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einstein's field eq
  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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