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How Einstein Got His Field Equations
S. Walters
In commemoration of General Relativity’s centennial
A
BSTRACT
. We study the pages in Albert Einstein’s 1916 landmark paper in the
Annalen der Physik
where he derived his ﬁeld equations for gravity. Einstein made two heuristic andphysically insightful steps. The ﬁrst was to obtain the ﬁeld equations in vacuum in a rather geometric fashion. The second step was obtaining the ﬁeld equations in the presence of matter from the ﬁeld equations in vacuum. (This transition is an essential principle in physics,much as the principle of local gauge invariance in quantum ﬁeld theory.) To this end, we goover some quick differential geometric background related to curvilinear coordinates, vectors,tensors, metric tensor, Christoffel symbols, Riemann curvature tensor, Ricci tensor, and seehow Einstein used geometry to model gravity.
This paper is a more detailed version of my talk given at the Math-PhysicsSymposium at UNBC on February 25, 2016. It is in reference to Einstein’spaper: A. Einstein, The Foundation of the General Theory of Relativity,
Annalen der Physik
, 49, 1916. (For an English translation see: H. A. Lorentz, A.Einstein, H. Minkowski, H. Weyl,
The Principle of Relativity
.) The paper has two sections. The ﬁrst section is a smash course on thesemi-Riemannian geometry tools needed to understand Einstein’s theory. The second section looks at Einstein’s derivation of his ﬁeld equations in vac-uum and in the presence of matter and/or electromagnetism as he workedthem out in his paper. This paper commemorates the 100th centennial of Einstein’s General The-ory of Relativity, which he ﬁnalized near the end of November 1915 and pub-lished in 1916.
Date
: Jan./Feb. 2016 (last update July 2016)
L
A
T E X File: EinsteinRelativityLectureFeb2016.tex
.2000
Mathematics Subject Classiﬁcation.
35Q76 83C05 35Q75 37N20 83-01 83-XX.
Key words and phrases.
Einstein ﬁeld equations, tensors, relativity, gravity, curvature, spacetime.
1
a r X i v : 1 6 0 8 . 0 5 7 5 2 v 1 [ p h y s i c s . h i s t - p h ] 1 9 A u g 2 0 1 6
2 S. Walters
1.
Semi-RiemannianGeometry
In this section we outline brieﬂy some of thebasicandstandardconceptsknownfromsemi-Riemannian geometry that are used inEinstein’s formulation of his theory of grav-ity.
Spacetime Curvilinear coordinates.
Spacetime coordinates are written usingsuperscripts simply as
x
µ
where
µ
=
1
,
2
,
3
,
4
. This is short for vectors
(
x
1
,
x
2
,
x
3
,
x
4
)
describ-ing points in spacetime with respect to a cer-tain coordinate system (cartesian, cylindri-cal, spherical, etc). The coordinates
x
1
,
x
2
,
x
3
refer to spatial coordinates and
x
4
to the timecoordinate. Anyothercoordinatesystemcan be written as
x
µ
. (An observer and the coor-dinate system used by him/her will be iden-tiﬁed.)
Einstein’s Summation Convention.
Any time an upper index is repeated as a lower index we are automatically summingover that index. For example, we write
4
∑
µ
=
1
V
µ
U
µ
≡
V
µ
U
µ
,
4
∑
µ
=
1
A
µν τµ
≡
A
µν τµ
.
(In such cases one drops the
∑
notation.) A vector ﬁeld
V
is a differentiable func-tion deﬁned on a certain region of space time whose values are “tangent” vectors to space-time (much as vectors one the sphere that are tangent to it at each point of a certainregion on it). There is a natural way to de-scribe
V
by means of its components
V
µ
rela-tive to a given coordinate system
x
µ
. (Essen-tially, each coordinate axis
x
µ
has a vector along it similar to the usual
ˆi
,
ˆ j
,
ˆk
vectors in3-space
R
3
, as illustrated in Figure 1.) The components of such vector ﬁelds are written with superscripts and called
contra variant
because their components in differ-ent coordinate systems –
V
µ
relative to
x
µ
and
V
µ
relative to
x
µ
– transform accordingto the rule
V
ν
=
∂
x
ν
∂
x
µ
V
µ
.
(1.1)
Covariant
vector ﬁelds are written usingsubscripts
V
µ
and they transform in a dualmanner via
V
ν
=
∂
x
µ
∂
x
ν
V
µ
.
(1.2)
Figure 1.
At each point
p
in space timeone has natural vectors
e
1
,
e
2
that are tan-gent to the curved coordinate axes, quiteanalogous to the
ˆi
,
ˆ j
,
ˆk
vectors in calculus.
Examples.
(1) If a test particle is moving in space andits spacetime coordinates are expressed interms of its proper time
τ
, i.e.
x
µ
=
x
µ
(
τ
)
,then its
velocity 4-vector
is deﬁned by
V
µ
=
dx
µ
d
τ
.
Deﬁning a particle’s velocity in this way isquite convenient since it gives us a “covari-ant” way of describing velocities. This is sosimply because from the chain rule from cal-culus we see that the velocity 4-vector trans-forms just like a contravariant vector ﬁeld:
dx
ν
d
τ
=
∂
x
ν
∂
x
µ
dx
µ
d
τ
.
(2) As a second example, consider a scalar ﬁeld
ϕ
=
ϕ
(
x
µ
) =
ϕ
(
x
ν
)
, and look at its gradi-ent vector ﬁeld
∇
ϕ
with components
V
ν
=
∂ϕ ∂
x
ν
.Given any other coordinate system
x
µ
, the
How Einstein Got His Field Equations 3
chain rule tells us hows the gradient com-ponents are related:
∂ϕ ∂
x
ν
=
∂
x
µ
∂
x
ν
∂ϕ ∂
x
µ
.
This is exactly how covariant vectors trans-form – as in equation (1.2). A
tensor
(ﬁeld) is like a vector (ﬁeld) except that it can have two or more index compo-nents, e.g.
A
µν
B
µν
C
µ νρ
R
ρτ κα
.
The number of indices is called the
rank
of the tensor. (So vectors have rank 1, scalar ﬁelds have rank 0, and
C
µ νρ
has rank 3.) The way tensor components transform be-tween different coordinate systems is just like equation (1.1) for each upper index, andlike equation (1.2) for each lower (covariant)index. Thus, for the rank 2 tensor
A
β α
, itscomponents in another coordinate system
A
ν µ
are related by
A
ν µ
=
∂
x
ν
∂
x
β
∂
x
α
∂
x
µ
A
β α
.
It is for such elegant transformation prop-erties of tensors that Einstein’s principle of generalcovariance(discussedinthenextsec-tion) requires that the laws of physics beexpressed in covariant form using tensors.Once such a law holds in one coordinatesystem the same form of the law holds inany other coordinate system. The most important rank 2 tensor is the
metric tensor
g
µν
because it contains theessential ingredients of gravity, distance,time, and curvature of spacetime. It has theessential property that it is symmetric
g
µν
=
g
νµ
.
Onecanhaveeithera
Riemannian
ora
semi-Riemannian
metric tensor depending on itssignature being
+ + ++
or
−
+ ++
, respec-tively. The
signature
of
g
µν
(treated as a 4 by 4 symmetric matrix with real entries) be-ing “
−
+++
” means that it has one one neg-ative eigenvalue and 3 positive eigenvalues.(The only negative eigenvalue is related tothe time coordinate.)InRelativityweareinterestedinsemi-Rie-mannian (or
Lorentzian
) metric tensors ex-empliﬁed by the
Minkowski metric
η
µν
=
1 0 0 00 1 0 00 0 1 00 0 0
−
1
(1.3)(used in Special Relativity) and its associ-ated element of
proper spacetime interval
ds
2
=
η
µν
dx
µ
dx
ν
=
−
d
τ
2
where
d
τ
2
is called the
proper time
(by anal-ogy with element of arc length in Riemann-ian geometry).In General Relativity, the spacetime inter- val extends to any coordinate system usingthe metric tensor components
g
µν
accordingto
ds
2
=
g
µν
dx
µ
dx
ν
=
−
d
τ
2
.
The essential property of proper time isthat it is an invariant for all coordinate sys-tems:
d
τ
2
=
d
τ
2
.
All observers (coordinate systems) will agreeon this quantity. (Again, this follows by thechain rule.)Since the metric tensor
g
µν
is an invertiblematrix, its inverse is written in superscript form
g
µν
. Thus using the usual rules of ma-trix multiplication one has
g
µν
g
νλ
=
δ
µ λ
where
δ
µ λ
istheKroneckerdeltafunction(whichis 1 when
µ
=
λ
and is 0 when
µ
=
λ
).Both forms of the metric tensor are usedto
raise and lower indices
on tensors. For
4 S. Walters
example,
A
ν
=
g
νµ
A
µ
,
g
ρµ
R
ρτ κα
=
R
τ µ κα
.
The
contraction
of a mixed tensor is ob-tained by setting an upper index and a lower indexequalandthensumming. Forinstance,if one contracts the rank 4 tensor
R
ρτ κα
withrespect to the indices
ρ
and
κ
, one gets therank 2 tensor
S
τ α
=
R
ρτ ρα
≡
4
∑
ρ
=
1
R
ρτ ρα
where the index
ρ
now disappeared.
1
Covariant Derivative.
Since spacetime isgenerally curved, the notion of the deriva-tive from calculus becomes curved also. It becomes what we call the
covariant deriva- tive
. The covariant derivative of a covariant vector ﬁeld
A
µ
is given by
A
µ
;
ν
=
∂
A
µ
∂
x
ν
−
Γ
λ µν
A
λ
where
Γ
λ µν
=
12
g
λρ
∂
g
ρν
∂
x
µ
+
∂
g
ρµ
∂
x
ν
−
∂
g
µν
∂
x
ρ
iscalledthe
Christoffel symbol
. Thecovari-ant derivative of a contravariant vector ﬁeldis quite similar except with a “
+
” in place of the “
−
”; thus
A
µ
;
ν
=
∂
A
µ
∂
x
ν
+
Γ
µ νλ
A
λ
.
For example, if the metric tensor
g
µν
isconstant in some coordinate system, then
Γ
’s are all 0 and so the covariant derivativereduces to the usual partial derivative. Theadvantage of the covariant derivative is that it gives rise to tensors: the covariant deriva-tive of tensors are tensors also.
1
One can contract between any upper index andany lower index. We can also “contract” between twolower or two upper indices after one raises or lowersone of them. It’s just like taking the trace of a matrix.
Einstein regarded the Christoffel symbols
Γ
λ µν
as“
the components of the gravitationalﬁeld
.” In his paper, he also referred to themetric tensor
g
µν
as “
describing the grav-itational ﬁeld in relation to the chosensystem of coordinates.
” So, in a manner of speaking, gravity affects how we take thederivative!
2
A test particle moving “freely” in vacuumunder the inﬂuence of the geometry of space(i.e., undergravity)obeysthe
geodesic equa-tion
d
2
x
ρ
d
τ
2
+
Γ
ρµν
dx
µ
d
τ
dx
ν
d
τ
=
0
(where
τ
is the particle’s proper time). Thisequation describes the “straight lines”(geodesics) of spacetime. Thus a ray of light or photons moving freely in (vacuum) space-time will follow a path that is the analogue of straight lines (or great circles on a sphere).For a slowly moving particle in low grav-ity, this equation essentially comes down toNewton’s second law
F
=
ma
. If
Γ
≡
0
(zerogravity)thegeodesicequationbecomes
d
2
x
ρ
d
τ
2
=
0
, i.e. zero acceleration or constant velocity,so it moves along a straight line (i.e., we’rein ﬂat spacetime). The
Riemann-Christoffel Curvature Ten-sor
is the rank 4 tensor
R
λ µνκ
=
∂ ∂
x
κ
Γ
λ µν
−
∂ ∂
x
ν
Γ
λ µκ
+
Γ
ηµν
Γ
λ κη
−
Γ
ηµκ
Γ
λ νη
.
Itgovernsallaspectofthecurvatureofspace-time. If it is zero everywhere then our space-time is
ﬂat
. This looks like a fairly com-plicated expression, but here’s a way to re-late it to something that is more familiar.From advanced calculus we know that par-tial derivatives commute on most functions
2
The same thing happens in QFT where the de-rivative is similarly modiﬁed by means of the 4-vector potential of the electromagnetic ﬁeld in order that theLagrangian remain invariant – e.g., see [3], Section10.3.

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