From SR to GR in an easy math (and physical) way

[lokofer]

From SR to GR in an "easy" math (and physical) way

Hello..i know that for example to go from Newtonian to SR you take:

$$\frac{du}{ds}=0 \rightarrow \nabla _{u} u=0$$

My question is ¿is there an "easy" form to go from SR to GR in the form:

$$\nabla _{u} u=0 \rightarrow \nabla R_{ab} =0$$
Or to get Einstein equation appart from using the Lagrangian formalism... using "Differential Geomertry" or similar...

 

[robphy]

Your first claim doesn't make much sense to me. Can you clarify?
Are you trying to describe the evolution of the universe as a geodesic in some abstract space?

 

[Entropia]

Yes, there is a way to go from special relativity (SR) to general relativity (GR) using differential geometry. First, let's review what SR and GR are. SR is a theory that describes the relationship between space and time, while GR is a theory that describes how gravity affects the shape of space and time. In SR, the fundamental equation is the Lorentz transformation, which relates the coordinates of an event in one inertial frame to the coordinates in another inertial frame. In GR, the fundamental equation is the Einstein field equations, which relate the curvature of spacetime to the distribution of matter and energy.

To go from SR to GR, we need to introduce the concept of curvature. In SR, we assume that spacetime is flat and use the Minkowski metric to describe it. In GR, we allow for the possibility that spacetime is curved and use the metric tensor to describe it. The metric tensor is a mathematical object that tells us how to measure distances and angles in a curved space. It is denoted by g and has 16 components, but in most cases, we only need to consider a subset of those components.

The key idea in GR is that the curvature of spacetime is related to the distribution of matter and energy through the Einstein field equations. These equations can be written in terms of the metric tensor as:

G_{ab} = 8\pi T_{ab}

where G_{ab} is the Einstein tensor, which describes the curvature of spacetime, and T_{ab} is the stress-energy tensor, which describes the distribution of matter and energy. The left-hand side of this equation is known as the Ricci curvature, and the right-hand side is known as the stress-energy tensor. This equation is the GR equivalent of the \nabla _{u} u=0 equation in SR.

To understand this equation, we need to introduce the concept of covariant derivatives. In SR, we use partial derivatives to describe how a quantity changes with respect to a coordinate. In GR, we use covariant derivatives, which take into account the curvature of spacetime. The covariant derivative of a tensor is denoted by \nabla and is defined as:

\nabla_{a} T_{bc} = \partial_{a} T_{bc} + \Gamma^{d}_{ab} T_{dc} + \Gamma^{d}_{ac} T_{bd}

where \Gamma