Spacetime And Geometry Carroll Solutions

[Efraine Ton]

In the general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of matter within it.[1]

Exact solutions for the EFE can only be found under simplifying assumptions such as symmetry. Special classes of exact solutions are most often studied since they model many gravitational phenomena, such as rotating black holes and the expanding universe. Further simplification is achieved in approximating the spacetime as having only small deviations from flat spacetime, leading to the linearized EFE. These equations are used to study phenomena such as gravitational waves.

where R μ ν \displaystyle R_\mu \nu is the Ricci curvature tensor, and R \displaystyle R is the scalar curvature. This is a symmetric second-degree tensor that depends on only the metric tensor and its first and second derivatives.

The EFE is a tensor equation relating a set of symmetric 4 4 tensors. Each tensor has 10 independent components. The four Bianchi identities reduce the number of independent equations from 10 to 6, leaving the metric with four gauge-fixing degrees of freedom, which correspond to the freedom to choose a coordinate system.

Although the Einstein field equations were initially formulated in the context of a four-dimensional theory, some theorists have explored their consequences in n dimensions.[9] The equations in contexts outside of general relativity are still referred to as the Einstein field equations. The vacuum field equations (obtained when Tμν is everywhere zero) define Einstein manifolds.

The above form of the EFE is the standard established by Misner, Thorne, and Wheeler (MTW).[11] The authors analyzed conventions that exist and classified these according to three signs ([S1] [S2] [S3]):

The third sign above is related to the choice of convention for the Ricci tensor: R μ ν = [ S 2 ] [ S 3 ] R α μ α ν \displaystyle R_\mu \nu =[S2]\times [S3]\times R^\alpha _\mu \alpha \nu

Reversing the trace again would restore the original EFE. The trace-reversed form may be more convenient in some cases (for example, when one is interested in weak-field limit and can replace g μ ν \displaystyle g_\mu \nu in the expression on the right with the Minkowski metric without significant loss of accuracy).

The inclusion of this term does not create inconsistencies. For many years the cosmological constant was almost universally assumed to be zero. More recent astronomical observations have shown an accelerating expansion of the universe, and to explain this a positive value of Λ is needed.[18][19] The cosmological constant is negligible at the scale of a galaxy or smaller.

The existence of a cosmological constant is thus equivalent to the existence of a vacuum energy and a pressure of opposite sign. This has led to the terms "cosmological constant" and "vacuum energy" being used interchangeably in general relativity.

Contracting the differential Bianchi identity R α β [ γ δ ; ε ] = 0 \displaystyle R_\alpha \beta [\gamma \delta ;\varepsilon ]=0 with gαβ gives, using the fact that the metric tensor is covariantly constant, i.e. gαβ;γ = 0, R γ β γ δ ; ε + R γ β ε γ ; δ + R γ β δ ε ; γ = 0 \displaystyle R^\gamma _\beta \gamma \delta ;\varepsilon +R^\gamma _\beta \varepsilon \gamma ;\delta +R^\gamma _\beta \delta \varepsilon ;\gamma =0

The nonlinearity of the EFE distinguishes general relativity from many other fundamental physical theories. For example, Maxwell's equations of electromagnetism are linear in the electric and magnetic fields, and charge and current distributions (i.e. the sum of two solutions is also a solution); another example is Schrdinger's equation of quantum mechanics, which is linear in the wavefunction.

The EFE reduce to Newton's law of gravity by using both the weak-field approximation and the slow-motion approximation. In fact, the constant G appearing in the EFE is determined by making these two approximations.

The solutions to the vacuum field equations are called vacuum solutions. Flat Minkowski space is the simplest example of a vacuum solution. Nontrivial examples include the Schwarzschild solution and the Kerr solution.

The solutions of the Einstein field equations are metrics of spacetime. These metrics describe the structure of the spacetime including the inertial motion of objects in the spacetime. As the field equations are non-linear, they cannot always be completely solved (i.e. without making approximations). For example, there is no known complete solution for a spacetime with two massive bodies in it (which is a theoretical model of a binary star system, for example). However, approximations are usually made in these cases. These are commonly referred to as post-Newtonian approximations. Even so, there are several cases where the field equations have been solved completely, and those are called exact solutions.[9]

One can also discover new solutions of the Einstein field equations via the method of orthonormal frames as pioneered by Ellis and MacCallum.[22] In this approach, the Einstein field equations are reduced to a set of coupled, nonlinear, ordinary differential equations. As discussed by Hsu and Wainwright,[23] self-similar solutions to the Einstein field equations are fixed points of the resulting dynamical system. New solutions have been discovered using these methods by LeBlanc[24] and Kohli and Haslam.[25]

The nonlinearity of the EFE makes finding exact solutions difficult. One way of solving the field equations is to make an approximation, namely, that far from the source(s) of gravitating matter, the gravitational field is very weak and the spacetime approximates that of Minkowski space. The metric is then written as the sum of the Minkowski metric and a term representing the deviation of the true metric from the Minkowski metric, ignoring higher-power terms. This linearization procedure can be used to investigate the phenomena of gravitational radiation.

Despite the EFE as written containing the inverse of the metric tensor, they can be arranged in a form that contains the metric tensor in polynomial form and without its inverse. First, the determinant of the metric in 4 dimensions can be written det ( g ) = 1 24 ε α β γ δ ε κ λ μ ν g α κ g β λ g γ μ g δ ν \displaystyle \det(g)=\tfrac 124\varepsilon ^\alpha \beta \gamma \delta \varepsilon ^\kappa \lambda \mu \nu g_\alpha \kappa g_\beta \lambda g_\gamma \mu g_\delta \nu using the Levi-Civita symbol; and the inverse of the metric in 4 dimensions can be written as: g α κ = 1 6 ε α β γ δ ε κ λ μ ν g β λ g γ μ g δ ν det ( g ) . \displaystyle g^\alpha \kappa =\frac \tfrac 16\varepsilon ^\alpha \beta \gamma \delta \varepsilon ^\kappa \lambda \mu \nu g_\beta \lambda g_\gamma \mu g_\delta \nu \det(g)\,.

Substituting this expression of the inverse of the metric into the equations then multiplying both sides by a suitable power of det(g) to eliminate it from the denominator results in polynomial equations in the metric tensor and its first and second derivatives. The action from which the equations are derived can also be written in polynomial form by suitable redefinitions of the fields.[26]

I'm going to start self-studying General Relativity from Sean Caroll's Spacetime and Geometry: An Introduction to General Relativity. I'd like to have a textbook on Differential Geometry/Calculus on Manifolds for me on the side.

I do like mathematical rigor, and I'd like a textbook whose focus caters to my need. Having said that, I don't want a exchaustive mathematics textbook (although I'd appreciate one) that'll hinder me from going back to the physics in a timely manner.

I looked for example at Lee's textbook but it seemed too advanced. I have done courses on Single and Multivariable Calculus, Linear Algebra, Analysis I and II and Topology but I'm not sure what book would be the most useful for me given that I have a knack of seeing all results formally.

I wanted to recommend Lee, but since you said it's too advanced... Well, to be fair, while his book is quite extensive, it is a very pedagogically written one too, so if you wish to study manifolds, at one point at least, you should read it.

This is essentially a 5-volume grimoire, however it builds everything up quite slowly and pedagogically, and makes an attempt to build a bridge between the old formalism (indices, coordinates, etc.) and the modern one

This is a very advanced book that is quite hard to read, so I'd suggest visiting this later. However, it is also quite essential. Despite the fact that this (two-volume) book is quite old, it is still the standard reference in the field. The contents of volume 1 is what would interest you more, probably, as the most of Riemannian geometry is being treated there.

Check out Barrett O'Neill's book on semi-Riemannian geometry. This book is written exactly for your purposes: it discusses manifolds with symmetric nonsingular metrics, and in particular spacetime metrics. There are even chapters on cosmology and the Schwarzchild metric.

Semi-Riemannian Geometry with Applications to General Relativity by Barrett O'Neill is my recommendation. He's very thorough, and doesn't skip the details, which is great for someone new to the subject. He introduces general relativity later on once he's covered all the necessary semi-Riemannian geometry

which is aimed at mathematics undergraduate students and develops semi-Riemannian geometry from scratch (only prerequisites are multi-variable calculus and linear algebra), and also presents the theory of general relativity in a definition-theorem-proof format. The contents of the book and a sample chapter can be viewed at the website above. As it contains solutions to all the exercises, it can be used for self-study.

c80f0f1006