Theoretical Physics Reference guide based on: How to become a GOOD Theoretical Physicist by Gerard 't Hooft

1. Languages

1. The acquisition-learning hypothesis is a hypothesis that forms part of Stephen Krashen's

theory of second language acquisition. It states that there are two independent ways in which we develop our linguistic skills: acquisition and learning. According to Krashen acquisition is more important than learning. https://en.wikipedia.org/wiki/Input_hypothesis

1. Learn through audiobook:
2. Learning:

1.1.English.

English is a prerequisite. All sciences nowadays are in English.

1. Learning:

1. Learning:

3. Classical Mechanics

3.5. Poisson’s brackets.

A function $$f(p, q, t)$$ of the phase space coordinates of the system and time as total time derivative

$\frac{d f}{d t}=\frac{\partial f}{\partial t}+\sum_{i}\left(\frac{\partial f}{\partial q_{i}} \dot{q}_{i}+\frac{\partial f}{\partial p_{i}} \dot{p}_{i}\right),$

often written as

$\frac{d f}{d t}=\frac{\partial f}{\partial t} + \{H, f\},$

where $$\{H, f\}$$ is the Poisson bracket. In canonical coordinates $$(q_i, p_i)$$ given (any) two functions $$f(p_i, q_i, t)$$ and $$g(p_i, q_i, t)$$ the Poisson bracket takes the form,

$\{f, g\}=\sum_{i=1}^{N}\left(\frac{\partial f}{\partial q_{i}} \frac{\partial g}{\partial p_{i}}-\frac{\partial f}{\partial p_{i}} \frac{\partial g}{\partial q_{i}}\right).$

The canonical coordinates satisfy the fundamental Poisson bracket relations

• $$\left\{q^{i}, q^{j}\right\}=0$$
• $$\left\{p_{i}, p_{j}\right\}=0$$
• $$\left\{q_{i}, p_{j}\right\}=\delta_{i j}$$

The Poisson bracket has the following properties:

1. $$\{f, g\} = -\{g, f\}$$,
2. $$\{f, c\} = 0$$,
3. $$\{f, q_i\} = -\frac{\partial f}{\partial p_i}$$,
4. $$\{f, p_i\} = \frac{\partial f}{\partial q_i}$$,
5. $$\{f_1+f_2, g\} = \{f_1, g\} + \{f_2, g\}$$,
6. $$\{f_1f_2, g\} = f_1\{f_2, g\} + \{f_1, g\}f_2$$,
7. $$\frac{\partial}{\partial t}\{f, g\} = \left\{\frac{\partial f}{\partial t}, g\right\} + \left\{f, \frac{\partial g}{\partial t}\right\}$$,

In case two, $$c$$ is a constant of the motion, also called an integral of the motion.

An important identity satisfied by the Poisson brackets is the Jacob identity

$\{f,\{g, h\}\} + \{g,\{h, f\}\} + \{h,\{f, g\}\} = 0.$

1. Hamilton's equations are

$\dot{q}=\frac{\partial H}{\partial p}\\ \dot{p}=-\frac{\partial H}{\partial q}$

with the Poisson bracket we have

$\dot{q} =\{q, H(q, p)\} \\ \dot{p} =\{p, H(q, p)\}$

5. Statistical Mechanics & Thermodynamics

5.7. Homogeneous and inhomogeneous systems

5.7.1. Pair correlation function or the radial correlation function

In computational mechanics and statistics mechanics, a radial distribution function (RDF), $$g(r)$$ describes how the density of surrounding matter varies as a function of the distance from a distinguished point. $$g(r)$$ is of fundamental importance in thermodynamics for macroscopic thermodynamics quantities can be calculated using $$g(r)$$.

1. The virial equation for the pressure:

$p = \rho k T-\frac{2\pi}{3kT}\rho^2\int r^3 u'(r)g(r, \rho,T) dr$

2. The energy equation:

$\frac{E}{NkT} = \frac{3}{2} + \frac{\rho}{2kT}\int 4\pi r^2 u(r)g(r, \rho, T) dr$

3. The compressibility equation:

$kT\left(\frac{\partial \rho}{\partial p}\right) = 1 + \rho \int[g(r) - 1] dr$

8. Computational Physic

8.1. Numerical Integration and Differentiation

8.1.1. Derivative by:

1. Finite difference https://en.wikipedia.org/wiki/Finite_difference_method

The derivative of a function $$f(x)$$ can be approximated by the forward difference based on the size $$h$$

$f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h} \approx \frac{f(x+h)-f(x)}{h} - \frac{h}{2}f^{\prime\prime}(\xi).$

First order of a Taylor expansion of $$f(x)$$ with truncation error $$O(h) = hf(\xi)^{\prime\prime}/2$$ linearly related to $$h$$. The approximation is poor unless $$h$$ is sufficient small. A final expression of this example and its order

$\frac{f(x+h)-f(x)}{h} = f^{\prime}(x) + O(h).$

The derivative can be also approximated by the backward difference, replacing $$h$$ by $$-h$$ in the Taylor expansion,

$f^{\prime}(x) \approx \frac{f(x)-f(x-h)}{h}.$

Subtracting the backward difference from the forward difference and dividing everything by $$2h$$, we find the central difference

$f^{\prime}(x) = \frac{f(x+h)-f(x-h)}{2 h}-\left(\frac{f^{\prime \prime \prime}(x)}{3 !} h^{2}+\frac{f^{(5)}(x)}{5 !} h^{4}+\cdots\right) \approx \frac{f(x+h)-f(x-h)}{2 h},$

with the second order truncation error $$O(h)$$. Adding the forward difference to the backward difference and solving for $$f(x)^{\prime\prime}$$ we can get the approximation for the second order derivative

$\boxed{f(x)^{\prime\prime} \approx \frac{f(x-h)-2 f(x)+f(x+h)}{h^{2}}}$

2. Interpolation

The derivative of a given function $$f(t)$$ can be approximated by that of its polynomial interpolation based on a set of $$n+1$$ points $$y_i = f(x_i), (i=0,\cdots, n)$$, such as the Lagrange polynomial interpolation (https://en.wikipedia.org/wiki/Polynomial_interpolation)

$f(x) \approx L_{n}(x)=\sum_{i=0}^{n} y_{i} l_{i}(x).$

Taking the derivative of $$f(x)$$,

$f^{\prime}(x) \approx L_{n}^{\prime}(x)=\sum_{i=0}^{n} y_{i} l_{i}^{\prime}(x)=\sum_{i=0}^{n} y_{i} d_{i}(x),$ where

$d_{i}(x)=\frac{d}{d x} l_{i}(x)=\frac{d}{d x}\left(\prod_{j=0, j \neq i}^{n} \frac{x-x_{j}}{x_{i}-x_{j}}\right).$

The error is

$E_{n}(x) = R_{n}^{\prime}(x) =\frac{1}{(n+1) !} \frac{d}{d x}\left(\left(f^{(n+1)}(\xi(x)) l(x)\right)\right.\\ = \frac{1}{(n+1) !}\left[l^{\prime}(x) f^{(n+1)}(\xi(x))+l(x) \frac{d}{d x} f^{(n+1)}(\xi(x))\right].$

The error is unknown in general since $$\xi$$ is not known explicit. But we can have for each node $$x_i$$ where $$w(x_i)$$,

$E_{n}(x)=R_{n}^{\prime}(x)=l^{\prime}(x) \frac{f^{(n+1)}(\xi(x))}{(n+1) !}.$

1. Special cases:
• $$n=1$$, linear interpolation based on $$n+1 = 2$$ points:

$\begin{array}{l} d_{0}=l_{0}^{\prime}(x)=\frac{d}{d x}\left(\frac{x-x_{1}}{x_{0}-x_{1}}\right)=\frac{1}{x_{0}-x_{1}}, \\ d_{1}=l_{1}^{\prime}(x)=\frac{d}{d x}\left(\frac{x-x_{0}}{x_{1}-x_{0}}\right)=\frac{1}{x_{1}-x_{0}}, \end{array}$

and

$f^{\prime}(x) \approx L_{1}^{\prime}(x)=y_{0} d_{0}(x)+y_{1} d_{1}(x)=\frac{y_{1}-y_{0}}{x_{1}-x_{0}}, \quad f^{\prime \prime}(x)=0.$

• $$n=2$$, quadratic interpolation based on $$n+1 = 3$$ points:

\begin{aligned} d_{0}(x) = l_{0}^{\prime}(x)=\frac{d}{d x}\left(\frac{\left(x-x_{1}\right)\left(x-x_{2}\right)}{\left(x_{0}-x_{1}\right)\left(x_{0}-x_{2}\right)}\right)=\frac{2 x-x_{1}-x_{2}}{\left(x_{0}-x_{1}\right)\left(x_{0}-x_{2}\right)}, \\ d_{1}(x) = l_{1}^{\prime}(x)=\frac{d}{d x}\left(\frac{(x-x_{0})(x-x_{2})}{\left(x_1-x_{0}\right)\left(x_1-x_{2}\right)}\right)=\frac{2 x-x_{0}-x_{2}}{\left(x_{1}-x_{0}\right)\left(x_{1}-x_{2}\right)}, \\ d_{2}(x) = l_{2}^{\prime}(x)=\frac{d}{d x}\left(\frac{\left(x-x_{0}\right)\left(x-x_{1}\right)}{\left(x_{2}-x_{0}\right)\left(x_{2}-x_{1}\right)}\right)=\frac{2 x-x_{0}-x_{1}}{\left(x_{2}-x_{0}\right)\left(x_{2}-x_{1}\right)}, \end{aligned}

and

\begin{aligned} f^{\prime}(x) \approx L_{2}^{\prime}(x) = \sum_{i=1}^{2} d_{i} y_{i} = \frac{2 x-x_{1}-x_{2}}{\left(x_{0}-x_{1}\right)\left(x_{0}-x_{2}\right)} y_{0}+\frac{2 x-x_{0}-x_{2}}{\left(x_{1}-x_{0}\right)\left(x_{1}-x_{2}\right)} y_{1}+\frac{2 x-x_{0}-x_{1}}{\left(x_{2}-x_{0}\right)\left(x_{2}-x_{1}\right)} y_{2}. \end{aligned}

If the 3 points are equally spaced with $$x_1=x, x_0 = x - h, x_2 = x + h$$ one can have

$f^{\prime}(x) \approx \frac{x-(x+h)}{2 h^{2}} y_{0}+\frac{2 x-(x-h)-(x+h)}{-h^{2}} y_{1}+\frac{x-(x-h)}{2 h^{2}}=\frac{1}{2 h}\left(y_{2}-y_{0}\right).$

with error $$O(h^2)$$.

The 2nd order derivative:

\begin{aligned} f^{\prime \prime}(x) \approx L_{2}^{\prime \prime}(x)=\frac{d}{d x} L_{2}^{\prime}(x) = \frac{d}{d x}\left[\frac{2 x-x_{1}-x_{2}}{\left(x_{0}-x_{1}\right)\left(x_{0}-x_{2}\right)} y_{0}+\frac{2 x-x_{0}-x_{2}}{\left(x_{1}-x_{0}\right)\left(x_{1}-x_{2}\right)} y_{1}+\frac{2 x-x_{0}-x_{1}}{\left(x_{2}-x_{0}\right)\left(x_{2}-x_{1}\right)} y_{2}\right] = 2\left[\frac{y_{0}}{\left(x_{0}-x_{1}\right)\left(x_{0}-x_{2}\right)}+\frac{y_{1}}{\left(x_{1}-x_{0}\right)\left(x_{1}-x_{2}\right)}+\frac{y_{2}}{\left(x_{2}-x_{0}\right)\left(x_{2}-x_{1}\right)}\right] \end{aligned}

If the 3 points are equally spaced we have

$f^{\prime}(x) \approx \frac{2 x-x_{1}-x_{2}}{2 h^{2}} y_{0}-\frac{2 x-x_{0}-x_{2}}{h^{2}} y_{1}+\frac{2 x-x_{0}-x_{1}}{2 h^{2}} y_{2},$

and

$f^{\prime}\left(x_{0}\right) = \frac{1}{2 h}\left(-3 y_{0}+4 y_{1}-y_{2}\right), \\ f^{\prime}\left(x_{1}\right) = \frac{1}{2 h}\left(-y_{0}+y_{2}\right), \\ f^{\prime}\left(x_{2}\right) = \frac{1}{2 h}\left(y_{0}-4 y_{1}+3 y_{2}\right).$

The second derivative is

$f^{\prime \prime}(x) \approx \frac{d}{d x}\left[\frac{2 x-x_{1}-x_{2}}{2 h^{2}} y_{0}-\frac{2 x-x_{0}-x_{2}}{h^{2}} y_{1}+\frac{2 x-x_{0}-x_{1}}{2 h^{2}} y_{2}\right]$ $\boxed{f^{\prime \prime}(x) = \frac{1}{h^{2}}\left(y_{0}-2 y_{1}+y_{2}\right)}.$

8.1.2. Boundary conditions:

1. Dirichlet Boundary Conditions:

The values at the edges are fixed at some constant values that we impose. In Finite Element Method, the Dirichlet boundary condition is defined by weighted-integral form of a differential equation. J. N. Reddy, SECOND-ORDER DIFFERENTIAL EQUATIONS IN ONE DIMENSION: FINITE ELEMENT MODELS, An Introduction to the Finite Element Method, 3rd Edition, pp. 110

2. Neumann Boundary Conditions:

The derivative (or discrete difference) at the edges are fixed at some constant values that we choose.

8.2. Programming languages

8.2.1. Julia

1. Packages
1. Latexify - https://github.com/korsbo/Latexify.jl

This is a package for generating LaTeX maths from julia objects.

2. SugarBLAS - https://github.com/lopezm94/SugarBLAS.jl

This package provides macros for BLAS functions representing polynomials

3. Flux - https://fluxml.ai/Flux.jl

Flux is an elegant approach to machine learning.

4. DifferentialEquations - https://diffeq.sciml.ai

This is a suite for numerically solving differential equations written in Julia and available for use in Julia, Python, and R.

5. Knet - https://denizyuret.github.io/Knet.jl

Knet (pronounced "kay-net") is the Koç University deep learning framework implemented in Julia by Deniz Yuret and collaborators. It supports GPU operation and automatic differentiation using dynamic computational graphs for models defined in plain Julia.

6. Symbolics - https://symbolics.juliasymbolics.org/dev

Symbolics.jl is a fast and modern Computer Algebra System (CAS) for a fast and modern programming language (Julia)

7. Pluto - https://github.com/fonsp/Pluto.jl

Simple reactive notebooks for Julia

8. Gadfly - https://gadflyjl.org/stable/

Gadfly is a system for plotting and visualization written in Julia. It is based largely on Hadley Wickhams's ggplot2 for R and Leland Wilkinson's book The Grammar of Graphics. It was Daniel C. Jones' brainchild and is now maintained by the community.

9. RCall - https://juliainterop.github.io/RCall.jl/stable/

RCall, allows the user to call R packages from within Julia.

8.2.2. R

1. ggplot2

ggplot2 is a system for declaratively creating graphics, based on The Grammar of Graphics. You provide the data, tell ggplot2 how to map variables to aesthetics, what graphical primitives to use, and it takes care of the details.

8.3. Solvers

8.3.1. Fortran solvers

1. VODE: stiff/nonstiff ODE systems, with direct linear solvers
2. VODPK: with Krylov linear solvers (GMRES)
3. NKSOL: Newton-Krylov solver - nonlinear algebraic systems
4. DASPK: DAE system solver (from DASSL)

8.3.2. C - organized in a single suite, SUNDIALS (Suite of Nonlinear and Differential/Algebraic Equation Solvers

1. CVODE
2. PVODE
3. KINSOL
4. IDA: rewrite of DASPK

8.3.3. Julia Solvers

1. DASSL

Solves stiff differential algebraic equations (DAE) using variable stepsize backwards finite difference formula (BDF).

DASKR (FORTRAN solver) is a derivative of the DASSL solver with root finding.

12. Nuclear Physics

12.1. Isotopes. Radio-activity. Fission and Fusion.

12.1.1. Geiger-Nuttall law

The Geinger-Nuttall relates the decay constant of a radioactive isotope with the energy of the alpha particles.

$\log _{10} \lambda=-a_{1} \frac{Z}{\sqrt{E}}+a_{2}$

where $$\lambda$$ is the decay constant, $$Z$$ the atomic number, $$E$$ the total kinetic energy and $$a_1$$ and $$a_2$$ are constants.

H. Geiger and J.M. Nuttall (1911) "The ranges of the α particles from various radioactive substances and a relation between range and period of transformation," Philosophical Magazine, Series 6, vol. 22, no. 130, pages 613-621. See also: H. Geiger and J.M. Nuttall (1912) "The ranges of α particles from uranium," Philosophical Magazine, Series 6, vol. 23, no. 135, pages 439-445.

G. Gamow (1928) "Zur Quantentheorie des Atomkernes" (On the quantum theory of the atomic nucleus), Zeitschrift für Physik, vol. 51, pages 204-212.

15. Special Relativity

These annotations are based on http://jacobi.luc.edu/index.html and http://jacobi.luc.edu/Useful.html i.e., the website of Prof. Robert A. McNees

Spacetime coordinates, 4-vectors, metric tensor $$g_{\mu\nu}$$, etc:

Spacetime is any mathematical model which fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum, within of special relativity it relies on the Minkowski geometry.

15.1. 4-vectors.

In relativistic expressions four-vectors are written as

$A^{\mu} = (A_0, \vec{A}) = (A_0, A^1, A^2, A^3).$

Four-vectors with superscript are called contravariant vectors (regarded as a column vector). Covariant vectors, denoted by subscript are defined as

$A_{\mu} = g_{\mu\nu}A^{\nu},$

with the metric tensor $$g_{\mu\nu}$$ that reads in Cartesian coordinates in Minkowski space.

$\eta_{\mu \nu}=\eta^{\mu \nu} = {\rm diag}(1, -1, -1, -1); \qquad \eta^{\mu}{ }_{\nu}=\delta^{\mu}{ }_{\nu} = {\rm diag}(1, 1, 1, 1)$

Raising and lowering indices:

$A^{\mu}=\eta^{\mu \nu} A_{\nu}; \qquad A_{\mu}=\eta_{\mu \nu} A^{\nu};$

Einstein summation convention:

$A^{\mu}B_{\mu} = A^0B_0 + A^{1}B_{1} + A^2B_2 + A^3B_3 = A^0B^0 - \vec{A}\cdot\vec{B}.$

The inverse of the metric tensor is given by

$g^{\mu\nu} = g_{\mu\nu}.$

A scalar product of two four-vectors is then defined by

$A \cdot B = A_{\mu}B^{\mu} = A^{\mu}B_{\mu} = A^0B^0 - \vec{A}\cdot\vec{B}.$

The invariant square of the four-vector is thus

$A^2 = A_{\mu}A^{\mu} = A_0^2 - \vec{A}^2.$

15.1.1. Spacetime coordinates:

$x^{\mu} = (x^0, \vec{x}) = (ct, \vec{x}); \qquad x_{\mu} = (ct, -\vec{x}); \qquad \vec{x} = (x^1, x^2, x^3) = (x, y, z);$

15.1.2. Four-momentum:

$p^{\mu} = (E/c, \vec{p}); \qquad p_{\mu} = (E/c, -\vec{p}); \qquad p^{\mu} = (p^1, p^2, p^3) = (p_x, p_y, p_z);$

15.1.3. Derivatives:

$\partial_{\mu} \equiv \frac{\partial}{\partial x^{\mu}} = \left(\frac{1}{c} \frac{\partial}{\partial t}, \vec\nabla\right); \qquad \partial^{\mu} \equiv \frac{\partial}{\partial x_{\mu}} = \left(\frac{1}{c} \frac{\partial}{\partial t}, -\vec\nabla\right);$

15.1.4. Four-momentum operator

$\hat{P}^{\mu} = i\partial^{\mu}.$

15.1.5. Lorentz-invariant d'Alembert operator

$\partial^{\mu}\partial_{\mu} = \frac{\partial^2}{\partial t^2} - \vec\nabla^2 = \Box.$

15.1.6. Four-current

$j^{\mu} = (\rho, \vec{j}).$

15.1.7. Electromagnetic potential

$A^{\mu} = (\phi, \vec{A}).$

15.2. The Lorentz transformation.

$A'_{\mu} = \Lambda_{\mu}{}^{\nu}A_{\nu}; \qquad A'^{\mu} = \Lambda^{\mu}{}_{\nu}A^{\nu}; \\ \eta^{\mu\nu} = \Lambda^{\mu}{}_{\alpha}\Lambda^{\nu}{}_{\beta}\,\eta^{\alpha\beta}; \qquad \eta_{\mu\nu} = \Lambda_{\mu}{}^{\alpha}\Lambda_{\nu}{}^{\beta}\,\eta_{\alpha\beta}.$

• Properties:
1. $\eta^{\mu \nu}=\Lambda_{\alpha}^{\mu} \Lambda_{\beta}^{\nu} \eta^{\alpha \beta}$
2. $\left|\operatorname{det} \Lambda_{\nu}^{\mu}\right|=1$
3. $\left(\Lambda_{1}\right)^{\mu} \alpha\left(\Lambda_{2}\right)_{\nu}^{\alpha}=\left(\Lambda_{12}\right)_{\nu}^{\mu}$
4. $\left|\Lambda_{0}^{0}\right| \geq 1$

16.15. WKB approximation, Extremum principle.

The WKB approximation (named after Wentzel, Kramers and Brillouin) is a method for obtaining an approximate solution to a time-independent differential equation. Applications include: calculating bound-state energies and tunneling rates through potential barriers.

Solving the Schrödinger equation: It is a non-linear second order differential equation (ODE), in general, no analytic solution, an approximation method is usually applied to tackle the problem. The WKB approximation makes an assumptions of a slowly varying potential. See Boxi Li's note https://www.thphys.uni-heidelberg.de/~wolschin/qms17_7s.pdf if the link is broken, you can access the copy in https://www.rvlobato.com/teaching/notes/pdfs/wkb.pdf, more about WKB applied to tunneling https://www.rvlobato.com/teaching/notes/pdfs/wkb_tunneling2.pdf`

16.18. Klein-Gordon equation

$\nabla^2\phi - m^2\phi = E^2\phi$

Lagrangian Walecka model

$\mathcal{L} = \frac{1}{2}(\partial_{\mu}\sigma\partial^{\mu} - m_{\sigma}^2\phi^2)$

17. Phenomenology/Astroparticles

17.1. Subatomic particles (mesons, baryons, photons, leptons, quarks) and cosmic rays

Particles classification

Particle Interaction
Baryons Strong
Weak
Electromagnetic
Leptons Weak
_ _
• Baryons -> Fermions with half-integral spin (1/2, 3/2, …), Composed of 3 quarks. The lowest mass baryons are the proton and the neutron.
• Meson -> Bosons with integral spin (0, 1, 2, 3, …). Composed of a quark + a anti-quarks.

18. General Relativity

18.1. 4-tensors

18.1.1. The metric tensor.

In local coordinates $$x^{\mu}$$ the metric can be written in the form

$g=g_{\mu \nu} d x^{\mu} \otimes d x^{\nu}.$

The factors $$dx^{\mu}$$ are one-form gradient of the scalar fields $$x^{\mu}$$. The coefficients $$g_{\mu\nu}$$ are a set of 16 real-valued functions. The metric is symmetric

$g_{\mu \nu} = g_{\nu \mu},$ giving 10 independents coefficients. Being $$dx^{\mu}$$ the components of an infinitesimal displacement four-vector, the metric determines the invariant square of an infinitesimal line element, interval, denoted as

$d s^{2}=g_{\mu \nu} d x^{\mu} d x^{\nu}.$

The interval $$ds^2$$ imparts information about the causal structure of the spacetime.

1. Timelike interval. If the spacetime interval is negative.
2. Lightlike interval. If the spacetime interval is zero.
3. Spacelike interval. If the spacetime interval is positive.

Under a change of coordinates $$x^{\mu} \rightarrow x^{\overline{\mu}}$$, the components transform as

$g_{\bar{\mu} \bar\nu}=\frac{\partial x^{\rho}}{\partial x^{\bar{\mu}}} \frac{\partial x^{\sigma}}{\partial x^{\bar{\nu}}} g_{\rho \sigma}=\Lambda_{\bar{\mu}}^{\rho} \Lambda_{\bar{\nu}}^{\sigma} g_{\rho \sigma}.$

1. Flat spacetime

The simplest of Lorentz manifold is flat spacetime which can be given as $$R^4$$ with coordinates,

$d s^{2}=-c^{2} d t^{2}+d x^{2}+d y^{2}+d z^{2}=\eta_{\mu \nu} d x^{\mu} d x^{\nu}$

18.1.2. Space-time curvature.

The metric $$g$$ completely determines the curvature of spacetime. On any semi-Riemannian manifold there is a unique connection $$\nabla$$ that is compatible with the metric and torsion-free. The Levi-Civita connection. The Christoffel symbols of this connection are given in terms of partial derivatives of the metric in local coordinates

$\Gamma^{\lambda}_{}{\mu \nu}=\frac{1}{2} g^{\lambda \rho}\left(\frac{\partial g_{\rho \mu}}{\partial x^{\nu}}+\frac{\partial g_{\rho \nu}}{\partial x^{\mu}}-\frac{\partial g_{\mu \nu}}{\partial x^{\rho}}\right)=\frac{1}{2} g^{\lambda \rho}\left(g_{\rho \mu, \nu}+g_{\rho \nu, \mu}-g_{\mu \nu, \rho}\right),$

the curvature of spacetime is then given by the Riemann curvature tensor which is defined in terms of the Levi-Civita connection $$\nabla$$,

$R^{\rho}{}_{\sigma \mu \nu} = \partial_{\mu} \Gamma^{\rho}{}_{\nu \sigma}-\partial_{\nu} \Gamma^{\rho}{}_{\mu \sigma} + \Gamma^{\rho}{}_{\mu \lambda} \Gamma^{\lambda}{}_{\nu \sigma} - \Gamma^{\rho}{}_{\nu \lambda} \Gamma^{\lambda}{}_{\mu \sigma}.$

• Ricci tensor

$R_{\mu \nu}=\delta^{\sigma}{ }_{\rho} R^{\rho}{}_{\mu \sigma \nu}.$

• Scalar curvature

$R = g^{\mu\nu}R_{\mu\nu}.$

• Schouten tensor

$S_{\mu \nu}=\frac{1}{d-1}\left(R_{\mu \nu}-\frac{1}{2 d} g_{\mu \nu} R\right); \qquad \nabla^{\nu} S_{\mu \nu}=\nabla_{\mu} S^{\nu}{}_{\nu}.$

• Weyl tensor

$C^{\lambda}{}_{\mu \sigma \nu} = R^{\lambda}{}_{\mu \sigma \nu} + g^{\lambda}{}_{\nu} S_{\mu \sigma} - g^{\lambda}{}_{\sigma} S_{\mu \nu} + g_{\mu \sigma} S^{\lambda}{}_{\nu} - g_{\mu \nu} S^{\lambda}{}_{\sigma}.$

• Commutators of Covariant Derivatives

${\left[\nabla_{\mu}, \nabla_{\nu}\right] A_{\lambda} = R_{\lambda \sigma \mu \nu} A^{\sigma}}; \qquad {\left[\nabla_{\mu}, \nabla_{\nu}\right] A^{\lambda} = R^{\lambda}{}_{\sigma \mu \nu} A^{\sigma}}.$

• Bianchi Identity

$\nabla_{\kappa} R_{\lambda \mu \sigma \nu}-\nabla_{\lambda} R_{\kappa \mu \sigma \nu}+\nabla_{\mu} R_{\kappa \lambda \sigma \nu}=0; \qquad \nabla^{\nu} R_{\lambda \mu \sigma \nu}=\nabla_{\mu} R_{\lambda \sigma}-\nabla_{\lambda} R_{\mu\sigma}; \qquad \nabla^{\nu} R_{\mu \nu}=\frac{1}{2} \nabla_{\mu} R.$

• Bianchi identity for Weyl

$\nabla^{\nu} C_{\lambda \mu \sigma \nu}=(d-2) \left(\nabla_{\mu} S_{\lambda\sigma}-\nabla_{\lambda} S_{\mu \sigma}\right); \qquad \nabla^{\lambda} \nabla^{\sigma} C_{\lambda\mu \sigma \nu} = \frac{d-2}{d-1}\left[\nabla^{2} R_{\mu\nu}-\frac{1}{2 d} g_{\mu \nu} \nabla^{2} R-\frac{d-1}{2 d} \nabla_{\mu} \nabla_{\nu}R - \left(\frac{d+1}{d-1}\right) R_{\mu}{}^{\lambda} R_{\nu \lambda} + C_{\lambda \mu \sigma \nu} R^{\lambda \sigma}+\frac{(d+1)}{d(d-1)} R R_{\mu \nu}+\frac{1}{d-1} g_{\mu \nu}\left(R^{\lambda \sigma} R_{\lambda \sigma}-\frac{1}{d} R^{2}\right)\right]$

• Einstein tensor

$G_{\mu\nu} = R_{\mu\nu} - \frac{1}{2}Rg_{\mu\nu}$

which always obey $$\nabla^{\mu}G_{\mu\nu}=0$$

18.2. Einstein’s gravity equation.

From Einstein tensor, one can have

$G_{\mu\nu} = \kappa T_{\mu\nu}$ the field equation for the metric. The right-hand side is a covariant expression of the energy-momentum density (symmetric (0, 2) tensor). The left-had side is a symmetric conserved (0 , 2) tensor from the metric and its derivatives.

18.3. The Schwarzschild spacetime - black hole.

The standard Schwarzschild coordinates are introduced by a chart applied to the manifold $$\cal{M}$$.

The metric tensor of the Lorentzian manifold $$\cal{M}$$

$g = \left( \frac{2 \, m}{r} - 1 \right) \mathrm{d} t\otimes \mathrm{d} t + \left( -\frac{1}{\frac{2 \, m}{r} - 1} \right) \mathrm{d} r\otimes \mathrm{d} r + r^{2} \mathrm{d} {\theta}\otimes \mathrm{d} {\theta} + r^{2} \sin\left({\theta}\right)^{2} \mathrm{d} {\phi}\otimes \mathrm{d} {\phi}$

• Christoffel symbols of $$g$$ with respect to the Schwarzschild-Droste coordinates (the nonzero and nonredundant ones) are

\begin{array}{lcl} \Gamma_{ \phantom{\, t} \, t \, r }^{ \, t \phantom{\, t} \phantom{\, r} } = -\frac{m}{2 \, m r - r^{2}} \\ \Gamma_{ \phantom{\, r} \, t \, t }^{ \, r \phantom{\, t} \phantom{\, t} } = -\frac{2 \, m^{2} - m r}{r^{3}} \\ \Gamma_{ \phantom{\, r} \, r \, r }^{ \, r \phantom{\, r} \phantom{\, r} } = \frac{m}{2 \, m r - r^{2}} \\ \Gamma_{ \phantom{\, r} \, {\theta} \, {\theta} }^{ \, r \phantom{\, {\theta}} \phantom{\, {\theta}} } = 2 \, m - r \\ \Gamma_{ \phantom{\, r} \, {\phi} \, {\phi} }^{ \, r \phantom{\, {\phi}} \phantom{\, {\phi}} } = {\left(2 \, m - r\right)} \sin\left({\theta}\right)^{2} \\ \Gamma_{ \phantom{\, {\theta}} \, r \, {\theta} }^{ \, {\theta} \phantom{\, r} \phantom{\, {\theta}} } = \frac{1}{r} \\ \Gamma_{ \phantom{\, {\theta}} \, {\phi} \, {\phi} }^{ \, {\theta} \phantom{\, {\phi}} \phantom{\, {\phi}} } = -\cos\left({\theta}\right) \sin\left({\theta}\right) \\ \Gamma_{ \phantom{\, {\phi}} \, r \, {\phi} }^{ \, {\phi} \phantom{\, r} \phantom{\, {\phi}} } = \frac{1}{r} \\ \Gamma_{ \phantom{\, {\phi}} \, {\theta} \, {\phi} }^{ \, {\phi} \phantom{\, {\theta}} \phantom{\, {\phi}} } = \frac{\cos\left({\theta}\right)}{\sin\left({\theta}\right)} \end{array}

20. Astrophysics

20.1. Compact stars

20.1.2. Neutron stars

• Reminiscent of stars with mass $$10\lesssim M/M_{\odot} \lesssim 25$$.
• In average the density is $$\rho\sim10^{14}\ {\rm g/cm^3}$$. Denser than an atomic nucleus deeper inside.
• One teaspoon of its material would have a mass over $$5.5\times10^{12}$$ kg. It is about 6000 cargo ships.
• The surface gravity is $$g\simeq10^{12}\ {\rm m/s^2}$$. That is 1011 times the gravity on Earth.
• The escape velocity on a NS $$100000 - 150000\ {\rm km/s}$$, i.e., one third of the light speed.
• Have magnetic field of $$10^8-10^{15}\ {\rm G}$$ (Earth magnetic field 1 G, refrigerator magnet 100 G, sunspots $$3\times10^3$$ G, magnetic resonance imaging $$7\times10^4$$ G, LHC $$12\times10^4$$ G)
• Electric field $$10^{14}-10^{18}\ {\rm V/cm}$$
• Temperature $$10^{6}-10^{11}\ {\rm K}$$ (sun temperature 5778 $$\sim10^3$$ K)
• The rotation can reach 716 Hz (716 times per second), the linear speed is about 0.25c.
1. Equation of state
1. Nucleonic
1. Non-relativistic
1. APR

Obtained from variational-method with modern nuclear potentials such as Argone (2-nucleon interaction) and Urbana (3-body interaction) potentials. Zero temperature and $β$-equilibrium $$npe\mu$$ matter. (Akmal et al. (1998)).

2. BBB

Obtained from the Brueckner-Hartree-Fock (BHF) framework as approximation of the Brueckner-Bethe-Goldstone (BBG) theory. Uses two-three particle potentials. (Baldo et al (1997)).

3. FPS

Modern version of the EoS by Friedman and Pandharipande. It is a Skyrme type model with a energy density functional (EDF) that considers a nucleon-nucleon interaction by the Urbana potential and phenomenological three-nucleons interaction. (Lorenz et al (1993)).

2. Relativistic
2. Massive pulsars
1. J1614-2230 ($$\approx2M_{\odot}$$)
2. J0348+0432 ($$\approx2M_{\odot}$$)
3. J0740+6620 ($$\approx2.14M_{\odot}$$)
• {\it NICER}

PSR J0030+0451

22. Supersymmetry/Super gravity

Created by: Ronaldo V. Lobato on 2020-11-11. Last Updated: 2021-12-21 mar 14:12