List of Physics Relativity Formulas Latex Code(First Part)

rockingdingo 2023-01-24 #physics #relativity #einstein

Navigation

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

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

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 $(\vec{x}^{'}, t^{'}) = (\vec{x}^{'}(\vec{x},t), t^{'}(\vec{x},t))$ leaves the wave equation invariant if c is invariant. The general form of the Lorentz transformation is given by $\vec{x}^{'} = \vec{x} +\frac{(\gamma-1)(\vec{x}\vec{v}) \vec{v}}{|v|^{2}} - \gamma \vec{v}t \\ t^{'} = \frac{\gamma(t-\vec{x}\vec{v})}{c^{2}}$ .

Related Documents

Physics Formulary

Lorentz Transformations

Related Videos
1.2 Red and Blue shift

Equation

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

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: $\vec{e_{v}}\vec{e_{r}}=\cos(\Phi)$ follows: $\frac{f^{'}}{f}=\gamma(1-\frac{v\cos(\Phi)}{c})$ .
- Gravitational redshift: $\frac{\Delta f}{f}=\frac{\kappa M}{rc^{2}}$ follows: $\frac{f^{'}}{f}=\gamma(1-\frac{v\cos(\Phi)}{c})$ .
- Red Shift result in cosmic background radiation as $\frac{\Delta f}{f}=\frac{\kappa M}{rc^{2}}$ follows: $\frac{\lambda_{0}}{\lambda_{1}}=\frac{R_{0}}{R_{1}}$
Related Documents
- Physics Formulary
- StackExchange:why-is-there-red-blue-shift-instead-of-light-dark-shift
Related Videos
1.3 The stress-energy tensor and the field tensor

Equation

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

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: $T_{\mu v}=(\varrho c^{2}+p)u_{p}u_{v}+pg_{\mu v}+\frac{1}{c^{2}}(F^{\mu}_{\alpha}F^{\alpha v} + \frac{1}{4}g^{\mu v}F^{\alpha\beta}F_{\alpha\beta})$
- The conservation laws can than be written as: $\triangledown_{v} T_{\mu v}=0$ . And the electromagnetic field tensor is given by $F_{\alpha\beta}=\frac{\partial{A_{\beta}}}{\partial{x^{\alpha}}} - \frac{\partial{A_{\alpha}}}{\partial{x^{\beta}}}$ .
- The equations of motion for a charged particle in an EM field become with the field tensor: $\frac{d p_{\alpha}}{d \tau}=qF_{\alpha\beta}u^{\beta}$
Related Documents
- Physics Formulary
Related Videos
1.4 General Relativity

Equation

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

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 $\tau$ or arc length $s$ as parameter. For particles with zero rest mass (photons), the use of a free parameter is required because for them holds $\mathrm{d}s=0$ . From $\delta \int \mathrm{d}s = 0$ the equations of motion can be derived as : $\frac{\mathrm{d}^{2} x^{\alpha}}{\mathrm{d} s^{2}}+\Gamma^{\alpha}_{\beta\gamma}\frac{\mathrm{d} x^{\beta}}{\mathrm{d} s}\frac{\mathrm{d} x^{\gamma}}{\mathrm{d} s}=0$
- 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
Related Documents
- Physics Formulary
Related Videos

1.5 Riemannian Tensor

Equation

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

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: $R^{\mu}_{v \alpha \beta}T^{v}=\triangledown_{\alpha}\triangledown_{\beta}T^{\mu}-\triangledown_{\beta}\triangledown_{\alpha}T^{\mu}$ , where the covariant derivate is given by $\triangledown_{j}a^{i}=\partial_{j}a^{i}+\Gamma^{i}_{jk}a^{k}$ and $\triangledown_{j}a_{i}=\partial_{j}a_{i}-\Gamma^{k}_{ij}a_{k}$ .
The Christoffel symbols is $\Gamma^{i}_{jk}=\frac{\partial^{2} \bar{x}^{l}}{\partial{x^{j}}\partial{x^{k}}}\frac{\partial{x^{i}}}{\partial{\bar{x}^{l}}}$

Navigation

1. Relativity

1.1 The Lorentz Transformation

Related Documents

Related Videos

1.2 Red and Blue shift

Related Documents

Related Videos

1.3 The stress-energy tensor and the field tensor

Related Documents

Related Videos

1.4 General Relativity

Related Documents

Related Videos

1.5 Riemannian Tensor

Related Documents

Related Videos

1.6 Einstein Field Equations

Related Documents

Related Videos