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The Einstein Field Equations (EFE) are a set of ten coupled, non-linear partial differential equations that form the core of general relativity. They describe the fundamental interaction of gravitation as a result of spacetime being curved by matter and energy. The equation is concisely written as [latex]G_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu}[/latex], relating spacetime geometry to its energy-momentum content.
These equations are the mathematical foundation of general relativity. In the equation [latex]G_{\mu\nu} + \Lambda g_{\mu\nu} = \kappa T_{\mu\nu}[/latex], the left side represents the geometry of spacetime, while the right side represents the matter and energy content within it. The Einstein tensor, [latex]G_{\mu\nu}[/latex], is a specific combination of the Ricci tensor and the scalar curvature, which are derived from the metric tensor [latex]g_{\mu\nu}[/latex]. The metric tensor itself defines all geometric properties of spacetime, such as distance, volume, and curvature. The term [latex]\Lambda[/latex] is the cosmological constant, originally introduced by Einstein to allow for a static universe and now associated with dark energy and cosmic acceleration.
On the right side, the stress-energy tensor, [latex]T_{\mu\nu}[/latex], is a mathematical object that describes the density and flux of energy and momentum in spacetime. It acts as the source of the gravitational field, analogous to how mass is the source of gravity in Newton’s theory. The constant [latex]\kappa = \frac{8\pi G}{c^4}[/latex] is the Einstein gravitational constant, which ensures that the theory’s predictions match Newtonian gravity in the weak-field, low-velocity limit.
由于这些方程的非线性特性，求解它们极其困难。这些方程表明，物质决定时空如何弯曲，而弯曲的时空又决定物质如何运动。这种反馈回路正是非线性的根源。目前已知的精确解析解寥寥无几，例如球形质量的史瓦西解。 黑洞）以及弗里德曼-勒梅特-罗伯逊-沃克（FLRW）度规，该度规适用于均匀、各向同性的宇宙，是现代宇宙学的基础。
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