Carbon Note 核心学堂 Carbon Note Academy

Practice: Mathematical and Physical Derivation of Einstein's Relativity and Spacetime Geometry

Einstein's theory of relativity represents one of the twin pillars of modern physics. It shattered the Newtonian view of absolute space and time, demonstrating that spacetime is dynamical and behaves as a curved, four-dimensional pseudo-Riemannian manifold.

This laboratory report derives Special Relativity's Lorentz Transformation matrix, the mass-energy equivalence $E=mc^2$, and introduces Riemann geometry and the Einstein Field Equations.

1. Conceptual Roadmap of Relativistic Derivations

Before looking at the tensor calculus, we map the logical progression of concepts:

graph TD
    Postulate[Relativity & Constancy of c] --> Lorentz[Lorentz Transformation Matrix]
    Lorentz --> Kinematics[Time Dilation & Length Contraction]
    Lorentz --> MassEnergy[Relativistic Momentum & E=mc^2]
    
    Equivalence[Equivalence Principle & Gen Relativity] --> Curvature[Spacetime Curvature & Metric Tensor]
    Curvature --> Geodesic[Geodesic Equations]
    Curvature --> FieldEqs[Einstein Field Equations G=8πT]

2. Special Relativity

Special Relativity is anchored upon two basic postulates:

1. The Laws of Physics are invariant in all inertial frames of reference (Principle of Relativity).
2. The speed of light in vacuum, $c$, is constant across all inertial frames (Constancy of Speed of Light).

A. Analytical Derivation of the Lorentz Transformation

Consider two inertial systems, $S$ and $S'$, where $S'$ moves relative to $S$ with constant velocity $v$ along the positive $x$-axis.

The expansion wavefront of a spherical light signal emitted from the origin at $t=t'=0$ satisfies:

$x^2 + y^2 + z^2 - c^2 t^2 = 0$

$x'^2 + y'^2 + z'^2 - c^2 t'^2 = 0$

Since axes perpendicular to relative motion remain unaffected, $y' = y$ and $z' = z$. We seek linear relations:

$x' = \gamma (x - vt)$

Applying the principle of relativity, the inverse transformation must be identical with a reversed velocity sign:

$x = \gamma (x' + vt')$

Substituting the first equation into the second yields:

$x = \gamma \left( \gamma(x - vt) + vt' \right) \quad \Rightarrow \quad vt' = \frac{x}{\gamma} - \gamma x + \gamma vt \quad \Rightarrow \quad t' = \gamma t + \frac{x}{v} \left( \frac{1}{\gamma} - \gamma \right)$

Substituting the terms for $x'$ and $t'$ back into the spherical wavefront constraint $x^2 - c^2 t^2 = x'^2 - c^2 t'^2$:

$x^2 - c^2 t^2 = \gamma^2 (x - vt)^2 - c^2 \gamma^2 \left( t + \frac{x}{v} \left( \frac{1}{\gamma^2} - 1 \right) \right)^2$

Equating coefficients of variables $x^2$ and $t^2$, we isolate the Lorentz Factor $gamma$:

$\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$

This yields the complete Four-Dimensional Lorentz Transformation Matrix (split column view of LaTeX code and math representation):

$\begin{pmatrix} c t' \\ x' \\ y' \\ z' \end{pmatrix} = \begin{pmatrix} \gamma & -\beta\gamma & 0 & 0 \\ -\beta\gamma & \gamma & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} c t \\ x \\ y \\ z \end{pmatrix} \quad \left( \text{where } \beta = \frac{v}{c} \right)$

B. Differential Derivation of Mass-Energy Equivalence $E = mc^2$

In relativistic dynamics, an object's mass increases as its velocity approaches $c$:

$m(v) = \gamma m_0 = \frac{m_0}{\sqrt{1 - \frac{v^2}{c^2}}}$

Where $m_0$ is the rest mass. Force is defined as the time derivative of momentum:

$F = \frac{\mathrm{d}p}{\mathrm{d}t} = \frac{\mathrm{d}(mv)}{\mathrm{d}t}$

Work done by force $F$ translates to relativistic kinetic energy $E_k$:

$\mathrm{d}E_k = F \cdot \mathrm{d}x = \frac{\mathrm{d}(mv)}{\mathrm{d}t} \mathrm{d}x = \mathrm{d}(mv) \frac{\mathrm{d}x}{\mathrm{d}t} = v \cdot \mathrm{d}(mv) = v^2 \mathrm{d}m + mv \mathrm{d}v$

We differentiate the squared mass relationship $m^2 (c^2 - v^2) = m_0^2 c^2$:

$\mathrm{d}\left[ m^2 (c^2 - v^2) \right] = 0$

$2m \mathrm{d}m (c^2 - v^2) - 2m^2 v \mathrm{d}v = 0 \quad \Rightarrow \quad \mathrm{d}m (c^2 - v^2) = mv \mathrm{d}v \quad \Rightarrow \quad c^2 \mathrm{d}m = v^2 \mathrm{d}m + mv \mathrm{d}v$

Substituting this substitution directly back into our kinetic energy equation yields:

$\mathrm{d}E_k = c^2 \mathrm{d}m$

Integrating from rest mass $m_0$ to current relativistic mass $m$:

$E_k = \int_{m_0}^{m} c^2 \mathrm{d}m = (m - m_0) c^2 = m c^2 - m_0 c^2$

The total energy $E$ is the sum of rest energy and kinetic energy:

$E = E_k + m_0 c^2 = m c^2$

This mathematically establishes Einstein's celebrated mass-energy equivalence equation. $\text{ Q.E.D.}$

3. General Relativity and Riemannian Geometry

General Relativity extends relativity to non-inertial reference frames, showing that gravity is not a physical force but the manifestation of spacetime curvature.

A. The Spacetime Metric Tensor

In curved spacetime, the infinitesimal interval $\mathrm{d}s^2$ is represented as:

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

Where $g_{\mu\nu}$ is the covariant metric tensor defining local geometry. In flat spacetime, this degenerates into the Minkowski metric $\eta_{\mu\nu}$.

B. Christoffel Symbols (Affine Connection)

Christoffel symbols characterize the connection, detailing how coordinate vectors change as they are parallel-transported across curved surfaces:

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

C. Geodesic Equations

Free-falling test particles follow straight-line paths (geodesics) on curved manifolds, governed by:

$\frac{\mathrm{d}^2 x^\mu}{\mathrm{d}\tau^2} + \Gamma^\mu_{\alpha\beta} \frac{\mathrm{d}x^\alpha}{\mathrm{d}\tau} \frac{\mathrm{d}x^\beta}{\mathrm{d}\tau} = 0$

Where $\tau$ is the proper time.

D. The Einstein Field Equations

The Einstein Field Equations tie geometry and energy together into a unified dynamical tensor statement (split column view of LaTeX code and math representation):

$G_{\mu\nu} \equiv R_{\mu\nu} - \frac{1}{2} R g_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu}$

Where:

• $R_{\mu\nu}$ is the Ricci Curvature Tensor.
• $R$ is the scalar curvature.
• $G$ is Newton's gravitational constant.
• $T_{\mu\nu}$ is the energy-momentum tensor.

As John Archibald Wheeler famously put it: "Spacetime tells matter how to move; matter tells spacetime how to curve."

4. Common Problems and Pitfalls (FAQ)

1. Why does my formula fail to render, showing red warning text or KaTeX parse errors for Greek letters like \mu or \nu?
   • Solution: This usually occurs due to typos or commands sticking together without spacing (e.g., writing \gamma_j\hatx instead of \gamma_j\hat{x} where \hatx is parsed as an invalid command). Always double-check standard LaTeX command spellings and ensure that braces are fully closed and matched.
2. Why does entering a percentage sign % or an underscore _ inside my equation cause the layout to break or go missing?
   • Solution: In LaTeX syntax, the percentage symbol % is a comment operator (which comments out/hides all text following it on that line), and _ is a subscript marker. To display these characters literally in your formulas, you must escape them with a backslash: use \% and \_.
3. How do I prevent extremely long equations (such as the expanded Einstein field equations) from running off the screen or overflowing on narrow viewports?
   • Solution: Extra-long formulas will break responsive layouts. It is highly recommended to wrap long equations in a split environment like \begin{aligned} ... \end{aligned} and manually insert breaks \\ at logical locations (like before addition or equality signs) to wrap lines beautifully on mobile screens.