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Although the change of coordinates is tricky, the metric is very straightforward, taking the form
$$ds^{2} = R^{2} \frac{dr^{2}}{r^{2}} + \frac{r^{2}}{R^{2}} \eta_{ij} dx^{i} dx^{j}$$
(4.28)
These coordinates don't cover the whole of AdS; instead they cover only one-half of the hyperboloid, restricted to X<sup>4</s... | {
"Header 1": "Geodesics in Anti-de Sitter",
"token_count": 1717,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
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The isometries of de Sitter and anti-de Sitter are simplest to see from their embeddings. The constraint([4.17\)](#page-159-0) that defines de Sitter space is invariant under the rotations of R<sup>1</sup>,<sup>4</sup> , and so de Sitter inherits the SO(1, 4) isometry group. Similarly, the constraint([4.26\)](#page-164... | {
"Header 1": "Geodesics in Anti-de Sitter",
"Header 3": "More Examples: de Sitter and anti-de Sitter",
"token_count": 1966,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
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We can do this by looking at the action for a massive particle (in the form (1.32))
$$S = \int d\tau \ g_{\mu\nu}(x) \frac{dx^{\mu}}{d\tau} \frac{dx^{\nu}}{d\tau}$$
Now consider the infinitesimal transformation
$$\delta x^{\mu} = K^{\mu}(x)$$
The action transforms as
$$\delta S = \int d\tau \, \partial_{\rho}... | {
"Header 1": "Geodesics in Anti-de Sitter",
"Header 3": "More Examples: de Sitter and anti-de Sitter",
"token_count": 2046,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
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However, it turns out that for our applications in Section [6](#page-238-0) the case of the vacuum Einstein equations Rµν = 0 is all we'll need.
#### 4.4 Asymptotics of Spacetime
The three solutions – Minkowski, de Sitter, and anti-de Sitter – have different spacetime curvature and differ in their symmetries. But t... | {
"Header 1": "Geodesics in Anti-de Sitter",
"Header 3": "More Examples: de Sitter and anti-de Sitter",
"token_count": 1792,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
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There are a number of interesting and deep stories associated to conformal transformations([4.34\)](#page-173-0). For example, there are a class of theories that are invariant under conformal transformations of Minkowski space; these so-called conformal field theories describe physics at a second order phase transition... | {
"Header 1": "Geodesics in Anti-de Sitter",
"Header 3": "4.4.2 Penrose Diagrams",
"token_count": 2027,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
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This means that, after a conformal compactification, ˜u and ˜v take values in
$$-\frac{\pi}{2} \le \tilde{u} \le \tilde{v} \le \frac{\pi}{2}$$
To draw a diagram corresponding to the spacetime([4.37\)](#page-178-0), we're going to have to ditch some dimensions. We chose not to depict the S 2 , and only show the ˜u a... | {
"Header 1": "Geodesics in Anti-de Sitter",
"Header 3": "4.4.2 Penrose Diagrams",
"token_count": 1980,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
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We then have
$$X^0 \approx R\epsilon \sinh(t/R)$$
and $X^4 \approx R\epsilon \cosh(t/R)$
We can now send ϵ → 0, keeping X<sup>0</sup> and X<sup>4</sup> finite provided that we also send t → ±∞. To do this, we must ensure that we keep the combination ϵ e<sup>±</sup>t/R finite. This means that we can identify the su... | {
"Header 1": "Geodesics in Anti-de Sitter",
"Header 3": "4.4.2 Penrose Diagrams",
"token_count": 678,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
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The global coordinates for anti-de Sitter space are [\(4.23](#page-161-0)),
$$ds^{2} = -\cosh^{2}\rho \, dt^{2} + R^{2}d\rho^{2} + R^{2}\sinh^{2}\rho \, d\Omega_{2}^{2}$$
with ρ ∈ [0, +∞). To construct the Penrose diagram, this time we introduce a "conformal radial coordinate" ψ, defined by
$$\frac{d\psi}{d\rho} ... | {
"Header 1": "Anti-de Sitter Space",
"token_count": 2046,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
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For example, we could equally well consider the theory
$$S_{\text{scalar}} = \int d^4x \sqrt{-g} \left( -\frac{1}{2} g^{\mu\nu} \nabla_{\mu} \phi \nabla_{\nu} \phi - V(\phi) - \frac{1}{2} \xi R \phi^2 \right)$$
(4.44)
for some constant ξ. This reduces to the flat space action([4.42\)](#page-186-0) when we take gµν ... | {
"Header 1": "Anti-de Sitter Space",
"token_count": 607,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
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We already met the action for Maxwell theory in Section [3.2.5](#page-119-0) as an example of integrating forms over manifolds. It is given by
$$S_{\text{Maxwell}} = -\frac{1}{2} \int F \wedge \star F = -\frac{1}{4} \int d^4 x \sqrt{-g} g^{\mu\rho} g^{\nu\sigma} F_{\mu\nu} F_{\rho\sigma}$$
(4.45)
with Fµν = ∂µA<sup... | {
"Header 1": "Anti-de Sitter Space",
"Header 3": "Maxwell Theory",
"token_count": 2038,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
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We take the action (4.43) and vary with respect to the metric. We will need the result $\delta\sqrt{-g} = -\frac{1}{2}\sqrt{-g} g_{\mu\nu} \delta g^{\mu\nu}$ from Section 4.1. We then find
$$\delta S_{\text{scalar}} = \int d^4 x \, \sqrt{-g} \left( \frac{1}{4} g_{\mu\nu} \nabla^{\rho} \phi \, \nabla_{\rho} \phi + \... | {
"Header 1": "Anti-de Sitter Space",
"Header 3": "Maxwell Theory",
"token_count": 1999,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
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(Often, we take Σ = R<sup>3</sup> so that B = S 2 <sup>∞</sup> × I with I an interval, and we only have to require that there are no currents at infinity.) This is the statement of charge conservation.
In Minkowski space, this same argument works just as well for P µ (Σ), meaning that we are able to assign conserved ... | {
"Header 1": "Anti-de Sitter Space",
"Header 3": "Maxwell Theory",
"token_count": 1424,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
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There are situations where spacetime does not have a Killing vector yet, intuitively, we would still like to associate something analogous to energy. This is where things start to get subtle.
A simple situation where this arises is two orbiting stars. It turns out that the resulting spacetime does not admit a timelik... | {
"Header 1": "Anti-de Sitter Space",
"Header 3": "Conserved Energy Without a Killing Vector?",
"token_count": 846,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
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In flat space, fermions transform in the spinor representation of the SO(1, 3) Lorentz group. Recall from the lectures on [Quantum Field Theory](http://www.damtp.cam.ac.uk/user/tong/qft.html) that we first introduce gamma matrices obeying the Clifford algebra
$$\{\gamma_a, \gamma_b\} = 2\eta_{ab} \tag{4.58}$$
Notic... | {
"Header 1": "4.5.6 Spinors",
"token_count": 1920,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
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If we know the kind of matter that fills spacetime, then we can just go ahead and solve the Einstein equations. However, we will often want to make more general statements about the allowed properties of spacetime without reference to any specific matter content. In this case, it is useful to place certain restrictions... | {
"Header 1": "4.5.7 Energy Conditions",
"token_count": 2030,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
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One can check that the extra condition (4.62) is satisfied for a scalar field. For a perfect fluid we have
$$J^{\mu} = -(\rho + P)(u \cdot X)u^{\mu} - PX^{\mu}$$
It's simple to check that the requirement JµJ <sup>µ</sup> ≤ 0 is simply ρ <sup>2</sup> ≥ P 2 .
The validity of the various energy conditions becomes ... | {
"Header 1": "4.5.7 Energy Conditions",
"token_count": 436,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
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There are surprisingly few phenomena in Nature where we need to solve the Einstein equations sourced by matter,
$$G_{\mu\nu} + \Lambda g_{\mu\nu} = 8\pi G T_{\mu\nu}$$
However there is one situation where the role of Tµν on the right-hand side is crucial: this is cosmology, the study of the universe as a whole.
#... | {
"Header 1": "4.6 A Taste of Cosmology",
"token_count": 2046,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
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What's left gives
$$R_{ij} = (\partial_0(a\dot{a}) + 3k - k + 3\dot{a}^2 - \dot{a}^2 - \dot{a}^2) \,\delta_{ij} + \mathcal{O}(x^2)$$
= $(a\ddot{a} + 2\dot{a}^2 + 2k) \,\delta_{ij} + \mathcal{O}(x^2)$
We now invoke the covariance argument to write
$$R_{ij} = (a\ddot{a} + 2\dot{a}^2 + 2k) \gamma_{ij} = \frac{1}{... | {
"Header 1": "4.6 A Taste of Cosmology",
"token_count": 1648,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
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The elegance of the Einstein field equations ensures that they hold a special place in the hearts of many physicists. However, any fondness you may feel for these equations will be severely tested if you ever try to solve them. The Einstein equations comprise ten, coupled partial differential equations. While a number ... | {
"Header 1": "5. When Gravity is Weak",
"token_count": 2031,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
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#### 5.1.1 Gauge Symmetry
Linearised gravity has a rather pretty gauge symmetry. This is inherited from the diffeomorphisms of the full theory. To see this, we repeat our consideration of infinitesimal diffeomorphisms from Section [4.1.3](#page-152-0). Under an infinitesimal change of coordinates
$$x^{\mu} \right... | {
"Header 1": "5. When Gravity is Weak",
"token_count": 1742,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
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Under certain circumstances, the linearised equations of general relativity reduce to the familiar Newtonian theory of gravity. These circumstances occur when we have a low-density, slowly moving distribution of matter.
For simplicity, we'll look at a stationary matter configuration. This means that we take
$$T_{00... | {
"Header 1": "5.1.2 The Newtonian Limit",
"token_count": 712,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
} |
A long time ago, in a galaxy far far away, two black holes collided. Here a "long time ago" means 1.3 billion years ago. And "far far away" means a distance of about 1.3 billion light years.
To say that this was a violent event is something of an understatement. One of the black holes was roughly 35 times heavier tha... | {
"Header 1": "5.2 Gravitational Waves",
"token_count": 524,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
} |
Gravitational waves propagate in vacuum, in the absence of any sources. This means that we need to solve the linearised equation
$$\Box \bar{h}_{\mu\nu} = 0 \tag{5.17}$$
One solution is provided by the gravitational wave
$$\bar{h}_{\mu\nu} = \text{Re}\left(H_{\mu\nu}\,e^{ik_{\rho}x^{\rho}}\right) \tag{5.18}$$
H... | {
"Header 1": "5.2.1 Solving the Wave Equation",
"token_count": 1728,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
} |
We then have
$$\frac{d^2 S^{\mu}}{dt^2} = R^{\mu}{}_{00\nu} S^{\nu}$$
The Riemann tensor in the linearised regime was previously computed in([5.3\)](#page-211-0)
$$R^{\mu}{}_{\rho\sigma\nu} = \frac{1}{2} \eta^{\mu\lambda} \left( \partial_{\sigma} \partial_{\rho} h_{\nu\lambda} - \partial_{\sigma} \partial_{\lambd... | {
"Header 1": "5.2.1 Solving the Wave Equation",
"token_count": 1449,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
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Gravitational wave detectors are interferometers. They bounce light back and forth between two arms, with the mirrors at either end playing the role of test masses.
If the gravitational wave travels perpendicular to the plane of the detector, it will shorten one arm and lengthen the other. With the arms aligned along... | {
"Header 1": "5.2.1 Solving the Wave Equation",
"Header 3": "Gravitational Wave Detectors",
"token_count": 2022,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
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Then we can further Taylor expand the current to write
$$T_{\mu\nu}(\mathbf{x}', t_{\text{ret}}) = T_{\mu\nu}(\mathbf{x}', t - r) + \dot{T}_{\mu\nu}(\mathbf{x}', t - r) \frac{\mathbf{x} \cdot \mathbf{x}'}{r} + \dots$$
(5.29)
We have two Taylor expansions, (5.28) and (5.29). At leading order in d/r we take the first... | {
"Header 1": "5.2.1 Solving the Wave Equation",
"Header 3": "Gravitational Wave Detectors",
"token_count": 2029,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
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As an example, consider two stars (or neutron stars, or black holes) each with mass M, separated by distance R, orbiting in the (x, y) plane. Using Newtonian gravity, the stars orbit with frequency
$$\omega^2 = \frac{2GM}{R^3} \tag{5.35}$$
If we treat these stars as point particles, then the energy density is simpl... | {
"Header 1": "5.2.1 Solving the Wave Equation",
"Header 3": "5.3.2 An Example: Binary Systems",
"token_count": 1932,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
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A source which emits gravitational waves will lose energy. We'd like to know how much energy is emitted. In other words, we'd like to understand how much energy is carried by the gravitational waves.
In the context of electromagnetism, it is fairly easy to calculate the analogous quantity. The energy current in elect... | {
"Header 1": "5.2.1 Solving the Wave Equation",
"Header 3": "5.3.4 Power Radiated: The Quadrupole Formula",
"token_count": 539,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
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When asked to construct an energy-momentum tensor for the metric perturbations, the first thing that springs to mind is to return to the Fierz-Pauli action (5.8). Viewed as an action describing a spin 2 field propagating in Minkowski space, we can then treat it as any other classical field theory and compute the energy... | {
"Header 1": "5.2.1 Solving the Wave Equation",
"Header 3": "A Quick and Dirty Approach: the Fierz-Pauli Action",
"token_count": 1999,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
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(You can find it in (101.6) of Landau and Lifshitz, volume 2 but it's unlikely to give you a sense of enlightenment.) The expression for the pseudo-tensor is slightly nicer in the linearised theory, but only slightly.
• The final approach is perhaps the least intuitive, but has the advantage that it gives a straightf... | {
"Header 1": "5.2.1 Solving the Wave Equation",
"Header 3": "A Quick and Dirty Approach: the Fierz-Pauli Action",
"token_count": 2004,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
} |
#### 5.3.5 Gravitational Wave Sources on the $\square$
We can do some quick, back-of-the-envelope calculations to get a sense for how much energy is emitted by a gravitational wave source. Assuming Newtonian gravity is a good approximation, two masses M, separated by a distance R, will orbit with frequency
$$\ome... | {
"Header 1": "5.2.1 Solving the Wave Equation",
"Header 3": "A Quick and Dirty Approach: the Fierz-Pauli Action",
"token_count": 1253,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
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We have already met the simplest black hole solution back in Section [1.3](#page-35-0): this is the Schwarzschild solution, with metric
$$ds^{2} = -\left(1 - \frac{2GM}{r}\right)dt^{2} + \left(1 - \frac{2GM}{r}\right)^{-1}dr^{2} + r^{2}(d\theta^{2} + \sin^{2}\theta \, d\phi^{2}) \tag{6.1}$$
It is not hard to show t... | {
"Header 1": "6.1 The Schwarzschild Solution",
"token_count": 277,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
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The Schwarzschild solution depends on a single parameter, M, which should be thought of as the mass of the black hole. This interpretation already follows from the relation to Newtonian gravity that we first discussed way back in Section [1.2](#page-17-0) where we anticipated that the g<sup>00</sup> component of the me... | {
"Header 1": "M is for Mass",
"token_count": 771,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
} |
The Schwarzschild solution([6.1\)](#page-238-0) is, it turns out, the unique spherically symmetric, asymptotically flat solution to the vacuum Einstein equations. This is known as the Birkhoff theorem. In particular, this means that the Schwarzschild solution does not just describe a black hole, but it describes the sp... | {
"Header 1": "6.1.1 Birkhoff 's Theorem",
"token_count": 2040,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
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To understand what's happening near the horizon r = 2GM, we can zoom in and look at the metric in the vicinity of the horizon. To do this, we write
$$r = 2GM + \eta$$
where we take η ≪ 2GM. We further take η > 0 which means that we're looking at the region of spacetime just outside the horizon. We then approximate ... | {
"Header 1": "6.1.1 Birkhoff 's Theorem",
"Header 3": "The Near Horizon Limit: Rindler Space",
"token_count": 1822,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
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As a first attempt to extend the Schwarzschild solution beyond the horizon, we replace t with t = v − r⋆(r). We have
$$dt = dv - dr_{\star} = dv - \left(1 - \frac{2GM}{r}\right)^{-1} dr$$
Making this substitution in the Schwarzschild metric([6.1\)](#page-238-0), we find the new metric
$$ds^{2} = -\left(1 - \frac{... | {
"Header 1": "Ingoing Eddington-Finkelstein Coordinates",
"token_count": 1982,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
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We have t = u + r⋆, so
$$dt = du + dr_{\star} = du + \left(1 - \frac{2GM}{r}\right)^{-1} dr$$
Making this substitution in the Schwarzschild metric([6.1\)](#page-238-0), we now find the metric
$$ds^{2} = -\left(1 - \frac{2GM}{r}\right)du^{2} - 2du\,dr + r^{2}\,d\Omega_{2}^{2}$$
(6.10)
This is the Schwarzschild s... | {
"Header 1": "Ingoing Eddington-Finkelstein Coordinates",
"token_count": 2046,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
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We have
$$r = 2GM \quad \Rightarrow \quad U = 0 \text{ or } V = 0$$
This tells us that the horizon is not one null surface, but two null surfaces, intersecting at the point U = V = 0. This agrees with what we learned from taking the near horizon limit where we encountered Rindler space. The null surface U = 0 is th... | {
"Header 1": "Ingoing Eddington-Finkelstein Coordinates",
"token_count": 1990,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
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To see this, note that there is a symmetry of (6.17) under ρ → G<sup>2</sup>M<sup>2</sup>/4ρ, which swaps the two asymptotic spacetimes, leaving the meeting point at ρ = GM/2 invariant. In this way, the metric (6.18) describes the two-sided Einstein-Rosen bridge shown in Figure [49.](#page-255-0)
The radius of the S ... | {
"Header 1": "Ingoing Eddington-Finkelstein Coordinates",
"token_count": 631,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
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As we explained in Section [4.4.2](#page-175-0), the best way to exhibit the causal structure of a spacetime is to draw the Penrose diagram. For the black hole, this is very similar to the Kruskal diagram: we simply straighten out a few lines.
Figure 51: The Penrose diagram for the Sch... | {
"Header 1": "Ingoing Eddington-Finkelstein Coordinates",
"Header 3": "The Penrose Diagram",
"token_count": 1570,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
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One important feature of the black hole remains: the singularity is shrouded behind the horizon. This means that the effects of the singularity cannot be felt by an asymptotic observer. We can ask: is this always the case? Or could we end up with a singularity
Figure 53:
which is not... | {
"Header 1": "Ingoing Eddington-Finkelstein Coordinates",
"Header 3": "Cosmic Censorship",
"token_count": 1141,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
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Throughout this section we have focussed on black holes in asymptotically Minkowski spacetime. It is not hard to find solutions corresponding to Schwarzschild black holes in de Sitter and anti-de Sitter spacetimes, solving the Einstein equations
$$R_{\mu\nu} = \Lambda g_{\mu\nu}$$
We have already done the hard work... | {
"Header 1": "Ingoing Eddington-Finkelstein Coordinates",
"Header 3": "6.1.6 Black Holes in (Anti) de Sitter",
"token_count": 2027,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
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So charged black holes have two horizons: an outer one at $r_{+}$ and an inner one at $r_{-}$ .
The presence of two roots changes the role played by the singularity. This is because the $g_{rr}$ metric component flips sign twice so that r is again a spatial coordinate by the time we get to $r < r_{-}$ . This su... | {
"Header 1": "Ingoing Eddington-Finkelstein Coordinates",
"Header 3": "6.1.6 Black Holes in (Anti) de Sitter",
"token_count": 2031,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
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However, this means that there is nothing inevitable about the singularity of the Reissner-Nordstr¨om black hole: there exist timelike worldlines that a test particle could follow that miss the singularity completely.
Such fortunate worldliners will ultimately end up in the upper-most quadrant of the right-hand diagr... | {
"Header 1": "Ingoing Eddington-Finkelstein Coordinates",
"Header 3": "6.1.6 Black Holes in (Anti) de Sitter",
"token_count": 2036,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
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Nonetheless, it's at least possible that there exist time independent solutions. This is in contrast to Schwarzschild or sub-extremal Reissner-Nordstr¨om black holes, where the attractive force means that two black holes must be orbiting each other, emitting gravitational waves in the process.
Given the complexity of... | {
"Header 1": "Ingoing Eddington-Finkelstein Coordinates",
"Header 3": "6.1.6 Black Holes in (Anti) de Sitter",
"token_count": 1754,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
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When ∆ = 0, the grr component of the metric diverges. Our previous experience with the Schwarzschild and Reissner-Nordstr¨om black holes suggests that these are coordinate singularities, and this turns out to be correct. We write the roots of ∆ as
$$\Delta = (r - r_+)(r - r_-)$$
with
$$r_{\pm} = GM \pm \sqrt{G^2M... | {
"Header 1": "6.3.2 The Global Structure",
"token_count": 2015,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
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Indeed, asymptotically, at r → ∞, K generates the geodesics of an observer stationary with respect to the black
Figure 61: The ergoregion outside the Kerr black hole.
hole. As we move closer to the black hole, with finite r, the integral curves of K are no longer geodesics since they... | {
"Header 1": "6.3.2 The Global Structure",
"token_count": 1993,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
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Applying this to the particle with $E_1 < 0$ that falls into the black hole, we have
$$E_1 \ge \Omega j_1 \tag{6.32}$$
In this sense, we necessarily extract more angular momentum than energy from the black hole. To see that this bound does indeed prohibit the formation of super-extremal rotating black holes, cons... | {
"Header 1": "6.3.2 The Global Structure",
"token_count": 795,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
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There is a grown-up version of the Penrose process in which fields scatter off a Kerr black hole, and return amplified. This effect is known as superradiance.
Here we sketch this phenomenon for a massless scalar field Φ. The energy-momentum tensor is [\(4.51\)](#page-191-0)
$$T_{\mu\nu} = \nabla_{\mu}\Phi\nabla_{\n... | {
"Header 1": "6.3.2 The Global Structure",
"Header 3": "Superradiance",
"token_count": 2046,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
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(It is K = ∂<sup>t</sup> in the usual coordinates.) The action for the scalar field is
$$S_{\text{scalar}} = \int d^4x \sqrt{-g} \, \frac{1}{2} \left( -g^{\mu\nu} \nabla_{\mu} \Phi \nabla_{\nu} \Phi - m^2 \Phi^2 \right)$$
$$= \int d^4x \sqrt{-g} \, \frac{1}{2} \left( -g^{tt} \partial_t \Phi \partial_t \Phi - 2g^{ti} ... | {
"Header 1": "6.3.2 The Global Structure",
"Header 3": "Superradiance",
"token_count": 778,
"source_pdf": "datasets/websources/Physics_v1/Physics/gr.pdf"
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**Fourth Edition**
David J. Griffiths *Reed College*
Executive Editor: Jim Smith Senior Project Editor: Martha Steele Development Manager: Laura Kenney Managing Editor: Corinne Benson Production Project Manager: Dorothy Cox Production Management and Compos... | {
"Header 1": "**INTRODUCTION TO ELECTRODYNAMICS**",
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| | Preface<br>Advertisement<br>Vector<br>Analysis<br>1.1<br>Vector<br>Algebra<br>1<br>1.1.1<br>Vector<br>Operations<br>1<br>1.1.2<br>Vector<br>Algebra:<br>Component<br>Form<br>4<br>1.1.3<br>Triple<br>Products<br>7<br>1.1.4<br>Position,<br>Displacement,<br>and<br>Separation<br>Vectors<br>8<br>1.1.5<br>How<br>Vectors<... | {
"Header 1": "**Contents**",
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This is a textbook on electricity and magnetism, designed for an undergraduate course at the junior or senior level. It can be covered comfortably in two semesters, maybe even with room to spare for special topics (AC circuits, numerical methods, plasma physics, transmission lines, antenna theory, etc.) A one-semester ... | {
"Header 1": "**Preface**",
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In the diagram below, I have sketched out the four great realms of mechanics:
| Classical<br>Mechanics<br>(Newton) | Quantum<br>Mechanics<br>(Bohr,<br>Heisenberg,<br>Schrödinger,<br>et<br>al.) |
|------------------------------------|-----------------------------------------------------------------------------|
| Spec... | {
"Header 1": "**Four Realms of Mechanics**",
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Mechanics tells us how a system will behave when subjected to a given *force.* There are just *four* basic forces known (presently) to physics: I list them in the order of decreasing strength:
Advertisement **xv**
- 1. Strong
- 2. Electromagnetic
- 3. Weak
- 4. Gravitational
The brevity of this list may surprise ... | {
"Header 1": "**Four Kinds of Forces**",
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In the beginning, **electricity** and **magnetism** were entirely separate subjects. The one dealt with glass rods and cat's fur, pith balls, batteries, currents, electrolysis, and lightning; the other with bar magnets, iron filings, compass needles, and the North Pole. But in 1820 Oersted noticed that an *electric* cu... | {
"Header 1": "**The Unification of Physical Theories**",
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The fundamental problem a theory of electromagnetism hopes to solve is this: I hold up a bunch of electric charges *here* (and maybe shake them around); what happens to some *other* charge, over *there?* The classical solution takes the form of a **field theory**: We say that the space around an electric charge is perm... | {
"Header 1": "**The Field Formulation of Electrodynamics**",
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- 1. Charge comes in two varieties, which we call "plus" and "minus," because their effects tend to cancel (if you have +q and -q at the same point, electrically it is the same as having no charge there at all). This may seem too obvious to warrant comment, but I encourage you to contemplate other possibilities: what i... | {
"Header 1": "**Electric Charge**",
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If you walk 4 miles due north and then 3 miles due east (Fig. 1.1), you will have gone a total of 7 miles, but you're *not* 7 miles from where you set out—you're only 5. We need an arithmetic to describe quantities like this, which evidently do not add in the ordinary way. The reason they don't, of course, is that **di... | {
"Header 1": "**1.1.1 Vector Operations**",
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In the previous section, I defined the four vector operations (addition, scalar multiplication, dot product, and cross product) in "abstract" form—that is, without reference to any particular coordinate system. In practice, it is often easier to set up Cartesian coordinates x, y, z and work with vector **components**. ... | {
"Header 1": "1.1.2 ■ Vector Algebra: Component Form",
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Since the cross product of two vectors is itself a vector, it can be dotted or crossed with a third vector to form a *triple* product.
(i) Scalar triple product: $\mathbf{A} \cdot (\mathbf{B} \times \mathbf{C})$ . Geometrically, $|\mathbf{A} \cdot (\mathbf{B} \times \mathbf{C})|$ is the volume of the parallelepipe... | {
"Header 1": "**1.1.3** ■ Triple Products",
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The location of a point in three dimensions can be described by listing its Cartesian coordinates (*x*, *y*,*z*). The vector to that point from the origin (O) is called the **position vector** (Fig. 1.13):
$$\mathbf{r} \equiv x\,\hat{\mathbf{x}} + y\,\hat{\mathbf{y}} + z\,\hat{\mathbf{z}}.\tag{1.19}$$
![](_page_26_... | {
"Header 1": "**1.1.4 Position, Displacement, and Separation Vectors**",
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The definition of a vector as "a quantity with a magnitude and direction" is not altogether satisfactory: What precisely does "direction" *mean*? This may seem a pedantic question, but we shall soon encounter a species of derivative that *looks* rather like a vector, and we'll want to know for sure whether it *is* one.... | {
"Header 1": "**1.1.4 Position, Displacement, and Separation Vectors**",
"Header 2": "1.1.5 ■ How Vectors Transform<sup>2</sup>",
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1.13) transform under inversion? [The cross-product of two vectors is properly called a **pseudovector** because of this "anomalous" behavior.] Is the cross product of two pseudovectors a vector, or a pseudovector? Name two pseudovector quantities in classical mechanics.
- (d) How does the scalar triple product of thre... | {
"Header 1": "**1.1.4 Position, Displacement, and Separation Vectors**",
"Header 2": "1.1.5 ■ How Vectors Transform<sup>2</sup>",
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Suppose we have a function of one variable: f(x). Question: What does the derivative, df/dx, do for us? Answer: It tells us how rapidly the function f(x) varies when we change the argument x by a tiny amount, dx:
$$df = \left(\frac{df}{dx}\right)dx. \tag{1.33}$$
In words: If we increment x by an infinitesimal amoun... | {
"Header 1": "**1.1.4 Position, Displacement, and Separation Vectors**",
"Header 2": "1.2.1 ■ \"Ordinary\" Derivatives",
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- (a) Where is the top of the hill located?
- (b) How high is the hill?
- (c) How steep is the slope (in feet per mile) at a point 1 mile north and one mile east of South Hadley? In what direction is the slope steepest, at that point?
- **Problem 1.13** Let $\boldsymbol{\nu}$ be the separation vector from a fixed p... | {
"Header 1": "**1.1.4 Position, Displacement, and Separation Vectors**",
"Header 2": "1.2.1 ■ \"Ordinary\" Derivatives",
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The gradient has the formal appearance of a vector, $\nabla$ , "multiplying" a scalar T:
$$\nabla T = \left(\hat{\mathbf{x}}\frac{\partial}{\partial x} + \hat{\mathbf{y}}\frac{\partial}{\partial y} + \hat{\mathbf{z}}\frac{\partial}{\partial z}\right)T. \tag{1.38}$$
(For once, I write the unit vectors to the *left*... | {
"Header 1": "1.2.3 ■ The Del Operator",
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From the definition of $\nabla$ we construct the divergence:
$$\nabla \cdot \mathbf{v} = \left(\hat{\mathbf{x}} \frac{\partial}{\partial x} + \hat{\mathbf{y}} \frac{\partial}{\partial y} + \hat{\mathbf{z}} \frac{\partial}{\partial z}\right) \cdot (v_x \hat{\mathbf{x}} + v_y \hat{\mathbf{y}} + v_z \hat{\mathbf{z}})$... | {
"Header 1": "**1.2.4** ■ The Divergence",
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#### **Solution**
$$\nabla \times \mathbf{v}_a = \begin{vmatrix} \hat{\mathbf{x}} & \hat{\mathbf{y}} & \hat{\mathbf{z}} \\ \partial/\partial x & \partial/\partial y & \partial/\partial z \\ -y & x & 0 \end{vmatrix} = 2\hat{\mathbf{z}},$$
and
$$\nabla \times \mathbf{v}_b = \begin{vmatrix} \hat{\mathbf{x}} & \hat... | {
"Header 1": "**1.2.4** ■ The Divergence",
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1.22 for the definition of $(\mathbf{A} \cdot \nabla)\mathbf{B}$ .
Problem 1.24 Derive the three quotient rules.
#### Problem 1.25
(a) Check product rule (iv) (by calculating each term separately) for the functions
$$\mathbf{A} = x \,\hat{\mathbf{x}} + 2y \,\hat{\mathbf{y}} + 3z \,\hat{\mathbf{z}}; \qquad \mat... | {
"Header 1": "**1.2.4** ■ The Divergence",
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*Check* it for function (b) in Prob. 1.11.
#### 1.3 ■ INTEGRAL CALCULUS
#### 1.3.1 ■ Line, Surface, and Volume Integrals
In electrodynamics, we encounter several different kinds of integrals, among which the most important are line (or path) integrals, surface integrals (or flux), and volume integrals.
(a) Line... | {
"Header 1": "**1.2.4** ■ The Divergence",
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#### **Solution**
Taking the sides one at a time:
(i) *x* = 2, *d***a** = *dy dz* **xˆ**, **v** · *d***a** = 2*xz dy dz* = 4*z dy dz*, so
$$\int \mathbf{v} \cdot d\mathbf{a} = 4 \int_0^2 dy \int_0^2 z \, dz = 16.$$
(ii) *x* = 0, *d***a** = −*dy dz* **xˆ**, **v** · *d***a** = −2*xz dy dz* = 0, so
$$\int \mat... | {
"Header 1": "**1.2.4** ■ The Divergence",
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You can do the three integrals in any order. Let's do *x* first: it runs from 0 to (1 − *y*), then *y* (it goes from 0 to 1), and finally *z* (0 to 3):
$$\int T d\tau = \int_0^3 z^2 \left\{ \int_0^1 y \left[ \int_0^{1-y} x \, dx \right] dy \right\} dz$$
$$= \frac{1}{2} \int_0^3 z^2 \, dz \int_0^1 (1-y)^2 y \, dy = \f... | {
"Header 1": "**Solution**",
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As always, $d\mathbf{l} = dx \,\hat{\mathbf{x}} + dy \,\hat{\mathbf{y}} + dz \,\hat{\mathbf{z}}; \, \nabla T = y^2 \,\hat{\mathbf{x}} + 2xy \,\hat{\mathbf{y}}.$
(i)
$$y = 0$$
; $d\mathbf{l} = dx \hat{\mathbf{x}}$ , $\nabla T \cdot d\mathbf{l} = y^2 dx = 0$ , so
$$\int_{\mathbf{i}} \nabla T \cdot d\mathbf{l} = 0.... | {
"Header 1": "**Solution**",
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The fundamental theorem for divergences states that:
$$\int_{\mathcal{V}} (\nabla \cdot \mathbf{v}) \, d\tau = \oint_{\mathcal{S}} \mathbf{v} \cdot d\mathbf{a}. \tag{1.56}$$
In honor, I suppose, of its great importance, this theorem has at least three special names: **Gauss's theorem**, **Green's theorem**, or simp... | {
"Header 1": "1.3.4 ■ The Fundamental Theorem for Divergences",
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**Corollary 2:** $\oint (\nabla \times \mathbf{v}) \cdot d\mathbf{a} = 0$ for any closed surface, since the boundary line, like the mouth of a balloon, shrinks down to a point, and hence the right side of Eq. 1.57 vanishes.
These corollaries are analogous to those for the gradient theorem. We will develop the par... | {
"Header 1": "1.3.4 ■ The Fundamental Theorem for Divergences",
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$$\int_0^\infty x e^{-x} \, dx.$$
#### **Solution**
The exponential can be expressed as a derivative:
$$e^{-x} = \frac{d}{dx} \left( -e^{-x} \right);$$
in this case, then, f(x) = x, $g(x) = -e^{-x}$ , and df/dx = 1, so
$$\int_0^\infty x e^{-x} \, dx = \int_0^\infty e^{-x} \, dx - x e^{-x} \Big|_0^\infty = -e... | {
"Header 1": "**Example 1.12.** Evaluate the integral",
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You can label a point P by its Cartesian coordinates (x, y, z), but sometimes it is more convenient to use **spherical** coordinates $(r, \theta, \phi)$ ; r is the distance from the origin (the magnitude of the position vector $\mathbf{r}$ ), $\theta$ (the angle down from the z axis) is called the **polar angle**, ... | {
"Header 1": "1.4.1 ■ Spherical Coordinates",
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1.39), so
$$d\mathbf{a}_1 = dl_\theta \, dl_\phi \, \hat{\mathbf{r}} = r^2 \sin\theta \, d\theta \, d\phi \, \hat{\mathbf{r}}.$$
On the other hand, if the surface lies in the xy plane, say, so that $\theta$ is constant (to wit: $\pi/2$ ) while r and $\phi$ vary, then
$$d\mathbf{a}_2 = dl_r \, dl_\phi \, \hat... | {
"Header 1": "1.4.1 ■ Spherical Coordinates",
"token_count": 1891,
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Also work out the inverse formulas, giving $\hat{\mathbf{x}}$ , $\hat{\mathbf{y}}$ , $\hat{\mathbf{z}}$ in terms of $\hat{\mathbf{r}}$ , $\hat{\boldsymbol{\theta}}$ , $\hat{\boldsymbol{\phi}}$ (and $\theta$ , $\boldsymbol{\phi}$ ).
- Problem 1.39
- (a) Check the divergence theorem for the function $\mathbf{v... | {
"Header 1": "1.4.1 ■ Spherical Coordinates",
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The cylindrical coordinates (*s*,φ,*z*) of a point *P* are defined in Fig. 1.42. Notice that φ has the same meaning as in spherical coordinates, and *z* is the same as Cartesian; *s* is the distance to *P from the z axis,* whereas the spherical coordinate *r* is the distance from the *origin*. The relation to Cartesian... | {
"Header 1": "**1.4.2 Cylindrical Coordinates**",
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Consider the vector function
$$\mathbf{v} = \frac{1}{r^2} \,\hat{\mathbf{r}}.\tag{1.83}$$
At every location, **v** is directed radially outward (Fig. 1.44); if ever there was a function that ought to have a large positive divergence, this is it. And yet, when you actually *calculate* the divergence (using Eq. 1.71)... | {
"Header 1": "**1.5.1** ■ The Divergence of $\\hat{\\mathbf{r}}/r^2$",
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$$\int_0^3 x^3 \delta(x-2) \, dx.$$
#### **Solution**
The delta function picks out the value of $x^3$ at the point x = 2, so the integral is $2^3 = 8$ . Notice, however, that if the upper limit had been 1 (instead of 3), the answer would be 0, because the spike would then be outside the domain of integration.
... | {
"Header 1": "**Example 1.14.** Evaluate the integral",
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Using Eq. 1.59, we transfer the derivative from **rˆ**/*r* <sup>2</sup> to (*r* <sup>2</sup> + 2):
$$J = -\int_{\mathcal{V}} \frac{\hat{\mathbf{r}}}{r^2} \cdot \left[ \nabla (r^2 + 2) \right] d\tau + \oint_{\mathcal{S}} (r^2 + 2) \frac{\hat{\mathbf{r}}}{r^2} \cdot d\mathbf{a}.$$
The gradient is
$$\nabla(r^2+2)=2r... | {
"Header 1": "**Solution 2**",
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- (a) **∇** × **F** = **0** everywhere.
- (b) **<sup>b</sup> <sup>a</sup> F** · *d***l** is independent of path, for any given end points.
- (c) **F** · *d***l** = 0 for any closed loop.
- (d) **F** is the gradient of some scalar function: **F** = −**∇***V*.
The potential is not unique—any constant can be added to ... | {
"Header 1": "**Solution 2**",
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**Divergence-less**(or "**solenoidal**") **fields**. The following conditions are equivalent:
- (a) **∇** · **F** = 0 everywhere.
- (b) **F** · *d***a** is independent of surface, for any given boundary line.
- (c) **F** · *d***a** = 0 for any closed surface.
- (d) **F** is the curl of some vector function: **F** = *... | {
"Header 1": "**Theorem 2**",
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**Problem 1.54** Check the divergence theorem for the function
$$\mathbf{v} = r^2 \cos \theta \,\, \hat{\mathbf{r}} + r^2 \cos \phi \,\, \hat{\boldsymbol{\theta}} - r^2 \cos \theta \, \sin \phi \,\, \hat{\boldsymbol{\phi}},$$
using as your volume one octant of the sphere of radius *R* (Fig. 1.48). Make sure you inc... | {
"Header 1": "**More Problems on Chapter 1**",
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[*Answer*: $\nabla \times (r^n \hat{\mathbf{r}}) = \mathbf{0}$ ]
**Problem 1.64** In case you're not persuaded that $\nabla^2(1/r) = -4\pi\delta^3(\mathbf{r})$ (Eq. 1.102 with $\mathbf{r}' = \mathbf{0}$ for simplicity), try replacing r by $\sqrt{r^2 + \epsilon^2}$ , and watching what happens as $\epsilon \to 0... | {
"Header 1": "**More Problems on Chapter 1**",
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#### 2.1 ■ THE ELECTRIC FIELD
#### 2.1.1 ■ Introduction
The fundamental problem electrodynamics hopes to solve is this (Fig. 2.1): We have some electric charges, $q_1, q_2, q_3, \ldots$ (call them **source charges**); what force do they exert on another charge, Q (call it the **test charge**)? The positions of th... | {
"Header 1": "**Electrostatics**",
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What is the force on a test charge *Q* due to a single point charge *q*, that is at *rest* a distance r away? The answer (based on experiments) is given by **Coulomb's law**:
$$\mathbf{F} = \frac{1}{4\pi\epsilon_0} \frac{q \, Q}{\imath^2} \hat{\mathbf{i}}. \tag{2.1}$$
The constant <sup>0</sup> is called (ludicrousl... | {
"Header 1": "**2.1.2 Coulomb's Law**",
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If we have *several* point charges $q_1, q_2, \ldots, q_n$ , at distances $v_1, v_2, \ldots, v_n$ from Q, the total force on Q is evidently
$$\mathbf{F} = \mathbf{F}_{1} + \mathbf{F}_{2} + \dots = \frac{1}{4\pi\epsilon_{0}} \left( \frac{q_{1}Q}{v_{1}^{2}} \hat{\mathbf{i}}_{1} + \frac{q_{2}Q}{v_{2}^{2}} \hat{\mathb... | {
"Header 1": "**Problem 2.1**",
"Header 2": "2.1.3 ■ The Electric Field",
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Our definition of the electric field (Eq. 2.4) assumes that the source of the field is a collection of discrete point charges $q_i$ . If, instead, the charge is distributed continuously over some region, the sum becomes an integral (Fig. 2.5a):
$$\mathbf{E}(\mathbf{r}) = \frac{1}{4\pi\epsilon_0} \int \frac{1}{n^2} \... | {
"Header 1": "**Problem 2.1**",
"Header 2": "2.1.4 ■ Continuous Charge Distributions",
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In principle, we are *done* with the subject of electrostatics. Equation 2.8 tells us how to compute the field of a charge distribution, and Eq. 2.3 tells us what the force on a charge *Q* placed in this field will be. Unfortunately, as you may have discovered in working Prob. 2.7, the integrals involved in computing *... | {
"Header 1": "**2.2.1 Field Lines, Flux, and Gauss's Law**",
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As it stands, Gauss's law is an *integral* equation, but we can easily turn it into a *differential* one, by applying the divergence theorem:
$$\oint_{S} \mathbf{E} \cdot d\mathbf{a} = \int_{V} (\mathbf{\nabla} \cdot \mathbf{E}) \, d\tau.$$
Rewriting *Q*enc in terms of the charge density ρ, we have
$$Q_{\rm enc... | {
"Header 1": "**2.2.1 Field Lines, Flux, and Gauss's Law**",
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Let's go back, now, and calculate the divergence of E directly from Eq. 2.8:
$$\mathbf{E}(\mathbf{r}) = \frac{1}{4\pi\epsilon_0} \int_{\text{all space}} \frac{\hat{\mathbf{z}}}{v^2} \rho(\mathbf{r}') d\tau'. \tag{2.15}$$
(Originally the integration was over the volume occupied by the charge, but I may as well exten... | {
"Header 1": "2.2.2 ■ The Divergence of E",
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I must interrupt the theoretical development at this point to show you the extraordinary power of Gauss's law, in integral form. When symmetry permits, it affords *by far* the quickest and easiest way of computing electric fields. I'll illustrate the method with a series of examples.
**Example 2.3.** Find the field o... | {
"Header 1": "2.2.2 ■ The Divergence of E",
"Header 2": "2.2.3 ■ Applications of Gauss's Law",
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Draw a Gaussian cylinder of length *l* and radius *s*. For this surface, Gauss's law states:
$$\oint_{\mathcal{S}} \mathbf{E} \cdot d\mathbf{a} = \frac{1}{\epsilon_0} Q_{\text{enc}}.$$
The enclosed charge is
$$Q_{\rm enc} = \int \rho \, d\tau = \int (ks')(s' \, ds' \, d\phi \, dz) = 2\pi kl \int_0^s s'^2 \, ds' =... | {
"Header 1": "**Solution**",
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} |
I'll calculate the curl of **E**, as I did the divergence in Sect. 2.2.1, by studying first the simplest possible configuration: a point charge at the origin. In this case
$$\mathbf{E} = \frac{1}{4\pi\epsilon_0} \frac{q}{r^2} \hat{\mathbf{r}}.$$
Now, a glance at Fig. 2.12 should convince you that the curl of this f... | {
"Header 1": "**2.2.4 The Curl of E**",
"token_count": 2058,
"source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf"
} |
2.21 and is largely a matter of convention.
**Problem 2.20** One of these is an impossible electrostatic field. Which one?
(a)
$$\mathbf{E} = k[xy\,\hat{\mathbf{x}} + 2yz\,\hat{\mathbf{y}} + 3xz\,\hat{\mathbf{z}}];$$
(b)
$$\mathbf{E} = k[y^2 \,\hat{\mathbf{x}} + (2xy + z^2) \,\hat{\mathbf{y}} + 2yz \,\hat{\mathbf... | {
"Header 1": "**2.2.4 The Curl of E**",
"token_count": 2025,
"source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf"
} |
For points outside the sphere (r > R),
$$V(r) = -\int_{\mathcal{O}}^{\mathbf{r}} \mathbf{E} \cdot d\mathbf{l} = \frac{-1}{4\pi\epsilon_0} \int_{\infty}^{r} \frac{q}{r'^2} dr' = \left. \frac{1}{4\pi\epsilon_0} \frac{q}{r'} \right|_{\infty}^{r} = \frac{1}{4\pi\epsilon_0} \frac{q}{r}.$$
To find the potential inside th... | {
"Header 1": "**2.2.4 The Curl of E**",
"token_count": 718,
"source_pdf": "datasets/websources/Physics_v1/Physics/griffiths_4ed.pdf"
} |
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