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A tensor of rank (m, n), also called a (m, n) tensor, is defined to be a scalar function of m one-forms and n vectors that is linear in all of its arguments. It follows at once that scalars are tensors of rank (0, 0), vectors are tensors of rank (1, 0) and one-forms are tensors of rank (0, 1).
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upper and lower free indices (not summed over) on each side of an equation must be the same. This holds true for any equation, not just the tensor transformation law. Remember also that upper indices can only be summed with lower indices; if you have two upper or lower indices that are the same, you goofed.
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ing with general relativity, which requires some practice to really get comfortable with. The methods are quite powerful and well worth the investment of e ort. One can only really hope to tackle the fascinating conceptual and mathematical challenges if one has rst gotten over the hurdle of understanding how to parse and manipulate the formalism.
general relativity are derived. A wide range of applications to physical situations follows, and the conclusion gives a brief discussion of classical field theory and the derivation of general relativity from a variational principle. Written for advanced undergraduate and graduate students, this approachable
- Preface
- Introduction
- 1.1 Basic concepts
- 1.2 Further Literature
- 2.1 The Lorentz transformation
- 2.1.1 Derivation from first principles
- 2.7 Mechanics of continuous matter
- 3.3.1 Contravariant vectors
- 3.3.2 Covariant vectors
- 3.4.2 Geodesics
- ∂Aμ
- 4.3.1 Symmetries of the Riemann tensor
- The Einstein equation
- + c−2 R2) ̇ sin2 θ
- R ̃ − R2 ̃
- 7.1 Appendix 1: Angular momentum and charge
- Aμν = s(μν)/|A| (7.26)
This course aims to provide some understanding of general relativity as a theory of gravity in terms of the geometric properties of spacetime. We proceed along the general line of thought formulated by Einstein in his original publications of the general theory of relativity. Only a few parts, including the treatment of the stress-energy tensor are...
This chapter displays some unsatisfactory aspects of Newtonian dynamics, even after introducing special relativity, which thus demonstrates the necessity of a theory of gravity and inertia. It provides a summary of some basic concepts and elements that play a role in the development of the theory, and gives some information on possibilities for fur...
The special theory of relativity offers a vast improvement over the older theories based on Galilean transformations. The new ‘classical’ mechanics which includes kinematics according to special relativity, satisfactorily explains many observed phe-nomena, including the Michelson-Morley experiment that indicated that the speed of light as measured ...
The general theory of relativity is not sufficiently simple to allow an introduction by means of a short text. A book or article on this subject may thus be subject to two dangers: first, the text may be too short so that it becomes incomprehensible; and second, it may become too long so that most potential readers are deterred in an early stage. A...
For the time being, we are dealing with Euclidean space plus one time direction. It is called flat spacetime, or Lorentzian spacetime. The extension of the Cartesian coordinate system with a time direction is also called Minkowski space. Furthermore we shall, in general, restrict ourselves to inertial frames. In these frames, unperturbed objects ar...
In order to formulate coordinate transformations between uniformly moving inertial frames of reference, Einstein gave up the notion of absolute simultaneity. However, he maintained this notion for observations done within the context of a given inertial frame. The Galilean transformations (2.1) have to be modified. How should this be done? Consider...
At this stage of the course, we do not yet know the precise equations describing how matter generates gravity. But it is certain that these equations shall contain the mass distribution, and they must be consistent with the principles of relativity formulated in the preceding pages. Naturally, under normal circumstances, the main source of gravity ...
We have already seen how to differentiate a scalar field and thus to define a vector field. One may differentiate a vector field but this does not, in general, lead to a tensor field. That is because, in addition to the vector field, also the metric may be position dependent. Let us investigate the transformation properties of the derivative of a c...
We shall also need to find an analogous expression for the covariant derivative of a covariant vector field. Ordinary differentiation yields
More generally, one can find the equation of motion of a test particle that is not subject to external forces without making use of a transformation to an inertial frame, by requiring that the particle follows a geodesic, also called a geodesic line or geodetic line. This is a straight line in a flat spacetime. A geodesic will be seen to remain a g...
and Aμ;ν ≡ ✪ ∂xν − Γτ μνAτ The definition of Rτ βγδ can thus be written more compactly as Rτ ≡ Γτ − Γτ + Γσ βγδ βδ,γ βγ,δ βδΓτ σγ − Γσ βγΓτ σδ Higher derivatives can be denoted similarly, for instance Aβ,γδ ≡ Aβ,γ,δ ≡ (Aβ,γ),δ =
The symmetries of the Riemann tensor are most clearly exposed in the fully covariant form
In this chapter we shall focus on the relation between the metric and the mass-energy-momentum distribution. Unfortunately, it is not possible to derive the desired equations without making new assumptions. This is by no means strange, for instance because also the Newtonian theory of gravity is based on an assumption for the equation for the gravi...
where ‘sufficient’ means that one can obtain all nonzero elements by translating to the fully covariant form of of these elements
and a further contraction yields the curvature scalar as
The Schwarzschild solution describes a centrally symmetric situation, and thus cor-responds with the gravitational field of an object with zero angular momentum. For a rotating object, the spherical symmetry is broken and only axial symmetry, i.e., rotational symmetry about for instance the z axis or in the φ direction, remains. It is thus interest...
where |A| is the determinant of Aμν, and s(μν) is the subdeterminant of Aμν, i.e., the determinant of the submatrix that remains when the μth row and the νth column of Aμν are erased, times a factor ±1 The expansion of the determinant |A| in column ν of Aμν leads to μ ν. ↔
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Weinberg, S. 1972, Gravitation and Cosmology. Principles and applications of the General Theory of Relativity, (New York: John Wiley) What is now the classic reference, but lacking any physical discussions on black holes, and almost nothing on the geometrical interpretation of the equations. The author is explicit
General Relativity is the classical theory that describes the evolution of systems under the e ect of gravity. Its history goes back to 1915 when Einstein postulated that the laws of gravity can be expressed as a system of equations, the so-called Einstein equations. In order
