and increased weight 𝑏 of the GHP scalar 𝑇𝑖1..𝑖𝑠 they act on.
Maxwell fields
In four dimensions, the Mariot–Robinson theorem (Theorem 7.4 of Ref. [27]) states that a zero vector field is multiply aligned with a (nonzero) algebraic special Maxwell field if, and only if, it is geodesic and shear-free. Therefore, according to the Goldberg-Sachs theorem, a vacuum spacetime admits such a Maxwell test field if, and only if, it is algebraically special.
Codimension-2 hypersurfaces
The simplest example of the latter behavior is a product spacetime, say 𝑑𝑆3×𝑆2, where every zero vector field tangent to 𝑑𝑆3 is a multiple WAND, regardless of whether it is geodesic or not [129]. Therefore, every geodesic null vector field in the constant curvature submanifolds of Theorem 3.4 is a geodesic multiple WAND of the entire spacetime, so Theorem 3.2 follows as a direct corollary of these two results.
Non-geodesity implies Type D
Therefore, all components of the non-zero acceleration weight Weyl tensor vanish. In the first case, this implies that ¯ℓ ∥𝑛, so the zero-weighted components of the Weyl tensor must have vanished in our original framework, meaning that the spacetime Weyl tensor vanishes equally.
Foliation by submanifolds
By assumption, this is a geodesic of the full spacetime, so the RHS of the above equation must vanish. But 𝑝 and 𝑈 are arbitrary, so 𝐾(𝑈, 𝑈) must vanish for any zero𝑈 tangent to𝒮, which implies that𝒮 is completely umbilical.□.
Results in five dimensions
We now see that the symmetries of the constant curvature submanifolds extend to symmetries of the full spacetime. Note that it is not obvious that the symmetries of the constant curvature submanifolds extend to the symmetries of spacetime.
Discussion
Recent work proposes an alternative generalization of the notion of type II space to higher dimensions. However, it can be shown [31] that Ψ(1)0 is gauge invariant if and only if ℓ is a repeated null principal direction of the background space.
Gauge-invariant variables
In section 4.2 we investigate the existence of gauge invariant quantities, and show that Ω(1) is gauge invariant if and only if the background spacetime is algebraically special. Finally, in Section 4.5 we discuss the possible applications of our results, leading to Chapter 5 where we will use our results to study perturbations of near-horizon geometries.
Decoupling of electromagnetic perturbations
Now the Maxwell equations cannot be used to eliminate the form 𝜿þ′(F+𝑓) without reintroducing 1-derivative terms of the form 𝜿k′. There are no Maxwell equations that can be used to eliminate k𝑖𝑓 without reintroducing new derivative terms of the form 𝝆þ′.
Decoupling of gravitational perturbations
Since we have no independent equation that will allow us to eliminate This is a necessary condition for decoupling; is also sufficient since we now have an equation in which the only perturbed Weyl components that appear are Ω. The resulting decoupled equation is:
Discussion
If so, under what circumstances does instability of the near-horizon geometry imply instability of the black hole as a whole. We argue that the instability of the scalar field in the near-horizon geometry implies instability in the entire spacetime of the black hole if this condition holds.
Decoupling and near-horizon geometries
We can again perform the separation of the perturbation equation for Ω and show that it reduces to the massive charged scalar equation in 𝐴𝑑𝑆2, which is satisfactory. Therefore, our condition for instability of the geometry near the horizon is the existence of an eigenvalue 𝜆 < −1/4.
Cohomogeneity-1 extreme MP black holes
1/4 is respected and there is no near-horizon geometry instability in this sector. Therefore we do not predict any full black hole instability in this case.
Instabilities from near-horizon geometries
This grows exponentially with time and therefore represents the instability of the scalar field behind the black hole. Therefore, there is an instability of the scalar field when the limit 𝐴𝑑𝑆2 BF is violated.
The Doubly-Spinning Black Ring Spacetime
Thus, for all allowed values of 𝜆 and 𝜈 we have that the point at the center of the ring lies outside the ergoregion. The effects cancel near the center of the ring, leaving a region that does not lie in the ergoregion.
Geodesic Structure
We have 𝐻(𝑦)< 0 everywhere inside the ergoregion, and 𝐻(𝑦) = 0 on the ergosurface, so the only turning point ˙𝑦 = 0 of the geodesic lies on the ergosurface. Instead, it is necessary to explicitly find turning points of the quartile 𝑈(𝑥) to find the range of 𝑥 values where 𝑈(𝑥) ≤ 0.
New Coordinate Systems
A sensible choice, which is guaranteed to be well defined, is to choose 𝑥0 = 0 and 𝑦0 as the turning point in the 𝑦 motion of the geodesic line, i.e. 𝜁(𝑦0) = 0. To obtain a new set of coordinates analogous to the single rotating case , we can hope to be able to set 𝑣 =𝑡−𝜂𝑡 where.
Hidden Symmetries
Note that the zero-energy geodesics in the 5-dimensional metric correspond exactly to the geodesics in the 4-dimensional metric (while the non-zero-energy ones are associated with the orbits of charged particles). In the double rotation case, it turns out that the conformal Killing tensor 𝐾𝑖𝑗 cannot be derived from the conformal Killing-Yan tensor.
Discussion and Outlook
Thus, as a direct corollary of Lemma 6.3, we see that the dimensional reduction of the black ring spacetime possesses a CKY tensor if and only if the ring is unirotating. As a result, we do not expect separability of the KG equation for the black ring to be possible.
Commutators
Bianchi equations
However, since we are now equipped with a property not enjoyed by 𝑛, we have broken the symmetry under the complement operation and must therefore explicitly write out all the equations. In this case, the priming symmetry is regained, and many of the equations below become unnecessary (and some of the remaining ones are further simplified).
Bianchi equations
Commutators
This is the obvious generalization of the standard Lichnerowicz operator on ℂℙ𝑁, with the Laplacian ˆ∇2 replaced by our loaded Laplacian ˆ𝒟2 (following [170]). The result of this is that we can expand general perturbations in terms of scalar, vector and tensor harmonics on ℂℙ𝑁, and the relevant eigenvalues of the Laplician ˆ𝒟2 are known (see [176] for further details).
Gravitational perturbations
In particular, the eigenvalues of 𝒪(2) will be the eigenvalues of the matrix describing the coupling between the different components of 𝑌𝑖𝑗. The eigenvalues of the standard Laplacian are thus related to the Bochner-Weitzenb¨ock identity on ℂℙ𝑁, which implies that.
Electromagnetic fields
Proof of Lemma 6.3
