# Derivation of Riemann tensor's first order-equation

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In summary, Schutz's second edition book explains that the equation for Riemann tensor to first order in ##h_{\mu\nu}## can be derived by using Eq. (8.12) and can be found on pages 189-192. This equation is first order in ##h## (and its derivatives) and can be obtained by assuming that ##h## and its derivatives are small and ignoring higher order terms. The derivation involves inserting the metric and Christoffel symbols into the expression for the curvature components.
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TL;DR Summary
Trying to derive Eq. (8.25) for Riemann tensor in the book of Schutz's in GR.
According to Schutz's second edition book, the equation for Riemann tensor to first order in ##h_{\mu\nu}## is:

$$R_{\alpha\beta\mu\nu}= \frac{1}{2}(h_{\alpha \nu,\beta \mu}+h_{\beta\mu,\alpha\nu}-h_{\alpha\mu,\beta \nu}-h_{\beta \nu , \alpha \mu})$$
which (as stated in the book) can be derived easily by using Eq. (8.12) which is:

$$g_{\alpha \beta}=\eta_{\alpha \beta}+h_{\alpha\beta}$$
All this is covered in pages 189-192.

How to derive the above identity? and how to derive the n-th order in ##h_{\mu\nu}## equation for Riemann tensor?
How come this equation is of first order in ##h_{\mu\nu}## if the terms in the above equation are second order derivatives of ##h##.

It is first order in ##h## (and its derivatives), not in the order of the derivatives. The assumption is that ##h## and its derivatives are small and you ignore terms that are of higher order in ##h##. The derivation is just inserting your metric into the expression for the Christoffel symbols (keeping only terms linear in ##h##) and then inserting the expression for the Christoffel symbols into the expression for the curvature components in terms of the Christoffel symbols (again, keeping only terms linear in ##h##).

MathematicalPhysicist

## 1. What is the Riemann tensor's first-order equation?

The Riemann tensor's first-order equation is a mathematical expression that describes the curvature of a space in terms of its metric. It is a set of four equations that relate the first derivatives of the metric tensor to the Riemann curvature tensor.

## 2. Why is the derivation of the Riemann tensor's first-order equation important?

The Riemann tensor's first-order equation is an essential tool in the study of general relativity and the theory of gravity. It allows us to understand the curvature of spacetime and its effects on the motion of objects in the universe.

## 3. How is the Riemann tensor's first-order equation derived?

The Riemann tensor's first-order equation is derived by applying the Bianchi identity, which relates the Riemann curvature tensor to the Ricci tensor and the scalar curvature. This identity is then used to simplify the Riemann tensor's second-order equation into a set of first-order equations.

## 4. What are some applications of the Riemann tensor's first-order equation?

The Riemann tensor's first-order equation is used in various fields, including cosmology, astrophysics, and high-energy physics. It is also used in practical applications, such as in the design of space-time sensors and in the analysis of gravitational waves.

## 5. Are there any limitations to the Riemann tensor's first-order equation?

While the Riemann tensor's first-order equation is a powerful tool, it has limitations in certain situations. For example, it does not take into account the effects of matter and energy on the curvature of spacetime. In these cases, more complex equations, such as the Einstein field equations, are needed to fully describe the curvature of the universe.

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