List of Physics Relativity Formulas Latex Code(First Part)

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In this blog, we will introduce most popuplar physics formulas in Relativity. We will also provide latex code of the equations. Topics include the Lorentz transformation, red and blue shift, general relativity, Riemannian Tensor and Einstein Field Equations, etc.

• 1. Relativity
• 1.1 The Lorentz Transformation
• 1.2 Red and Blue shift
• 1.3 The stress-energy tensor and the field tensor
• 1.4 General Relativity
• 1.5 Riemannian Tensor
• 1.6 Einstein Field Equations

1. Relativity

• 1.1 The Lorentz Transformation

  Equation

  
  
  

  Latex Code

  (\vec{x}^{'}, t^{'}) = (\vec{x}^{'}(\vec{x},t), t^{'}(\vec{x},t)) \\ 
  \vec{x}^{'} = \vec{x} +\frac{(\gamma-1)(\vec{x}\vec{v}) \vec{v}}{|v|^{2}} - \gamma \vec{v}t \\ 
  t^{'} = \frac{\gamma(t-\vec{x}\vec{v})}{c^{2}} \\ 
  \gamma = \frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}} \\ 
  \frac{\partial^{2}}{\partial{x^{2}}} + \frac{\partial^{2}}{\partial{y^{2}}} + \frac{\partial^{2}}{\partial{z^{2}}} - \frac{1}{c^{2}} \frac{\partial^{2}}{\partial{t^{2}}} = \frac{\partial^{2}}{\partial{{x^{'}}^{2}}} + \frac{\partial^{2}}{\partial{{y^{'}}^{2}}} + \frac{\partial^{2}}{\partial{{z^{'}}^{2}}} - \frac{1}{c^{2}} \frac{\partial^{2}}{\partial{{t^{'}}^{2}}}
          

  Explanation

  Latex code for the lorentz transformation Equations. I will briefly introduce the notations in this formulation. The Lorentz transformation leaves the wave equation invariant if c is invariant. The general form of the Lorentz transformation is given by .
  

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  • Lorentz Transformations

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• 1.2 Red and Blue shift

  Equation

  
  
  

  Latex Code

  \vec{e_{v}}\vec{e_{r}}=\cos(\Phi) \\
  \frac{f^{'}}{f}=\gamma(1-\frac{v\cos(\Phi)}{c}) \\
  \frac{\Delta f}{f}=\frac{\kappa M}{rc^{2}} \\
  \frac{\lambda_{0}}{\lambda_{1}}=\frac{R_{0}}{R_{1}}
          

  Explanation

  Latex code for the the red and blue shift. I will briefly introduce the notations in this formulation.

  • Motion: follows: .
  • Gravitational redshift: follows: .
  • Red Shift result in cosmic background radiation as follows:
  
  

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  • StackExchange:why-is-there-red-blue-shift-instead-of-light-dark-shift

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• 1.3 The stress-energy tensor and the field tensor

  Equation

  
  
  

  Latex Code

  T_{\mu v}=(\varrho c^{2}+p)u_{p}u_{v}+pg_{\mu v}+\frac{1}{c^{2}}(F^{\mu}_{\alpha}F^{\alpha v} + \frac{1}{4}g^{\mu v}F^{\alpha\beta}F_{\alpha\beta}) \\
  \triangledown_{v} T_{\mu v}=0 \\
  F_{\alpha\beta}=\frac{\partial{A_{\beta}}}{\partial{x^{\alpha}}} - \frac{\partial{A_{\alpha}}}{\partial{x^{\beta}}} \\
  \frac{d p_{\alpha}}{d \tau}=qF_{\alpha\beta}u^{\beta}
  

  Explanation

  Latex code for the stress-energy tensor and the field tensor. I will briefly introduce the notations in this formulation.

  • The stress-energy tensor is given by:
  • The conservation laws can than be written as: . And the electromagnetic field tensor is given by .
  • The equations of motion for a charged particle in an EM field become with the field tensor:
  
  

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• 1.4 General Relativity

  Equation

  
  
  

  Latex Code

  \frac{\mathrm{d}^{2} x^{\alpha}}{\mathrm{d} s^{2}}+\Gamma^{\alpha}_{\beta\gamma}\frac{\mathrm{d} x^{\beta}}{\mathrm{d} s}\frac{\mathrm{d} x^{\gamma}}{\mathrm{d} s}=0
  

  Explanation

  Latex code for the principles of general relativity. I will briefly introduce the notations in this formulation.

  • The geodesic postulate: free falling particles move along geodesics of space-time with the proper time or arc length as parameter. For particles with zero rest mass (photons), the use of a free parameter is required because for them holds . From the equations of motion can be derived as :
  • The principle of equivalence: inertial mass â?¡ gravitational mass->gravitation is equivalent with a curved space-time were particles move along geodesics.
  • By a proper choice of the coordinate system it is possible to make the metric locally flat in each point
  
  

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• 1.5 Riemannian Tensor

  Equation

  
  
  

  Latex Code

  R^{\mu}_{v \alpha \beta}T^{v}=\triangledown_{\alpha}\triangledown_{\beta}T^{\mu}-\triangledown_{\beta}\triangledown_{\alpha}T^{\mu} \\
  \triangledown_{j}a^{i}=\partial_{j}a^{i}+\Gamma^{i}_{jk}a^{k} \\
  \triangledown_{j}a_{i}=\partial_{j}a_{i}-\Gamma^{k}_{ij}a_{k} \\
  \Gamma^{i}_{jk}=\frac{\partial^{2} \bar{x}^{l}}{\partial{x^{j}}\partial{x^{k}}}\frac{\partial{x^{i}}}{\partial{\bar{x}^{l}}}
          

  Explanation

  Latex code for the riemannian tensor. I will briefly introduce the notations in this formulation.

  • The Riemann tensor is defined as: , where the covariant derivate is given by and .
  • The Christoffel symbols is
  
  

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• 1.6 Einstein Field Equations

  Equation

  
  
  

  Latex Code

  G^{\alpha\beta}:=R^{\alpha\beta}-\frac{1}{2}g^{\alpha\beta}R \\
  G_{\alpha\beta}=\frac{8\pi \kappa}{c^{2}}T_{\alpha\beta}
          

  Explanation

  Latex code for the Einstein Field Equations. I will briefly introduce the notations in this formulation.

  • The Einstein tensor is given by: , and is the Ricci scalar.
  • With the variational principles , the Einstein field equations can be derived: , and
  
  

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