# Encyclopedia of Geodesy

Living Edition
| Editors: Erik Grafarend

# Operational Significance of the Deviation Equation in Relativistic Geodesy

Living reference work entry
DOI: https://doi.org/10.1007/978-3-319-02370-0_164-1

## Definitions

Deviation equation. Second-order differential equation for the 4-vector which measures the distance between reference points on neighboring world lines in spacetime manifolds.

Relativistic geodesy. Science representing the Earth (or any planet), including the measurement of its gravitational field, in a four-dimensional curved spacetime using differential-geometric methods in the framework of Einstein’s theory of gravitation (general relativity).

## Introduction

How does one measure the gravitational field in Einstein’s theory? What is the foundation of relativistic gradiometry? The deviation equation gives answers to these fundamental questions.

In Einstein’s theory of gravitation, i.e., general relativity, the gravitational field manifests itself in the form of the Riemannian curvature tensor R abcd (Synge, 1960). This 4th-rank tensor can be defined as a measure of the noncommutativity of the parallel transport process of the underlying spacetime manifold (Synge and Schild, 1978...
This is a preview of subscription content, log in to check access.

## References

1. Ciufolini, I., and Demianski, M., 1986. How to measure the curvature of space-time. Physical Review D, 34, 1018.
2. Levi-Civita, T., 1926. Sur l’écart géodésique. Mathematische Annalen, 97, 291.
3. Pirani, F. A. E., 1956. On the physical significance of the Riemann tensor. Acta Physica Polonica, 15, 389.Google Scholar
4. Puetzfeld, D., and Obukhov, Yu. N., 2016. Generalized deviation equation and determination of the curvature in General Relativity. Physical Review D, 93, 044073.Google Scholar
5. Synge, J. L., 1926. The first and second variations of the length integral in Riemannian space. Proceedings of the London Mathematical Society, 25, 247.
6. Synge, J. L., 1927. On the geometry of dynamics. Philosophical Transactions of the Royal Society of London. Series A, 226, 31.
7. Synge, J. L., 1960. Relativity: The General Theory. Amsterdam: North-Holland.Google Scholar
8. Synge, J. L., and Schild, A., 1978. Tensor Calculus. New York: Dover.Google Scholar
9. Szekeres, P., 1965. The gravitational compass. Journal of Mathematical Physics, 6, 1387.