On Invariant Operations on a Manifold with a Linear Connection and an Orientation

We prove a theorem that describes all possible tensor-valued natural operations in the presence of a linear connection and an orientation in terms of certain linear representations of the special linear group. As an application of this result, we prove a characterization of the torsion and curvature operators as the only natural operators that satisfy the Bianchi identities.

Approaches to Spherically Symmetric Solutions in f(T) Gravity

We study properties of static spherically symmetric solutions in f(T) gravity. Based on our previous work on generalizing Bianchi identities for this kind of theory, we show how this search for solutions can be reduced to the study of two relatively simple equations. One of them does not depend on the function f and therefore describes the properties of such solutions in any f(T) theory. Another equation is the radial one and, if a possible solution is chosen, it allows the discovery of which function f is suitable for it. We use these equations to find exact and perturbative solutions for arbitrary and specific choices of f.

A brief review of general relativity

The first chapter in Part II contains basic information on general relativity and all necessary notations and formulas. The consideration is concise but rather brief; it is not meant to replace a textbook on general relativity. Covariant derivatives, curvature tensors, Bianchi identities, covariant equations, the classical limit and Einstein equations are covered. The fact that the Schwarzschild solution and the cosmological solution contain singularities, which can be interpreted as indicating a need for quantum gravitational theory, is addressed. In addition, the applicability of general relativity and Planck units are discussed.

Does general relativity highlight necessary connections in nature?

The dynamics of general relativity is encoded in a set of ten differential equations, the so-called Einstein field equations. It is usually believed that Einstein’s equations represent a physical law describing the coupling of spacetime with material fields. However, just six of these equations actually describe the coupling mechanism: the remaining four represent a set of differential relations known as Bianchi identities. The paper discusses the physical role that the Bianchi identities play in general relativity, and investigates whether these identities—qua part of a physical law—highlight some kind of a posteriori necessity in a Kripkean sense. The inquiry shows that general relativistic physics has an interesting bearing on the debate about the metaphysics of the laws of nature.

Preliminaries

This chapter discusses the main quantities, equations, and basic tools needed in the following chapters. It is the main reference kit providing all main null structure and null Bianchi equations, in general null frames, in the context of axially symmetric polarized spacetimes. The chapter translates the null structure and null Bianchi identities associated to an S-foliation in the reduced picture. It starts with general, Z-invariant, S-foliation, before considering the special case of geodesic foliations. The chapter then looks at perturbations of Schwarzschild and invariant quantities. It investigates how the main Ricci and curvature quantities change relative to frame transformations, i.e., linear transformations which take null frames into null frames. Finally, the chapter presents wave equations for the invariant quantities.

Minimal gauge invariant couplings at order $$\alpha '^3$$: NS–NS fields

Removing the field redefinitions, the Bianchi identities and the total derivative freedoms from the general form of gauge invariant NS–NS couplings at order $$\alpha '^3$$ α ′ 3 , we have found that the minimum number of independent couplings is 872. We find that there are schemes in which there is no term with structures $$R,\,R_{\mu \nu },\,\nabla _\mu H^{\mu \alpha \beta }$$ R , R μ ν , ∇ μ H μ α β , $$ \nabla _\mu \nabla ^\mu \Phi $$ ∇ μ ∇ μ Φ . In these schemes, there are sub-schemes in which, except one term, the couplings can have no term with more than two derivatives. In the sub-scheme that we have chosen, the 872 couplings appear in 55 different structures. We fix some of the parameters in type II supersting theory by its corresponding four-point functions. The coupling which has term with more than two derivatives is constraint to be zero by the four-point functions.