Metric tensor in general relativity

Importance of Metric Tensor

1. Metric tensor in General Relativity by Halo Anwar Abdulkhalaq May, 2016 University of Sulaimani School of Science education Physics department

2. Overview Introduction Euclidian Metric Minkowski Metric Metric in General relativity 1- General form 2- Schwarzschild Metric 3- Some more Metrics Use for Einstein field equations Summary

3. Introduction Metric tensor (Metric) is an important quantity in General relativity. Sometimes it is thought of as alternative of Newton’s gravitational potential . It is in fact the geometrical representation of space or space-time. Metric is considered to be basic block of Einstein's equations of field. It can be calculated from the line element (the distance between two points in space or space-time).

4. Euclidian Metric To understand metric it is useful to start with simplest line element. Distance between two points in two or three dimension in space is called Euclidian geometry: And in 3-D: The metric components of the above line element are (1,1,1), they are the coefficients of the coordinates ( dx , dy , dz ).

5. Minkowski Metric This is the combination between Euclidean 3-D space with time. The line element here represents the distance between two events: This can be written in general form as: Hence: , Here conventional units have been used, where (c=1) and Einstein summation convention been applied.

6. So the Minkowski metric is (-1,1,1,1). And it can be written as a 4x4 matrix:

7. Metric in General relativity General form: , In matrix notation: In flat geometry:

8. 2- Schwarzschild Metric: It is written as: It is actually driven from spherical coordinate line element. The metric components are:

9. 3- Some more Metrics: Eddington-Finkelstein Kruskal-Szekeres Friedman-Robertson-Walker

10. Einstein field equations Field equations are set of equations relate geometry to matter. According to these equations gravity is geometry. They are as follow: where is Einstein tensor given by: To solve these equations we need to follow this pattern:

11. The ricci tensor is written in terms of the spacetime connection: The spacetime connection is directly written in terms of the metric tensor:

12. Summary Metric tensor is key factor in Einstein's theory of general relativity. It can be calculated from line element. By having metric tensor the spacetime connection can be calculated. From spacetime connection Ricci tensor and Ricci scalar are determined. Finally, from Ricci tensor and scalar Einstein tensor will be evaluated which is final step of solving field equations.

13. References T . Clifton, P.G. Ferreira, A. Padilla, and C. Skordis . Modied gravity and cosmology. Phys.Rept ., 513:1, 2012. S.M.Carroll . Spacetime and Geometry: An Introduction to General Rel-ativity . Chicago, Addison Wesley, 2004. C.W. Misner , K.S. Throne, and J.A. Wheeler. Gravitation. Freeman,1973. H. Stephani . Relativity: An Introduction to Special and General Relativity. 3id edition, 2004. B.F.Schutz . A First Course in General Relativity. Cambridge University Press, 2nd edition, 2009 . http://gfm.cii.fc.ul.pt/events/lecture_series/general_relativity/gfm-general_relativity-lecture4.pdf

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