2024年2月26日 · The Einstein Field Equations (EFE) are a set of ten interrelated differential equations that form the core of Einstein's general theory of relativity. These equations describe how matter and energy determine the curvature of spacetime, providing a mathematical framework to relate spacetime geometry to its energy-matter content.

2018年5月13日 · Now we have the coframe field ansatz we can express the metric in terms of it and calculate the Einstein tensor. Then we can solve the Einstein Field Equations. Generating the Einstein tensor isn’t difficult, just boring. You need all the Christoffel symbols (most are zero but you need to check) and all the independent components of the ...

2015年4月15日 · Of course depending on what type of tensor ##X## is it will have other indeces as well, if ##X## is a vecotr filed, then ##\nabla X## is a ##(1,1)## tensor field and so on. Similar threads Cosmological Constant, Einstein equation Quick Question

2011年8月19日 · The way to get the divergence of the Einstein tensor is to start with the second Bianchi identity of the Riemann tensor, and then contract the upper index with one of the lower ones. Aug 20, 2011 #3

2014年8月23日 · This note develops a formula for the ##G^{00}## component of the Einstein tensor in the FLRW coordinate system for a homogeneous and isotropic spacetime. We use the convention that tensor indices ##i, j## or ##k## are used only for spatial dimensions (indices 1, 2 and 3), whereas any other pronumeral (whether Greek or Roman) in an index ...

2015年3月9日 · The Riemann tensor contains more information than the Ricci tensor. The Bianchi identity is an identity that holds for the Riemann tensor (and there is an associated one for the Einstein tensor) that basically says some sum of some permutation of the (covariant) derivatives of the Riemann tensor gives you 0.

2016年8月14日 · Hello I am little bit confused about calculating Ricci tensor for schwarzschild metric: So we have Ricci flow equation,∂ t g μν =-2R μν. And we have metric tensor for schwarzschild metric: Diag((1-r s /r),(1-r s]/r)-1,(r 2),(sin 2 Θ) and ∂ t g μν =0 so 0=-2R μν and we get that R μν =0.But R μν should not equal to zero for ...

2013年8月16日 · To answer the OP title the Einstein tensor can be defined as the Ricci tensor minus a expression which divergence be equal to Ricci tensor's divergence(by virtue of the symmetries of the Riemann tensor manifest in the second Bianchi identity), since we want to obtain a divergence-free tensor for physical (conservation) reasons.

2023年4月18日 · The Schwarzschild solution is a vacuum solution the stress-energy tensor is everywhere zero; hence so is the Einstein tensor. Note that the Einstein tensor depends on the Ricci tensor, which is a summarisation of the Riemann curvature tensor and does not include all parts of the Riemann. The remaining parts are the Weyl curvature, which is non ...

2012年10月30日 · The Einstein Tensor, also known as the Einstein curvature tensor, is a mathematical object used in Einstein's theory of general relativity to describe the curvature of spacetime. It is a combination of the Ricci tensor and the scalar curvature and is essential in understanding the gravitational field equations.