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The Stirling formula is an approximation for n! that is good at large values of n.
$$n! = 1 \cdot 2 \cdot 3 \cdots (n-1) \cdot n \tag{E.1}$$
$$\ln(n!) = \underbrace{\ln 1}_{0} + \ln 2 + \ln 3 + \dots + \ln(n-1) + \ln(n)$$
(E.2)
Note that the function ln x is nearly flat for large va... | {
"Header 1": "Statistical Mechanics",
"Header 2": "Stirling's Approximation",
"token_count": 546,
"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
} |
You know that a sum can be approximated by an integral. How accurate is that approximation? The Euler-Maclaurin formula gives the corrections.
$$\begin{split} \sum_{k=0}^{n-1} f(k) &\approx \int_0^n f(x) \, dx \\ &- \frac{1}{2} [f(n) - f(0)] + \frac{1}{12} [f'(n) - f'(0)] - \frac{1}{720} [f'''(n) - f'''(0)] \\ &+ \fr... | {
"Header 1": "Statistical Mechanics",
"Header 2": "The Euler-Maclaurin Formula and Asymptotic Series",
"token_count": 349,
"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
} |
The function
$$\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}$$
for $s > 1$ ,
called the Riemann zeta function, has applications to number theory, statistical mechanics, and quantal chaos.
#### History
From Simmons:
No great mind of the past has exerted a deeper influence on the mathematics of the twentieth ce... | {
"Header 1": "Statistical Mechanics",
"Header 2": "Ramblings on the Riemann Zeta Function",
"token_count": 1633,
"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
} |
The zeta function can be evaluated exactly for positive even integer arguments:
$$\zeta(m) = \frac{(2\pi)^m |B_m|}{2m!}$$
for $m = 2, 4, 6, \dots$
Here the B<sup>m</sup> are the Bernoulli numbers, defined through
$$\frac{x}{e^x - 1} = \sum_{m=0}^{\infty} \frac{B_m}{m!} x^m.$$
The first few Bernoulli numbers ar... | {
"Header 1": "Statistical Mechanics",
"Header 2": "Ramblings on the Riemann Zeta Function",
"Header 3": "Exact values for the zeta function",
"token_count": 1062,
"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
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You know from as far back as your introductory mechanics course that some problems are difficult given one choice of coordinate axes and easy or even trivial given another. (For example, the famous "monkey and hunter" problem is difficult using a horizontal axis, but easy using an axis stretching from the hunter to the... | {
"Header 1": "Statistical Mechanics",
"Header 2": "Tutorial on Matrix Diagonalization",
"token_count": 274,
"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
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There is a difference between an entity and its name. For example, a tree is made of wood, whereas its name "tree" made of ink. One way to see this is to note that in German, the name for a tree is "Baum", so the name changes upon translation, but the tree itself does not change. (Throughout this tutorial, the term "tr... | {
"Header 1": "Statistical Mechanics",
"Header 2": "H.1 What's in a name?",
"token_count": 1692,
"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
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A tensor, like a vector, is a geometrical entity that may be described ("named") through components, but a d-dimensional tensor requires d 2 rather than d components. Tensors are less familiar and more difficult to visualize than vectors, but they are neither less important nor "less physical". We will introduce tensor... | {
"Header 1": "Statistical Mechanics",
"Header 2": "H.3 Tensors in two dimensions",
"token_count": 1924,
"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
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A three-dimensional tensor is represented in component form by a 3 × 3 matrix with nine entries. If the tensor is symmetric, there are six independent elements. . . three on the diagonal and three off-diagonal. The components of a tensor in three dimensions change with coordinate system according to
$$\mathsf{T}' = \... | {
"Header 1": "Statistical Mechanics",
"Header 2": "H.4 Tensors in three dimensions",
"token_count": 445,
"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
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A d-dimensional tensor is represented by a d×d matrix with d 2 entries. If the tensor is symmetric, there are d independent on-diagonal elements and d(d−1)/2 independent off-diagonal elements. The tensor components will change with coordinate system in the now-familiar form
$$\mathsf{T}' = \mathsf{RTR}^{\dagger},\tag... | {
"Header 1": "Statistical Mechanics",
"Header 2": "H.5 Tensors in d dimensions",
"token_count": 575,
"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
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Section H.3 considered $2 \times 2$ matrices as representations of tensors. This section gains additional insight by considering $2 \times 2$ matrices as representations of linear transformations. It demonstrates how diagonalization can be useful and gives a clue to an efficient algorithm for diagonalization.
A l... | {
"Header 1": "Statistical Mechanics",
"Header 2": "H.5 Tensors in d dimensions",
"Header 3": "H.6 Linear transformations in two dimensions",
"token_count": 1526,
"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
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If a vector x is acted upon by a linear transformation B, then the output vector
$$\mathbf{x}' = \mathsf{B}\mathbf{x} \tag{H.36}$$
will usually be skew to the original vector x. However, for some very special vectors it might just happen that x 0 is parallel to x. Such vectors are called "eigenvectors". (This is a ... | {
"Header 1": "Statistical Mechanics",
"Header 2": "H.7 What does \"eigen\" mean?",
"token_count": 485,
"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
} |
We saw in section H.3 that for any 2 × 2 symmetric matrix, represented in its initial basis by, say,
$$\left(\begin{array}{cc} a & b \\ b & c \end{array}\right), \tag{H.40}$$
a simple rotation of axes would produce a new coordinate system in which the matrix representation is diagonal:
$$\left(\begin{array}{cc} d... | {
"Header 1": "Statistical Mechanics",
"Header 2": "H.8 How to diagonalize a symmetric matrix",
"token_count": 2031,
"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
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It says that for some matrix $M = T - \lambda 1$ , we have
$$M\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix}. \tag{H.57}$$
You know right away one vector (x, y) that satisfies this equation, namely (x, y) = (0, 0). And most of the time, this is the *only* vector that satisfies the equa... | {
"Header 1": "Statistical Mechanics",
"Header 2": "H.8 How to diagonalize a symmetric matrix",
"token_count": 1977,
"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
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Anyone who has worked even one of the problems in section H.8 knows that diagonalizing a matrix is no picnic: there's a lot of mundane arithmetic involved and it's very easy to make mistakes. This is a problem ripe for computer solution. One's first thought is to program a computer to solve the problem using the same t... | {
"Header 1": "Statistical Mechanics",
"Header 2": "H.9 A glance at computer algorithms",
"token_count": 369,
"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
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Many of the matrices that arise in applications are symmetric and hence the results of the previous sections are the only ones needed. But every once in a while you do encounter a non-symmetric matrix and this section gives you a guide to treating them. It is just an introduction and treats only 2 × 2 matrices.
Given... | {
"Header 1": "Statistical Mechanics",
"Header 2": "H.10 A glance at non-symmetric matrices and the Jordan form",
"token_count": 1965,
"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
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Instead of scaling the y-axis by a factor of 12, we can scale it by a factor of 1/b, and produce a new matrix representation of the form
$$\begin{pmatrix} a & 1 \\ 0 & a \end{pmatrix}.$$
(H.94)
Where is the information in this case? In the initial coordinate system, the four elements of the matrix contain four inde... | {
"Header 1": "Statistical Mechanics",
"Header 2": "H.10 A glance at non-symmetric matrices and the Jordan form",
"token_count": 596,
"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
} |
Effective teaching does not simply teach students what is correct—it also insures that students do not believe what is incorrect. There are a number of prevalent misconceptions in statistical mechanics. For example, an excellent history of statistical mechanics is titled The Kind of Motion We Call Heat. This title is w... | {
"Header 1": "Statistical Mechanics",
"Header 2": "Catalog of Misconceptions",
"token_count": 658,
"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
} |
$$E(S, V, N)$$
$$dE = T dS - p dV + \mu dN$$
$$F(T, V, N) = E - TS$$
$$dF = -S dT - p dV + \mu dN$$
$$H(S, p, N) = E + pV$$
$$dH = T dS + V dp + \mu dN$$
$$G(T, p, N) = F + pV$$
$$dG = -S dT + V dp + \mu dN$$
$$\Pi(T, V, \mu) = F - \mu N = -pV$$
$$d\Pi = -S dT - p dV - N d\mu$$
p(T, µ) [intensive qu... | {
"Header 1": "Statistical Mechanics",
"Header 2": "Thermodynamic Master Equations",
"token_count": 218,
"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
} |
Isothermal compressibility:
$$\kappa_T = -\frac{1}{V} \left( \frac{\partial V}{\partial p} \right)_{T,N} = \frac{1}{\rho} \left( \frac{\partial \rho}{\partial p} \right)_T = \frac{1}{\rho^2} \left( \frac{\partial^2 p}{\partial \mu^2} \right)_T$$
Master thermodynamic equation:
$$dF = -S dT - p dV - M dH + \sum_{i}... | {
"Header 1": "Statistical Mechanics",
"Header 2": "**Useful Formulas**",
"token_count": 464,
"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
} |
| adiabatic compressibility, 87, 105 | second-order, 95 | | | | | | |
|------------------------------------------------------|----------------------------------------------------|--|--|--|--|--|--|
| analogy | compres... | {
"Header 1": "Statistical Mechanics",
"Header 2": "Index",
"token_count": 2016,
"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
} |
level), 152 |
| politics, 124 | Stefan-Boltzmann law, 101 |
| polymer, 129 | sum over all states, 110 |
| pressure, 7, 45, 179 ... | {
"Header 1": "Statistical Mechanics",
"Header 2": "Index",
"token_count": 720,
"source_pdf": "datasets/websources/Physics_v1/Physics/book.pdf"
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Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, Wilberforce Road, Cambridge, CB3 OBA, UK
http://www.damtp.cam.ac.uk/user/tong/gr.html [d.tong@damtp.cam.ac.uk](mailto:d.tong@damtp.cam.ac.uk)
#### Recommended Books and Resources
There are many decent text books on genera... | {
"Header 1": "David Tong",
"token_count": 404,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
} |
| 0. | | Introduction | | 1 |
|----|-----------------------|-----------------------------------|------------------------------------------|----|
| 1. | | | Geodesics in Spacetime ... | {
"Header 1": "Contents",
"token_count": 1434,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
} |
| | | The Einstein Equations | 140 |
| | 4.1 | | The Einstein-Hilbert Action | 140 |
| | | 4.1.1 | An Aside on Dimensional Analysis | 144 |
| | | 4.1.2 | The Cosmological Constant | 145 |
| | | 4.1.3 | Diffeomorphisms ... | {
"Header 1": "Contents",
"token_count": 1834,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
} |
General relativity is the theory of space and time and gravity. The essence of the theory is simple: gravity is geometry. The effects that we attribute to the force of gravity are due to the bending and warping of spacetime, from falling cats, to orbiting spinning planets, to the motion of the cosmos on the grandest sc... | {
"Header 1": "0. Introduction",
"token_count": 2046,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
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We can simply estimate the size of relativistic effects in gravity. What follows is really nothing more than dimensional analysis, with a small story attached to make it sound more compelling. Consider a planet in orbit around a star of mass M. If we assume a circular orbit, the speed of the planet is easily computed b... | {
"Header 1": "When is a Relativistic Theory of Gravity Important",
"token_count": 606,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
} |
Classical theories of physics involve two different objects: particles and fields. The fields tell the particles how to move, and the particles tell the fields how to sway. For each of these, we need a set of equations.
In the theory of electromagnetism, the swaying of the fields is governed by the Maxwell equations,... | {
"Header 1": "1. Geodesics in Spacetime",
"token_count": 232,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
} |
Our tool of choice throughout these lectures is the action. The advantage of the action is that it makes various symmetries manifest. And, as we shall see, there are some deep symmetries in the theory of general relativity that must be maintained. This greatly limits the kinds of equations which we can consider and, ul... | {
"Header 1": "The Principle of Least Action",
"token_count": 2004,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
} |
We can make this obvious by rewriting this equation as
$$g_{ik}\ddot{x}^k + \frac{1}{2} \left( \frac{\partial g_{ik}}{\partial x^j} + \frac{\partial g_{ij}}{\partial x^k} - \frac{\partial g_{jk}}{\partial x^i} \right) \dot{x}^j \dot{x}^k = 0$$
(1.6)
Finally, there's one last manoeuvre: we multiply the whole equatio... | {
"Header 1": "The Principle of Least Action",
"token_count": 1442,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
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The above description of R<sup>3</sup> in polar coordinates allows us to immediately describe a situation in which the space is truly curved: motion on the two-dimensional sphere S 2 . This is achieved simply by setting the radial coordinate r to some constant value, say r = R. We can substitute this constraint into th... | {
"Header 1": "A Slightly Less Trivial Example: S<sup>2</sup>",
"token_count": 278,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
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Let's start by considering a particle moving in Minkowski spacetime R<sup>1</sup>,<sup>3</sup> . We'll work with Cartestian coordinates x <sup>µ</sup> = (ct, x, y, z) and the Minkowski metric
$$\eta_{\mu\nu} = \text{diag}(-1, +1, +1, +1)$$
The distance between two neighbouring points labelled by x <sup>µ</sup> and ... | {
"Header 1": "1.2 Relativistic Particles",
"Header 3": "1.2.1 A Particle in Minkowski Spacetime",
"token_count": 1557,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
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There's a crucial difference between moving in Euclidean space and moving in Minkowski spacetime. You're not obliged to move in Euclidean space. You can just stop if you want to. In contrast, you can never stop moving in a timelike direction in Minkowski spacetime. You will, sadly, always be dragged inexorably towards ... | {
"Header 1": "1.2 Relativistic Particles",
"Header 3": "1.2.2 Why You Get Old",
"token_count": 1940,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
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We then write the action
$$S_1 = \int_{\sigma_1}^{\sigma_2} d\sigma \left[ -mc\sqrt{-\eta_{\mu\nu}\frac{dx^{\mu}}{d\sigma}\frac{dx^{\nu}}{d\sigma}} - qA_{\mu}(x)\dot{x}^{\mu} \right]$$
(1.20)
where q is some number, associated to the particle, that characterises the strength with which it couples to the new term $... | {
"Header 1": "1.2 Relativistic Particles",
"Header 3": "1.2.2 Why You Get Old",
"token_count": 2047,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
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First we need to determine the trajectory of a constantly accelerating observer. This was a problem that we addressed already in our first lectures on Special Relativity (see Section 7.4.6 of those notes). Here we give a different, and somewhat quicker, derivation.
We will view things from the perspective of an ine... | {
"Header 1": "1.2 Relativistic Particles",
"Header 3": "1.2.2 Why You Get Old",
"token_count": 2020,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
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To illustrate this, let's return to the situation in which you wake, weightless in an elevator, trying to figure out if you're floating in space or plummeting to your death. How can you tell?
Well, you could wait and find out. But suppose you're impatient. The equivalence principle says that there is no local exper... | {
"Header 1": "1.2 Relativistic Particles",
"Header 3": "1.2.2 Why You Get Old",
"token_count": 1205,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
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There is another measurable consequence of the gravitational time dilation. To see this, let's return to Alice on the ground and Bob, above, in his hot air balloon. Bob is kind of annoying and starts throwing peanuts at Alice. He throws peanuts at time intervals ∆TB. Alice receives these peanuts (now travelling at cons... | {
"Header 1": "Gravitational Redshift",
"token_count": 1984,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
} |
We started in Section 1.1.1 with a non-relativistic action
$$S = \int dt \, \frac{m}{2} \, g_{ij}(x) \dot{x}^i \dot{x}^j$$
and found that it gives rise to the geodesic equation (1.7).
However, to describe relativistic physics in spacetime, we've learned that we need to incorporate reparameterisation invariance in... | {
"Header 1": "Gravitational Redshift",
"token_count": 648,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
} |
Physics was born from our attempts to understand the motion of the planets. The problem was largely solved by Newton, who was able to derive Kepler's laws of planetary motion from the gravitational force law. This was described in some detail in our first lecture course on [Dynamics and Relativity.](http://www.damtp.ca... | {
"Header 1": "1.3 A First Look at the Schwarzschild Metric",
"token_count": 1972,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
} |
To see this, we write
$$V_{\text{eff}}(r) = \frac{c^2}{2} - \frac{GM}{r} + \frac{l^2}{2r^2} - \frac{l^2GM}{r^3c^2}$$
The non-relativistic limit is, roughly, c <sup>2</sup> → ∞. This means that we drop the final term in the potential that scales as 1/r<sup>3</sup> . (Since c is dimensionful, it is more accurate to s... | {
"Header 1": "1.3 A First Look at the Schwarzschild Metric",
"token_count": 2032,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
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Note also that, in contrast to the Newtonian case, the angular momentum barrier is now finite: no matter how large the angular momentum, a particle with enough energy (in the form of ingoing radial velocity) will always be able to cross the barrier, at which point it plummets towards r = 0. We will say more about thi... | {
"Header 1": "1.3 A First Look at the Schwarzschild Metric",
"token_count": 2024,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
} |
Instead, we can invoke the elliptic formula [\(1.44](#page-40-0)) which tells us that the minimum r<sup>−</sup> and maximum distance r<sup>+</sup> is given by
$$r_{\pm} = \frac{l^2}{GM} \frac{1}{1 \mp e} \quad \Rightarrow \quad l^2 = GMr_{+}(1 - e)$$
(1.49)
from which we get the precession
$$\delta = \frac{6\pi G... | {
"Header 1": "1.3 A First Look at the Schwarzschild Metric",
"token_count": 1796,
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Then this equation becomes
$$\frac{d^2u_1}{d\phi^2} + u_1 = -\frac{GM}{l^2}(1 - 3e\cos\phi)$$
which has the solution
$$u_1 = \frac{GM}{l^2} \left( -1 + \frac{3e}{2} \phi \sin \phi \right)$$
The precession of the perihelion occurs when
$$\frac{du}{d\phi} = 0 \quad \Rightarrow \quad -e\sin\phi + \frac{3e\alpha}... | {
"Header 1": "1.3 A First Look at the Schwarzschild Metric",
"token_count": 1349,
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It is straightforward to extend the results above to determine the null geodesics in the Schwarzschild metric. We continue to use the equations of motion derived from Suseful in [\(1.35](#page-36-0)). But this time we replace the constraint([1.38\)](#page-37-0) with the null version([1.34\)](#page-35-0), which reads
... | {
"Header 1": "1.3.5 Light Bending",
"token_count": 2039,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
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Before we claim success, we should check to see if the relativistic result([1.55](#page-51-0)) differs from the Newtonian prediction for light bending. Strictly speaking, there's an ambiguity in the Newtonian prediction for the gravitational force on a massless particle. However, we can invoke the principle of equivale... | {
"Header 1": "1.3.5 Light Bending",
"Header 3": "Newtonian Scattering of Light",
"token_count": 795,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
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Gravity is geometry. To fully understand this statement, we will need more sophisticated tools and language to describe curved space and, ultimately, curved spacetime. This is the mathematical subject of differential geometry and will be introduced in this section and the next. Armed with these new tools, we will then ... | {
"Header 1": "2. Introducing Differential Geometry",
"token_count": 302,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
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The stage on which our story will play out is a mathematical object called a manifold. We will give a precise definition below, but for now you should think of a manifold as a curved, n-dimensional space. If you zoom in to any patch, the manifold looks like R<sup>n</sup> . But, viewed more globally, the manifold may ha... | {
"Header 1": "2.1 Manifolds",
"token_count": 1167,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
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We now come to our main character: an n-dimensional manifold is a space which, locally, looks like R<sup>n</sup> . Globally, the manifold may be more interesting than R<sup>n</sup> , but the idea is that we can patch together these local descriptions to get an understanding for the entire space.
Definition: An n-dime... | {
"Header 1": "2.1.2 Differentiable Manifolds",
"token_count": 1976,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
} |
Here we define the map ϕ<sup>2</sup> : O<sup>2</sup> → R<sup>2</sup> using the coordinates
$$x = -\sin\theta'\cos\phi'$$
, $y = \cos\theta'$ , $z = \sin\theta'\sin\phi'$
with θ ′ ∈ (0, π) and ϕ ∈ (0, 2π). Again this is a map to an open subset of R<sup>2</sup> . We have O1∪O<sup>2</sup> = S <sup>2</sup> while, on ... | {
"Header 1": "2.1.2 Differentiable Manifolds",
"token_count": 2037,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
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But, as we will see, the placement of indices up or down will tell us something and all sums will necessarily have one index up and one index down. This is a convention that we met already in Special Relativity where the up/downness of the index changes minus signs. Here it has a more important role that we will see as... | {
"Header 1": "2.1.2 Differentiable Manifolds",
"token_count": 2046,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
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\frac{\partial f}{\partial x^{\mu}} \right|_p = X^{\mu} \left. \frac{\partial \tilde{x}^{\nu}}{\partial x^{\mu}} \right|_{\phi(p)} \left. \frac{\partial f}{\partial \tilde{x}^{\nu}} \right|_p$$
where we've used the chain rule. (Actually, we've been a little quick here. You can be more careful by introducing the funct... | {
"Header 1": "2.1.2 Differentiable Manifolds",
"token_count": 1411,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
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Given two vector fields X, Y ∈ X(M), we can't multiply them together to get a new vector field. Roughly speaking, this is because the product XY is a second order differential operator rather than a first order operator. This reveals itself in a failure of Leibnizarity for the object XY ,
$$XY(fg) = X(fY(g) + Y(f)g) ... | {
"Header 1": "The Commutator",
"token_count": 2003,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
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#### Push-Foward and Pull-Back
Suppose that we have a map $\varphi: M \to N$ between two manifolds M and N which we will take to be a diffeomorphism. This allows us to import various structures on one manifold to the other.
For example, if we have a function on f : N → R, then we can construct a new function th... | {
"Header 1": "The Commutator",
"token_count": 2005,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
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We then have
$$\mathcal{L}_X(Y^{\mu}\partial_{\mu}) = X^{\nu} \frac{\partial Y^{\mu}}{\partial x^{\nu}} \partial_{\mu} - Y^{\mu} \frac{\partial X^{\nu}}{\partial x^{\mu}} \partial_{\nu}$$
But this is precisely the structure of the commutator. We learn that the Lie derivative acting on vector fields is given by
$$... | {
"Header 1": "The Commutator",
"token_count": 301,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
} |
For any vector space V , the dual vector space V ∗ is the space of all linear maps from V to R.
This is a standard mathematical construction, but even if you haven't seen it before it should resonate with something you know from quantum mechanics. There we have states in a Hilbert space with kets |ψ⟩ ∈ H and a dual H... | {
"Header 1": "The Commutator",
"Header 3": "2.3 Tensors",
"token_count": 1801,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
} |
In Section [2.2.4](#page-71-0), we explained how to construct the Lie derivative, which differentiates a vector field in the direction of a second vector field X. This same idea can be adapted to one-forms.
Under a map φ : M → N, we saw that a vector field X on M can be pushed forwards to a vector field φ∗X on N. In ... | {
"Header 1": "The Commutator",
"Header 3": "2.3.2 The Lie Derivative Revisited",
"token_count": 2041,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
} |
Given a tensor S of rank (p,q) and a tensor T of rank (r,s), we can form the *tensor product*, $S \otimes T$ which a new tensor of rank (p+r,q+s), defined by
$$S \otimes T(\omega_1, \dots, \omega_p, \eta_1, \dots, \eta_r, X_1, \dots, X_q, Y_1, \dots, Y_s)$$
= $S(\omega_1, \dots, \omega_p, X_1, \dots, X_q)T(\eta_... | {
"Header 1": "The Commutator",
"Header 3": "2.3.2 The Lie Derivative Revisited",
"token_count": 2028,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
} |
This construction is called the wedge product, and is defined by
$$(\omega \wedge \eta)_{\mu_1\dots\mu_p\nu_1\dots\nu_q} = \frac{(p+q)!}{p!q!} \omega_{[\mu_1\dots\mu_p} \eta_{\nu_1\dots\nu_q]}$$
where the [. . .] in the subscript tells us to anti-symmetrise over all indices. For example, given ω, η ∈ Λ 1 (M), we ca... | {
"Header 1": "The Commutator",
"Header 3": "2.3.2 The Lie Derivative Revisited",
"token_count": 856,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
} |
We learned in Section [2.3.1](#page-75-0) how to construct a one-form df from a function f. In a coordinate basis, this one-form has components([2.17\)](#page-76-0),
$$df = \frac{\partial f}{\partial x^{\mu}} dx^{\mu}$$
We can extend this definition to higher forms. The exterior derivative is a map
$$d: \Lambda^p... | {
"Header 1": "The Commutator",
"Header 3": "2.4.1 The Exterior Derivative",
"token_count": 2022,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
} |
The electromagnetic gauge field A<sup>µ</sup> = (ϕ, A) should really be thought of as the components of a one-form on spacetime R<sup>4</sup> . (Here I've set c = 1.) We write
$$A = A_{\mu}(x)dx^{\mu}$$
Taking the exterior derivative yields a 2-form F = dA, given by
$$F = \frac{1}{2} F_{\mu\nu} dx^{\mu} \wedge dx... | {
"Header 1": "The Electromagnetic Field",
"token_count": 2024,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
} |
For example, the first law of thermodynamics is written as
$$dE = dQ + dW$$
Here dE is the infinitesimal change of energy in the system. The first law of thermodynamics, as written above, states that this decomposes into the heat flowing into the system dQ and the work done on the system dW.
Why the stupid notati... | {
"Header 1": "The Electromagnetic Field",
"token_count": 409,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
} |
The exterior derivative is a map which squares to zero, d <sup>2</sup> = 0. It turns out that one can have a lot of fun with such maps. We will now explore a little bit of this fun.
First a repeat of definitions we met already: a p-form is closed if dω = 0 everywhere. A p-form is exact if ω = dη everywhere for some η... | {
"Header 1": "2.4.3 A Sniff of de Rham Cohomology",
"token_count": 568,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
} |
Consider the one-dimensional manifold M = R. We can take a one-form ω = f(x)dx. This is always closed because it is a top form. It is also exact. We introduce the function
$$g(x) = \int_0^x dx' \ f(x')$$
Then ω = dg.
Now consider the topologically more interesting one-dimensional manifold S 1 , which we can view ... | {
"Header 1": "2.4.3 A Sniff of de Rham Cohomology",
"Header 3": "An Example",
"token_count": 248,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
} |
On M = R<sup>2</sup> , the Poincar´e lemma ensures that all closed forms are exact. However, things change if we remove a single point and consider R<sup>2</sup> − {0, 0}. Consider the one-form defined by
$$\omega = -\frac{y}{x^2 + y^2}dx + \frac{x}{x^2 + y^2}dy$$
This is not a smooth one-form on R<sup>2</sup> beca... | {
"Header 1": "Another Example",
"token_count": 557,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
} |
We denote the set of all closed p-forms on a manifold M as Z p (M). Equivalently, Z p (M) is the kernel of the map d : Λ<sup>p</sup> (M) → Λ <sup>p</sup>+1(M).
We denote the set of all exact p-forms on a manifold M as B<sup>p</sup> (M). Equivalently, B<sup>p</sup> (M) is the range of d : Λ<sup>p</sup>−<sup>1</sup> (M... | {
"Header 1": "Betti Numbers",
"token_count": 603,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
} |
To start, we need to orient ourselves. A volume form, or orientation on a manifold of dimension dim(M) = n is a nowhere-vanishing top form v. Any top form has just a single component and can be locally written as
$$v = v(x) dx^1 \wedge \ldots \wedge dx^n$$
where we require v(x) ̸= 0. If such a top form exists every... | {
"Header 1": "Integrating over Manifolds",
"token_count": 692,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
} |
We don't have to integrate over the full manifold M. We can integrate over some lower dimensional submanifold.
A manifold Σ with dimension k < n is a submanifold of M if we can find a map ϕ : Σ → M which is one-to-one (which ensures that Σ doesn't intersect itself in M) and ϕ<sup>∗</sup> : Tp(Σ) → Tϕ(p)(M) is one-to-... | {
"Header 1": "Integrating over Manifolds",
"Header 3": "Integrating over Submanifolds",
"token_count": 357,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
} |
Until now, we have considered only smooth manifolds. There is a slight generalisation that will be useful. We define a manifold with boundary in the same way as a manifold, except the charts map ϕ : O → U where U is an open subset of R<sup>n</sup><sup>+</sup> = {{x 1 , . . . , x<sup>n</sup>} such that x <sup>n</sup> ≥ ... | {
"Header 1": "2.4.5 Stokes' Theorem",
"token_count": 327,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
} |
First, consider n = 1 with M the interval I. We introduce coordinates x ∈ [a, b] on the interval. The 0-form ω = ω(x) is simply a function and dω = (dω/dx)dx. In this case, the two sides of Stokes' theorem can be evaluated to give
$$\int_{M} d\omega = \int_{a}^{b} \frac{d\omega}{dx} dx \quad \text{and} \quad \int_{\p... | {
"Header 1": "The Mother of all Integral Theorems",
"token_count": 713,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
} |
We have yet to meet the star of the show. There is one object that we can place on a manifold whose importance dwarfs all others, at least when it comes to understanding gravity. This is the metric.
The existence of a metric brings a whole host of new concepts to the table which, collectively, are called Riemannian g... | {
"Header 1": "3. Introducing Riemannian Geometry",
"token_count": 647,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
} |
For most applications of differential geometry, we are interested in manifolds in which all diagonal entries of the metric are positive. A manifold equipped with such a metric is called a Riemannian manifold. The simplest example is Euclidean space R<sup>n</sup> which, in Cartesian coordinates, is equipped with the met... | {
"Header 1": "3.1.1 Riemannian Manifolds",
"token_count": 1492,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
} |
First, the metric gives us a natural isomorphism between vectors and covectors, g : Tp(M) → T ∗ p (M) for each p, with the one-form constructed from the contraction of g and a vector field X.
In a coordinate basis, we write X = X<sup>µ</sup>∂µ. This is mapped to a one-form which, because this is a natural isomorphism... | {
"Header 1": "3.1.1 Riemannian Manifolds",
"Header 3": "The Metric as an Isomophism",
"token_count": 1883,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
} |
#### The Hodge Dual
On an oriented manifold M, we can use the totally anti-symmetric tensor ϵ<sup>µ</sup>1,...,µ<sup>n</sup> to define a map which takes a p-form ω ∈ Λ p (M) to an (n − p)-form, denoted (⋆ ω) ∈ Λ n−p (M), defined by
$$(\star \omega)_{\mu_1\dots\mu_{n-p}} = \frac{1}{p!} \sqrt{|g|} \,\epsilon_{\mu_1... | {
"Header 1": "3.1.1 Riemannian Manifolds",
"Header 3": "The Metric as an Isomophism",
"token_count": 1191,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
} |
We can combine d and d † to construct the Laplacian, △ : Λ<sup>p</sup> (M) → Λ p (M), defined as
$$\triangle = (d+d^{\dagger})^2 = dd^{\dagger} + d^{\dagger}d$$
where the second equality follows because d <sup>2</sup> = d † <sup>2</sup> = 0. The Laplacian can be defined on both Riemannian manifolds, where it is pos... | {
"Header 1": "3.1.4 A Sniff of Hodge Theory",
"token_count": 1970,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
} |
Using the properties of the connection, we can write a general covariant derivative of a vector field as
$$\nabla_X Y = \nabla_X (Y^{\mu} e_{\mu})$$
$$= X(Y^{\mu}) e_{\mu} + Y^{\mu} \nabla_X e_{\mu}$$
$$= X^{\nu} e_{\nu} (Y^{\mu}) e_{\mu} + X^{\nu} Y^{\mu} \nabla_{\nu} e_{\mu}$$
$$= X^{\nu} \left( e_{\nu} (Y^{\... | {
"Header 1": "3.1.4 A Sniff of Hodge Theory",
"token_count": 1892,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
} |
To construct this, we will insist that the connection obeys Leibniz in the modified sense that
$$\nabla_X(\omega(Y)) = (\nabla_X \omega)(Y) + \omega(\nabla_X Y)$$
But ω(Y ) is simply a function, which means that we can also write this as
$$\nabla_X(\omega(Y)) = X(\omega(Y))$$
Putting these together gives
$$(\... | {
"Header 1": "3.1.4 A Sniff of Hodge Theory",
"token_count": 841,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
} |
Even though the connection is not a tensor, we can use it to construct two tensors. The first is a rank (1, 2) tensor T known as torsion. It is defined to act on X, Y ∈ X(M) and ω ∈ Λ 1 (M) by
$$T(\omega; X, Y) = \omega(\nabla_X Y - \nabla_Y X - [X, Y])$$
The other is a rank (1, 3) tensor R, known as curvature. It ... | {
"Header 1": "3.2.2 Torsion and Curvature",
"token_count": 329,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
} |
To demonstrate that T and R are indeed tensors, we need to show that they are linear in all arguments. Linearity in ω is straightforward. For the others, there are some small calculations to do. For example, we must show that T(ω; fX, Y ) = fT(ω; X, Y ). To see this, we just run through the definitions of the various o... | {
"Header 1": "3.2.2 Torsion and Curvature",
"Header 3": "Checking Linearity",
"token_count": 1939,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
} |
Written in an orgy of anti-symmetrised notation, this calculation gives
$$\begin{split} \nabla_{[\mu}\nabla_{\nu]}Z^{\sigma} &= \partial_{[\mu}(\nabla_{\nu]}Z^{\sigma}) + \Gamma^{\sigma}_{[\mu|\lambda|}\nabla_{\nu]}Z^{\lambda} - \Gamma^{\rho}_{[\mu\nu]}\nabla_{\rho}Z^{\sigma} \\ &= \partial_{[\mu}\partial_{\nu]}Z^{\s... | {
"Header 1": "3.2.2 Torsion and Curvature",
"Header 3": "Checking Linearity",
"token_count": 1881,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
} |
Consider $S^2$ with radius r and the round metric
$$ds^2 = r^2(d\theta^2 + \sin^2\theta \, d\phi^2)$$
We can extract the Christoffel symbols from those of flat space in polar coordinates (1.10). The non-zero components are
$$\Gamma^{\theta}_{\phi\phi} = -\sin\theta\cos\theta \ , \quad \Gamma^{\phi}_{\theta\phi}... | {
"Header 1": "Another Example: The Sphere $S^2$",
"token_count": 284,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
} |
Gauss' Theorem, also known as the divergence theorem, states that if you integrate a total derivative, you get a boundary term. There is a particular version of this theorem in curved space that we will need for later applications.
As a warm-up, we have the following result:
Lemma: The contraction of the Christoffe... | {
"Header 1": "Another Example: The Sphere $S^2$",
"Header 3": "3.2.4 The Divergence Theorem",
"token_count": 1896,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
} |
Let's briefly turn to some physics. We take the manifold M to be spacetime. In classical field theory, the dynamical degrees of freedom are objects that take values at each point in M. We call these objects fields. The simplest such object is just a function which, in physics, we call a scalar field.
As we described ... | {
"Header 1": "3.2.5 The Maxwell Action",
"token_count": 1006,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
} |
To define electric and magnetic charges, we integrate over submanifolds. For example, consider a three-dimensional spatial submanifold Σ. The electric charge in Σ is defined to be
$$Q_e = \int_{\Sigma} \star J$$
It's simple to check that this agrees with our usual definition Q<sup>e</sup> = R d <sup>3</sup>x J<sup>... | {
"Header 1": "3.2.5 The Maxwell Action",
"Header 3": "Electric and Magnetic Charges",
"token_count": 806,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
} |
$$d\star J=0\quad\Leftrightarrow\quad\nabla_{\mu}J^{\mu}=0$$
**Proof:** We have
$$\nabla_{\mu}J^{\mu} = \partial_{\mu}J^{\mu} + \Gamma^{\mu}_{\mu\rho}J^{\rho} = \frac{1}{\sqrt{-g}}\partial_{\mu}\left(\sqrt{-g}J^{\mu}\right)$$
where, in the second equality, we have used our previous result (3.21): $\Gamma^{\mu}_{... | {
"Header 1": "Claim:",
"token_count": 777,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
} |
Although we have now met a number of properties of the connection, we have not yet explained its name. What does it connect?
The answer is that the connection connects tangent spaces, or more generally any tensor vector space, at different points of the manifold. This map is called *parallel transport*. As we stresse... | {
"Header 1": "Claim:",
"Header 3": "3.3 Parallel Transport",
"token_count": 2045,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
} |
This time we fall short, leaving
$$\frac{1}{4}n^2(n+1)^2 - \frac{1}{6}n^2(n+1)(n+2) = \frac{1}{12}n^2(n^2-1)$$
unaccounted for. This, therefore, is the number of ways to characterise the second derivative of the metric in a manner that cannot be undone by coordinate transformations. Indeed, it is not hard to show t... | {
"Header 1": "Claim:",
"Header 3": "3.3 Parallel Transport",
"token_count": 2027,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
} |
\frac{dZ^{\mu}}{d\tau} \right|_{\tau=0} \delta \tau + \frac{1}{2} \left. \frac{d^2 Z^{\mu}}{d\tau^2} \right|_{\tau=0} \delta \tau^2 + \mathcal{O}(\delta \tau^3)$$
**Figure 24:** Parallel transporting a vector $Z_p$ along two different paths does not give the same answer.
From (3.32... | {
"Header 1": "Claim:",
"Header 3": "3.3 Parallel Transport",
"token_count": 2042,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
} |
The calculation above is closely related to the idea of holonomy. Here, one transports a vector around a closed curve C and asks how the resulting vector compares to the original. This too is captured by the Riemann tensor. A particularly simple example of non-trivial holonomy comes from parallel transport of a vector ... | {
"Header 1": "Claim:",
"Header 3": "3.3 Parallel Transport",
"token_count": 1740,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
} |
It is simple to determine the geodesics on the sphere S <sup>2</sup> of radius r. Using the Christoffel symbols([3.19\)](#page-116-0), the geodesic equations are
$$\frac{d^2\theta}{d\tau^2} = \sin\theta\cos\theta \left(\frac{d\phi}{d\tau}\right)^2 \quad \text{and} \quad \frac{d^2\phi}{d\tau^2} = -2\frac{\cos\theta}{\... | {
"Header 1": "An Example: the Sphere S<sup>2</sup> Again",
"token_count": 2039,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
} |
First, given a rank (1, 3) tensor, we can always construct a rank (0, 2) tensor by contraction. If we start with the Riemann tensor, the resulting object is called the Ricci tensor. It is defined by
$$R_{\mu\nu} = R^{\rho}{}_{\mu\rho\nu}$$
The Ricci tensor inherits its symmetry from the Riemann tensor. We write R... | {
"Header 1": "An Example: the Sphere S<sup>2</sup> Again",
"token_count": 2029,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
} |
This follows from the first of two *Cartan structure relations*:
Claim: For a torsion free connection,
$$d\hat{\theta}^a + \omega^a{}_b \wedge \hat{\theta}^b = 0 \tag{3.46}$$
**Proof:** We first look at the second term.
$$\omega^a{}_b \wedge \hat{\theta}^b = \Gamma^a_{cb} \left( e^c{}_\mu dx^\mu \right) \wedge ... | {
"Header 1": "An Example: the Sphere S<sup>2</sup> Again",
"token_count": 2006,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
} |
The exterior derivatives are simply
$$d\hat{\theta}^0 = f'\,dr \wedge dt \ , \ d\hat{\theta}^1 = 0 \ , \ d\hat{\theta}^2 = dr \wedge d\theta \ , \ d\hat{\theta}^3 = \sin\theta\,dr \wedge d\phi + r\cos\theta\,d\theta \wedge d\phi$$
The first Cartan structure relation, $d\hat{\theta}^a = -\omega^a{}_b \wedge \hat{\t... | {
"Header 1": "An Example: the Sphere S<sup>2</sup> Again",
"token_count": 2048,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
} |
. , N. In fact, as we saw above, this "vector" is really a one-form. The novelty is that it's a Lie algebra-valued one-form.
Mathematicians don't refer to A<sup>µ</sup> as a gauge potential. Instead, they call it a connection (on a fibre bundle). This relationship becomes clearer if we look at how A<sup>µ</sup> chang... | {
"Header 1": "An Example: the Sphere S<sup>2</sup> Again",
"token_count": 520,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
} |
All our fundamental theories of physics are described by action principles. Gravity is no different. Furthermore, the straight-jacket of differential geometry places enormous restrictions on the kind of actions that we can write down. These restrictions ensure that the action is something intrinsic to the metric itself... | {
"Header 1": "4. The Einstein Equations",
"Header 3": "4.1 The Einstein-Hilbert Action",
"token_count": 347,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
} |
We would like to determine the Euler-Lagrange equations arising from the action([4.1\)](#page-146-0). We do this in the usual way, by starting with some fixed metric gµν(x) and seeing how the action changes when we shift
$$g_{\mu\nu}(x) \to g_{\mu\nu}(x) + \delta g_{\mu\nu}(x)$$
Writing the Ricci scalar as R = g µν... | {
"Header 1": "4. The Einstein Equations",
"Header 3": "Varying the Einstein-Hilbert Action",
"token_count": 2032,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
} |
Requiring that the action is extremised, so δS = 0, we have the equations of motion
$$G_{\mu\nu} := R_{\mu\nu} - \frac{1}{2}Rg_{\mu\nu} = 0 \tag{4.4}$$
where Gµν is the Einstein tensor defined in Section [3.4.1.](#page-137-0) These are the Einstein field equations in the absence of any matter. In fact they simplify... | {
"Header 1": "4. The Einstein Equations",
"Header 3": "Varying the Einstein-Hilbert Action",
"token_count": 1525,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
} |
The Einstein-Hilbert action (with cosmological constant) is the simplest thing we can write down but it is not the only possibility, at least if we allow for higher derivative terms. For example, there are three terms that contain four derivatives of the metric,
$$S_{4-\text{deriv}} = \int d^4x \sqrt{-g} \left( c_1 R... | {
"Header 1": "4. The Einstein Equations",
"Header 3": "Higher Derivative Terms",
"token_count": 2033,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
} |
In contrast, symmetries of the action are those variations δgµν for which δS = 0 for any choice of metric. Since diffeomorphisms are (gauge) symmetries, we know that the action is invariant under changes of the form (4.7). Using the fact that Gµν is symmetric, we must have
$$\delta S = 2 \int d^4x \sqrt{-g} \, G^{\mu... | {
"Header 1": "4. The Einstein Equations",
"Header 3": "Higher Derivative Terms",
"token_count": 890,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
} |
We now turn to the Einstein equations with Λ > 0. Once again, there are many solutions. Since it's a pain to solve the Einstein equations, let's work with an ansatz that we've already seen. Suppose that we look for solutions of the form
$$ds^{2} = -f(r)^{2}dt^{2} + f(r)^{-2}dr^{2} + r^{2}(d\theta^{2} + \sin^{2}\theta... | {
"Header 1": "4.2.1 de Sitter Space",
"token_count": 2040,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
} |
We will have to wait until Section [4.4.2](#page-175-0) to get a full understanding of the physics behind this. But we can make some progress by writing the de Sitter metric in different coordinates. In fact, it turns out that there's a rather nice way of embedding de Sitter space as a sub-manifold of R<sup>1</sup>,<su... | {
"Header 1": "4.2.1 de Sitter Space",
"Header 3": "de Sitter Embeddings",
"token_count": 2013,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
} |
Because the anti-de Sitter metric [\(4.22](#page-161-0)) falls in the general class([4.9](#page-156-0)), we can import the geodesic equations that we derived for de Sitter space. The radial trajectory of a massive particle moving in the θ = π/2 plane is again governed by
$$\dot{r}^2 + V_{\text{eff}}(r) = E^2 \tag{4.2... | {
"Header 1": "Geodesics in Anti-de Sitter",
"token_count": 1961,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
} |
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