Introduction

Fundamentals　(Under construction)

Let us overview General Relativity, Metric-Affine Gravity, and Gauge Theories of Gravity, from the viewpoint of connections.
Suppose that \(M\) is a differentiable manifold, \(g\) is a metric tensor with a Lorentzian signature, and \(\nabla\) is an affine connection. We consider a structure \((M,g,\nabla)\).

General Relativity (Under construction)

General Relativity describes gravity as the curvature of spacetime, which is shaped by matter and energy.
The underlying geometry is equipped with the Levi-Civita connection, \(\overset{\circ}{\nabla}\), which satistfies the following two conditions:

\[ \overset{\circ}{\nabla}_{X}Y - \overset{\circ}{\nabla}_{Y}X = [X,Y] \tag{1} \]
and
\[ (Xg)(Y,Z) = g(\overset{\circ}{\nabla}_{X}Y,Z) + g(Y,\overset{\circ}{\nabla}_{X}Z) \,, \tag{2} \]
where \(X,Y,Z \in TM\). In a local coordinate system \(x^{\mu}\), the components of the Levi-Civita connection is given by
\[ \overset{\circ}{\nabla}_{\partial_{\mu}}\partial_{\nu} = \overset{\circ}{\Gamma}{}^{\rho}_{\mu\nu}\partial_{\rho} \,. \tag{3} \]
Explicitly, \(\overset{\circ}{\Gamma}{}^{\rho}_{\mu\nu}\) is expressed in
\[ \overset{\circ}{\Gamma}{}^{\rho}_{\mu\nu} = \frac{1}{2}g^{\rho\sigma}( \partial_{\mu}{g_{\nu\sigma}} + \partial_{\nu}{g_{\mu\sigma}} - \partial_{\sigma}{g_{\mu\nu}} ) \,, \tag{4} \]
where \(g_{\mu\nu}\) is the components of the metric tensor, \(g=g_{\mu\nu}dx^{\mu}\otimes dx^{\nu}\), and \(g^{\mu\nu}\) is the inverse components of \(g_{\mu\nu}\). Then, \(\overset{\circ}{\nabla}\) constitues the curvature tensor as follows:
\[ \overset{\circ}{R}(X,Y)Z = \overset{\circ}{\nabla}_{X}\overset{\circ}{\nabla}_{Y}Z - \overset{\circ}{\nabla}_{Y}\overset{\circ}{\nabla}_{X}Z + \overset{\circ}{\nabla}_{[X,Y]}Z \,. \tag{5} \]
In the local coordinates \(x^{\mu}\), the components of the curvature tensor is given by
\[ \overset{\circ}{R}{}^{\rho}{}_{\sigma\mu\nu} = \partial_{\mu}\overset{\circ}{\Gamma}{}^{\rho}_{\nu\sigma} - \partial_{\nu}\overset{\circ}{\Gamma}{}^{\rho}_{\mu\sigma} + \overset{\circ}{\Gamma}{}^{\rho}_{\mu\lambda}\overset{\circ}{\Gamma}{}^{\lambda}_{\nu\sigma} - \overset{\circ}{\Gamma}{}^{\rho}_{\nu\lambda}\overset{\circ}{\Gamma}{}^{\lambda}_{\mu\sigma} \,. \tag{6} \]
Contracting the first and the third indices, we obtain the Ricci tensor, \(\overset{\circ}{R}{}_{\mu\nu}=\overset{\circ}{R}{}^{\rho}{}_{\mu\rho\nu}\). Further contraction leads to the Ricci scalar, \(\overset{\circ}{R}=g^{\mu\nu}\overset{\circ}{R}{}_{\mu\nu}\).
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Metric-Affine Gravity (Under construction)

Metric-affine gravity treats the spacetime metric and affine connection as independent geometric fields. This framework allows spacetime geometry to possess curvature, torsion, and non-metricity.

Gauge Theories of Gravity (Under construction)

Gauge theories of gravity formulate gravity using gauge fields associated with local spacetime symmetries, following the gauge principle familiar from particle physics.

Bibles for exploring our research field

Below are some of my favorite textbooks that may help beginners explore this research field.

Textbooks highlighted in purple cover advanced topics and are suitable for graduate students and professional researchers.

Mathematics

  1. 集合・位相入門 , 松坂和夫, 岩波書店.
  2. 代数入門（新装版） , 堀田良之, 裳華房.
  3. 多様体入門（新装版） , 松島与三, 裳華房.
  4. Series on Knots and Everything, Volume 4: Gauge Fields, Knots and Gravity , J. Baez and J. P. Muniain, World Scientific.
  5. Geometry, Topology and Physics , M. Nakahara, CRC Press.
  6. Foundations of Differential Geometry, Volume 1 and Volume 2 , S. Kobayashi and K. Nomizu, Wiley.

General Relativity

  1. Gravitation: Foundations and Frontiers , T. Padmanabhan, Cambridge University Press.
  2. Spacetime and Geometry: An Introduction to General Relativity , Sean M. Carroll, Cambridge University Press, 2019.
  3. General Relativity , R. Wald, University of Chicago Press.
  4. The Large Scale Structure of Space-Time , S. W. Hawking and G. F. R. Ellis, Cambridge University Press.

Quantum Theory

  1. 量子力学 , 砂川重信, 岩波書店.
  2. Modern Quantum Mechanics, Revised Edition , J. J. Sakurai, Addison-Wesley.
  3. >An Introduction to Quantum Field Theory , M. E. Peskin and D. V. Schroeder, Avalon Publishing.
  4. Quantum Field Theory in Curved Spacetime: Quantized Fields and Gravity , L. Parker and D. Toms, Cambridge University Press.
  5. The Quantum Theory of Fields, Volume 1 , Volume 2 , and Volume 3 , S. Weinberg, Cambridge University Press.

Cosmology

  1. 現代宇宙論 ― 時空と物質の共進化 , 松原隆彦, 東京大学出版会.
  2. 現代宇宙論講義 ― 基礎から系統的な理解を目指して , 辻川信二, サイエンス社.
  3. Modern Cosmology , S. Dodelson, Academic Press.
  4. Physical Foundations of Cosmology , V. Mukhanov, Cambridge University Press.
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