1) The metric tensor is a fundamental quantity in general relativity that represents the geometry of spacetime. It can be derived from the line element and used to calculate distances between points in spacetime. 2) Some important metrics include the Euclidean metric, Minkowski metric for flat spacetime, and Schwarzschild metric for a spherical mass. The Schwarzschild metric has components that depend on the mass of the object. 3) Einstein's field equations relate the Einstein tensor, which depends on properties of spacetime curvature derived from the metric tensor, to the stress-energy tensor representing matter and energy in spacetime. Solving the field equations involves determining the metric tensor and using it to calculate curvature properties.

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).
222
yxs 
222
dydxds 

2222
dzdydxds 

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.
22222
dzdydxdtds 
232221202
)()()()( dxdxdxdxds 

 dxdxds 2
3,2,1,0

6. • So the Minkowski metric is (-1,1,1,1).
• And it can be written as a 4x4 matrix:













1000
0100
0010
0001


7. Metric in General relativity
1. General form:
,
In matrix notation:
In flat geometry:

 dxdxgds 2
3,2,1,0













33323130
23222120
13121110
03020100
gggg
gggg
gggg
gggg
g
 g

8. 2- Schwarzschild Metric:
• It is written as:
• It is actually driven from spherical coordinate
line element.
• The metric components are:
222222122
sin)
2
1()
2
1(  drdrdr
r
M
dt
r
M
ds  






















22
2
1
sin000
000
00)
2
1(0
000)
2
1(
r
r
r
M
r
M
g

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:
 GTG 8
G
 RgRG
2
1



 GRgds 2

11. • The ricci tensor is written in terms of the
spacetime connection:
• The spacetime connection is directly written in
terms of the metric tensor:











  ,,R
)(
2
1
,,, 

 gggg 

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

14. Thanks for your attendance
and
Happy to hear your questions