fundamentals of general relativity - gsiweb-docs.gsi.de/~struck/hp/wsem/genrel_talk.pdf ·...

Fundamentals of General Relativity

Jürgen Struckmeier

GSI Helmholtzzentrum für Schwerionenforschung GmbH, Darmstadt, GermanyFIAS Frankfurt Institute for Advanced Studies, Frankfurt am Main, Germany

HICforFAIR-Workshop

“Current Topics in Accelerator-, Astro-, and Plasmaphysics”

Institute for Applied PhysicsGoethe University, Frankfurt am Main

Riezlern, 12–18 March 2017

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Outline

1 Basic observations regarding General RelativityEquivalence principleTidal forcesBending of light

2 Non-Euclidean geometryCurved surfacesParallel transport of a vectorRiemann curvature tensor, metric tensor

3 Outline of the theory of General RelativityAction principleEinstein equationsSchwarzschild solution of the Einstein equations

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Basic observations regarding General Relativity Equivalence principle

Equivalence principle

Observation 1: all bodies fall at the same rate in a gravitationalfield.Observation 2: an experimenter is (locally) not able to tell whetherhe is in a gravitational field or being accelerated through emptyspace.Conclusion: a gravitational field is locally equivalent to anaccelerating frame of reference.

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Basic observations regarding General Relativity Tidal forces

Tidal forces in accelerating frames

Observation 2’: an experimenter in a box is(locally) not able to tell whether he is in freespace or freely accelerating in a gravitationalfield.Observation 3: if the box accelerates over alarge distance towards a center of mass, thetwo test bodies approach one another the experimenter encounters a tidal force.Conclusion: a gravitational field is globallynot equivalent to an accelerating frame ofreference.The gravitational field appears to produce acurvature of space.

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Basic observations regarding General Relativity Bending of light

Optics

Observation 4: if light travels in an inertial frame, it travels along astraight line across the box (a).Observation 5: if the box is accelerating upwards, the lightdescribes a parabolic path, hence will be detected nearer thebottom of the box (b).Conclusion: the (local) equivalence principle then requires:

Light also follows a curved path in a gravitational field.

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Optics

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Optics

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Non-Euclidean geometry Curved surfaces

Non-Euclidean geometry: types of surfaces

If we inscribe circles on a plane, on a sphere, and on a saddle, we findfor the circles’ circumferences C and areas A as functions of the radiusa:

Plane: C = 2πa A = πa2 flat (zero curvature),Sphere: C < 2πa A < πa2 curved (positive curvature),Saddle: C > 2πa A > πa2 curved (negative curvature).

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Non-Euclidean geometry

Also, the sum Σ of the angles of a triangle depends on the geometry ofthe respective surface:

Plane: Σ = 180◦,Sphere: Σ > 180◦,Saddle: Σ < 180◦.

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Is our space flat?

One of the first experiments to scrutinize whether the space weare living in is actually flat was worked out by Carl Friedrich Gaußin 1821-1825.Of course, within the error tolerances no deviation from 180◦ foran Euclidean geometry triangle was observed.As we know today, the deflection of a light beam on earth,traveling over a distance of 100 km is about 10−3 mm.

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Is our space flat?

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Is our space flat?

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Non-Euclidean geometry Parallel transport of a vector

Parallel transport of a vector

In Euclidean space, two vectors are parallel if their Cartesiancomponents are the same (case (a)).In case (b), the space is a cylinder, which may be unrolled tomake it into a plane. This way it becomes case (a).In case (c), the result of a parallel transport becomespath-dependent.

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Parallel transport of a vector on a sphere

Consider the parallel transport of a vector round the closed pathPNQRP. The vector starts out at P as Vi and arrives back at P as Vf,where obviously Vi 6= Vf.

The difference δV = Vf − Vi gives a measure of the curvature of thepath enclosed.

The quantity “curvature” will now be investigated thoroughly.

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Non-Euclidean geometry Riemann curvature tensor, metric tensor

General coordinate systems

In a general, non-Cartesian coordinate system, the base vectors eα ofa frame of reference may depend on the set of independent variablesxµ, hence eα = eα(xµ).The derivative of eα resides in the space spanned by the eα, hencecan be expressed as a linear combination of the set of base vectors,

∂eα

∂xµ= Γb

αµeb.

As always in this talk, we make use of the summation convention,which means that summation over b is understood.The coefficients Γβ

αµ are referred to as the Christoffel symbols orconnection coefficients. So the real number Γβ

αµ is simply the βcomponent of the vector ∂eα/∂xµ. In a Cartesian coordinate system, all Christoffel symbols vanish.

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∂eα

∂xµ= Γb

αµeb.

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∂eα

∂xµ= Γb

αµeb.

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Simple example: polar basis vectors

We switch from Cartesian to polar basis vectors

er = cos θ ex + sin θ ey eθ = −r sin θ ex + r cos θ ey .

The derivatives of the base vectors are then∂er

∂θ= − sin θex + cos θ ey =

eθ,∂er

∂r= 0

∂eθ

∂θ= −r cos θex − r sin θ ey = −rer

∂eθ

∂r= − sin θex + cos θ ey =

The eight (23) Christoffel symbols follow as

Γrrr = 0, Γθ

rr = 0, Γrθr = 0, Γθ

θr =1r

Γrrθ = 0, Γθ

rθ =1r, Γr

θθ = −r , Γθθθ = 0.

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eθ,∂er

∂r= 0

∂eθ

Γrrr = 0, Γθ

θr =1r

Γrrθ = 0, Γθ

rθ =1r, Γr

θθ = −r , Γθθθ = 0.

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eθ,∂er

∂r= 0

∂eθ

Γrrr = 0, Γθ

θr =1r

Γrrθ = 0, Γθ

rθ =1r, Γr

θθ = −r , Γθθθ = 0.

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Vector derivative, parallel transport

The µ-derivative ∇µV of a vector V = V aea in a general coordinatesystem has now the representation

∇µV =∂V a

∂xµea + V a ∂ea

∂xµ=∂V a

∂xµea +

∂xµV b

(∂V a

∂xµ+ Γa

bµV b)

The condition for the parallel transport of V along the coordinate xµ isthen

∇µV = 0 ⇐⇒ ∂V ν

∂xµ= −Γν

bµV b.

The deviation δV = Vf − Vi due to a parallel transport of V along aclosed path measures the curvature of the underlying space. We cannow set up the general measure for that curvature, namely theRiemann curvature tensor.

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Vector derivative, parallel transport

The µ-derivative ∇µV of a vector V = V aea in a general coordinatesystem has now the representation

∇µV =∂V a

∂xµea + V a ∂ea

∂xµ=∂V a

∂xµea +

∂xµV b

(∂V a

∂xµ+ Γa

bµV b)

The condition for the parallel transport of V along the coordinate xµ isthen

∇µV = 0 ⇐⇒ ∂V ν

∂xµ= −Γν

bµV b.

The deviation δV = Vf − Vi due to a parallel transport of V along aclosed path measures the curvature of the underlying space. We cannow set up the general measure for that curvature, namely theRiemann curvature tensor.

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Parallel transport along a closed path (in 2D)

We parallel transport the vector Vi = V ai ea around the infinitesimal

closed loop ABCDA. So, going from A to B means x2 = b = const.and x1(A) = a, x1(B) = a + δa, with c = 1,2, hence

V ν(B)− V νi (A) =

∫ x1(B)

∂V ν

∂x1 dx1 = −∫ a+δa

c 1V cdx1.

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Parallel transport along a closed path

Going from B to C means x1 = a + δa = const., hence

V ν(C)− V ν(B) = −∫ b+δb

c 2V cdx2.

Going from C to D means x2 = b + δb = const., hence

V ν(D)− V ν(C) = −∫ a

a+δaΓν

c 1V cdx1.

Finally, going from D to A means x1 = a = const., hence

V νf (A)− V ν(D) = −

b+δbΓν

c 2V cdx2.

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The sum over the four segments yields

δV ν = V νf (A)− V ν

i (A) =

−∫ a+δa

a[Γν

c 1V c]x2=b dx1 −∫ b+δb

b[Γν

c 2V c]x1=a+δa dx2

∫ a+δa

a[Γν

c 1V c]x2=b+δb dx1 +

∫ b+δb

b[Γν

c 2V c]x1=a dx2.

To first order, the integrands can be combined in pairs in terms of aTaylor series that is truncated after the linear term

[Γνc 1V c]x2=b+δb − [Γν

c 1V c]x2=b = δb∂

∂x2 (Γνc 1V c)

[Γνc 2V c]x1=a+δa − [Γν

c 2V c]x1=a = δa∂

∂x1 (Γνc 2V c) .

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The sum over the four segments yields

δV ν = V νf (A)− V ν

i (A) =

−∫ a+δa

a[Γν

c 1V c]x2=b dx1 −∫ b+δb

b[Γν

c 2V c]x1=a+δa dx2

∫ a+δa

a[Γν

c 1V c]x2=b+δb dx1 +

∫ b+δb

b[Γν

c 2V c]x1=a dx2.

To first order, the integrands can be combined in pairs in terms of aTaylor series that is truncated after the linear term

[Γνc 1V c]x2=b+δb − [Γν

c 1V c]x2=b = δb∂

∂x2 (Γνc 1V c)

[Γνc 2V c]x1=a+δa − [Γν

c 2V c]x1=a = δa∂

∂x1 (Γνc 2V c) .

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The sum over the four segments then simplifies to

δV ν = δb∫ a+δa

∂x2 (Γνc 1V c) dx1 − δa

∫ b+δb

∂x1 (Γνc 2V c) dx2

≈ δaδb[∂

∂x2 (Γνc 1V c)− ∂

∂x1 (Γνc 2V c)

], dx1 ≈ δa, dx2 ≈ δb

= δaδb[∂Γν

c 1∂x2 −

∂Γνc 2

∂x1 − Γνd 1Γd

c 2 + Γνd 2Γd

= δaδb Rνc 21 V c ,

where in the last step the xµ-derivatives of the V c were replaced byΓc

dµV d . We conclude:

The space is flat (δV ν = 0) exactly if all coefficients Rνc 21 vanish.

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Riemann-Christoffel curvature tensor

The procedure is generalized to a space of dimension n by replacingthe fixed indexes 2 and 1 by open indexes µ, λ = 1, . . . ,n.So generally, a space is flat exactly if all components of the Riemanntensor Rν

βµλ vanish

Rνβµλ =

∂Γνβλ

∂xµ−∂Γν

∂xλ+ Γν

aµΓaβλ − Γν

aλΓaβµ.

Due to symmetries, Rνβµλ has only 20 independent components.

A space is flat if and only if all components of the Riemann tensorvanish

Rνβµλ = 0 ⇔ flat space.

We may contract Rνβµλ by letting ν = µ = n and summing over all n.

This yields the symmetric Ricci tensor Rβλ = Rnβnλ

Rβλ ≡ Rnβnλ = 0 6⇔ flat space.

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βµλ vanish

Rνβµλ =

∂Γνβλ

∂xµ−∂Γν

∂xλ+ Γν

aµΓaβλ − Γν

aλΓaβµ.

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βµλ vanish

Rνβµλ =

∂Γνβλ

∂xµ−∂Γν

∂xλ+ Γν

aµΓaβλ − Γν

aλΓaβµ.

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Metric tensor

Suppose we are given two Vectors A = Aaea and B = Bbeb in ageneral coordinate system. Their scalar product A · B is then definedas the scalar quantity

A · B = (Aaea) · (Bbeb) = AaBb(ea · eb) = gabAaBb,

wherein we have defined the metric tensor (or briefly: the metric)

gαβ = eα · eβ ⇒ gαβ = gβα.

In a Cartesian system, the metric is simply the unit matrix

(gαβ) = diag(1,1, . . .), gαβ = δαβ,

whereas in a Minkowski space

(gαβ) ≡ (ηαβ) = diag(−1,1,1,1),

so thatA · B = −A0B0 + A1B1 + A2B2 + A3B3.

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Metric tensor

Suppose we are given two Vectors A = Aaea and B = Bbeb in ageneral coordinate system. Their scalar product A · B is then definedas the scalar quantity

A · B = (Aaea) · (Bbeb) = AaBb(ea · eb) = gabAaBb,

wherein we have defined the metric tensor (or briefly: the metric)

gαβ = eα · eβ ⇒ gαβ = gβα.

In a Cartesian system, the metric is simply the unit matrix

(gαβ) = diag(1,1, . . .), gαβ = δαβ,

whereas in a Minkowski space

(gαβ) ≡ (ηαβ) = diag(−1,1,1,1),

so thatA · B = −A0B0 + A1B1 + A2B2 + A3B3.

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Dual vector, infinitesimal distance

The metric enables us to convert the contravariant vector components(denoted by an upper index) into the covariant (dual) vectorcomponents (denoted by a lower index)

Aβ = gβaAa ⇒ A · B = AbBb.

The square ds2 of the infinitesimal distance between two points in ageneral Riemannian space is given by

ds2 = gab(x) dxadxb.

The components of the metric may depend on the spacetimecoordinates. Example: spherical space coordinates

ds2 = −c2dt2 + dr2 + r2dθ2 + r2 sin2 θ dφ2

(gαβ) = diag(−1,1, r2, r2 sin2 θ), x0 = ct , x1 = r , x2 = θ, x3 = φ.

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Curvature scalarThe identity

gµkgkν = δµν

defines the inverse metric (gµν) to (gµν).For a torsion-free space the correlation between the metric and theChristoffel symbols is given by the Levi-Civita connection

Γνβµ = 1

2gνk(∂gβk

∂xµ+∂gµk

∂xβ−∂gβµ

If the metric is given, the Christoffel symbols can be deduced. The Christoffel symbols then only serve as ancillary variables toexpress the correlation between metric and curvature tensors.The curvature scalar (Ricci scalar) is the contraction of the Ricci tensorRβλ with the metric tensor gλβ, hence the trace of Rβ

Rβλgβλ = Rλλ = R

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gµkgkν = δµν

Γνβµ = 1

2gνk(∂gβk

∂xµ+∂gµk

∂xβ−∂gβµ

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gµkgkν = δµν

Γνβµ = 1

2gνk(∂gβk

∂xµ+∂gµk

∂xβ−∂gβµ

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Outline of the theory of General Relativity Action principle

Hilbert action

The goal is to set up differential equations for the metric gµν(x) for agiven physical system described by a Lagrangian LM .

It was Hilbert’s idea to derive the Einstein equation from the actionprinciple

∫(LR + LM)

√−g d4x !

= 0 for δgµν 6= 0.

with g the determinant of the metric gµν and√−g d4x the invariant

volume element (under δgµν).

The Lagrangian LR that describes the “free” gravitational field (i.e. forthe classical vacuum LM ≡ 0) must be postulated.Both Lagrangians are coupled through their common dependence onthe metric gµν(x).

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Hilbert action

∫(LR + LM)

√−g d4x !

= 0 for δgµν 6= 0.

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Hilbert action

∫(LR + LM)

√−g d4x !

= 0 for δgµν 6= 0.

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Hilbert action

Under a general transformation of the reference frame x 7→ y themetric gik (x) transforms as

Gµν(y) = gik (x)∂x i

∂yµ

∂yν.

Taking the determinants of both sides, we find with g = det(gµν),G = det(Gµν), and det(ηµν) = −1√

−G =√−g∣∣∣∣∂x∂y

∣∣∣∣ , ∣∣∣∣∂x∂y

∣∣∣∣ =∂(x0, . . . , x3)

∂(y0, . . . , y3).

Under a change of the integration variables, the volume element d4xtransforms as

∣∣∣∣∂x∂y

∣∣∣∣d4y ⇒ d4x√−g = d4y

√−G.

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Hilbert action

∂yµ

∂yν.

−G =√−g∣∣∣∣∂x∂y

∣∣∣∣ , ∣∣∣∣∂x∂y

∣∣∣∣ =∂(x0, . . . , x3)

∂(y0, . . . , y3).

∣∣∣∣∂x∂y

∣∣∣∣d4y ⇒ d4x√−g = d4y

√−G.

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Hilbert action

∂yµ

∂yν.

−G =√−g∣∣∣∣∂x∂y

∣∣∣∣ , ∣∣∣∣∂x∂y

∣∣∣∣ =∂(x0, . . . , x3)

∂(y0, . . . , y3).

∣∣∣∣∂x∂y

∣∣∣∣d4y ⇒ d4x√−g = d4y

√−G.

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Outline of the theory of General Relativity Einstein equations

Einstein equations with energy/matter

Following the principle of Ockham’s razor, the “Gravity Lagrangian” LRwas postulated by Hilbert to be the Ricci scalar

LR =R2κ

2κgµνRµν

with the coupling constant, κ to be determined to yield the Newtongravity in the weak gravity limit.

Working out the Euler-Lagrange equation for gµν with this particularLR, one finds

Rµν − 12Rgµν = κTµν ,

Tµν =2√−g

∂(LM√−g)

∂gµν

the symmetric energy-momentum tensor associated with LM.24 / 37

Einstein equations with energy/matter

Following the principle of Ockham’s razor, the “Gravity Lagrangian” LRwas postulated by Hilbert to be the Ricci scalar

LR =R2κ

2κgµνRµν

with the coupling constant, κ to be determined to yield the Newtongravity in the weak gravity limit.

Working out the Euler-Lagrange equation for gµν with this particularLR, one finds

Rµν − 12Rgµν = κTµν ,

Tµν =2√−g

∂(LM√−g)

∂gµν

the symmetric energy-momentum tensor associated with LM.24 / 37

Einstein equations for a source-free region

Setting Tµν ≡ 0, we encounter the Einstein equation for a source-freeregion of a gravitational field

Rµν − 12Rgµν = 0.

Contracting with gµν gives

gµνRµν − 12Rgµνgµν = R − 1

2Rδµµ = R − 2R = 0 R = 0,

as the trace of a 4× 4 unit matrix is δµµ = 4. So the conditions

Rµν = 0 (≡ 10 coupled differential equations for gµν!)

are the final vacuum field equations.As the gravitational field acts as its own source, this equation is, strictlyspeaking, only an approximation.

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Einstein equations for a source-free region

Albert Einstein at the same state ofaffairs in his talk.

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Outline of the theory of General Relativity Schwarzschild solution of the Einstein equations

Exterior Schwarzschild solution

Solving the vacuum field equations Rµν = 0 for a static and sphericallysymmetric body at rest is referred to as the exterior Schwarzschildsolution of the Einstein equations.In flat spacetime the metric writes in spherical coordinates

ds2 = −c2dt2 + dr2 + r2(

dθ2 + sin2 θdφ2).

The Schwarzschild ansatz for the infinitesimal distance ds2 must havethe general form

ds2 = −e2ξ(r)c2dt2 + e2η(r)dr2 + r2(

dθ2 + sin2 θdφ2)

with ξ(r), η(r) as yet arbitrary functions to be determined.The procedure is now

∣∣Ansatz ⇒ Γa

bc ⇒ Rcd = 0 ⇒ gµν

∣∣final.

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ds2 = −c2dt2 + dr2 + r2(

ds2 = −e2ξ(r)c2dt2 + e2η(r)dr2 + r2(

dθ2 + sin2 θdφ2)

∣∣final.

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ds2 = −c2dt2 + dr2 + r2(

ds2 = −e2ξ(r)c2dt2 + e2η(r)dr2 + r2(

dθ2 + sin2 θdφ2)

∣∣final.

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Schwarzschild metric

So, the Ansatz expressions for the metric tensor and its inverse are thediagonal matrices

(gµν) = diag(− e2ξ(r),e2η(r), r2, r2 sin2 θ

)(gµν) = diag

(− e−2ξ(r),e−2η(r),1/r2,1/(r2 sin2 θ)

This can be translated into the following expressions for the connectioncoefficients Γν

βµ. For instance, Γ100 is obtained as

Γ100 = 1

2g1k(��∂g0k

∂x0 +��∂g0k

∂x0 −∂g00

)= −1

2g11∂g00

∂r= −1

2e−2η(r) ddr

(−e2ξ(r)

e2ξ(r)−2η(r).

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Similarly the other connection coefficients are obtained as

Γ010 = Γ0

01 =dξdr

Γ111 =

dηdr, Γ1

22 = −re−2η, Γ133 = −r sin2 θe−2η

Γ212 = Γ2

21 = Γ313 = Γ3

31 =1r, Γ2

33 = − sin θ cos θ

Γ323 = Γ3

32 = cot θ, all other coefficients are =0.

These quantities are now inserted into the vacuum equations

Rµν =∂Γb

∂xb −∂Γb

∂xν+ Γb

a bΓaµν − Γb

a νΓaµ b = 0.

We thus obtain differential equations for the Ansatz functions ξ(r) andη(r).

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Similarly the other connection coefficients are obtained as

Γ010 = Γ0

01 =dξdr

Γ111 =

dηdr, Γ1

22 = −re−2η, Γ133 = −r sin2 θe−2η

Γ212 = Γ2

21 = Γ313 = Γ3

31 =1r, Γ2

33 = − sin θ cos θ

Γ323 = Γ3

32 = cot θ, all other coefficients are =0.

These quantities are now inserted into the vacuum equations

Rµν =∂Γb

∂xb −∂Γb

∂xν+ Γb

a bΓaµν − Γb

a νΓaµ b = 0.

We thus obtain differential equations for the Ansatz functions ξ(r) andη(r).

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This way, three essential equations emerge

R00 =[ξ′′ + (ξ′)

2 − ξ′η′ + 2ξ′/r]

e2ξ(r)−2η(r) = 0

R11 = −ξ′′ − (ξ′)2

+ ξ′η′ + 2η′/r = 0

R22 =(−1− rξ′ + rη′

)e−2η(r) + 1 = 0.

In the first equation, the expression brackets must be zero. Adding thesecond equation, this gives immediately

ξ′ + η′ = 0 ⇒ ξ(r) + η(r) = d = const.

For r →∞ the Schwarzschild metric must approach the Minkowskimetric (ηµν) = diag

(− 1,1, r2, r2 sin2 θ

). Thus

limr→∞

ξ(r) = 0, limr→∞

η(r) = 0 ⇒ d = 0, ξ(r) = −η(r).

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R00 =[ξ′′ + (ξ′)

2 − ξ′η′ + 2ξ′/r]

e2ξ(r)−2η(r) = 0

R11 = −ξ′′ − (ξ′)2

+ ξ′η′ + 2η′/r = 0

R22 =(−1− rξ′ + rη′

)e−2η(r) + 1 = 0.

ξ′ + η′ = 0 ⇒ ξ(r) + η(r) = d = const.

(− 1,1, r2, r2 sin2 θ

). Thus

limr→∞

ξ(r) = 0, limr→∞

η(r) = 0 ⇒ d = 0, ξ(r) = −η(r).

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R00 =[ξ′′ + (ξ′)

2 − ξ′η′ + 2ξ′/r]

e2ξ(r)−2η(r) = 0

R11 = −ξ′′ − (ξ′)2

+ ξ′η′ + 2η′/r = 0

R22 =(−1− rξ′ + rη′

)e−2η(r) + 1 = 0.

ξ′ + η′ = 0 ⇒ ξ(r) + η(r) = d = const.

(− 1,1, r2, r2 sin2 θ

). Thus

limr→∞

ξ(r) = 0, limr→∞

η(r) = 0 ⇒ d = 0, ξ(r) = −η(r).

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The third equation (R22 = 0) now simplifies to

1 =(1 + 2rξ′

)e2ξ =

(re2ξ

)⇒ re2ξ = r − rs

with rs an integration constant. We thus finally obtain

e2ξ(r) = 1− rs

r, e2η(r) =

(1− rs

)−1,

and hence the Schwarzschild metric (which is exact!)

(gµν) = diag(−(

1− rs

)−1, r2, r2 sin2 θ

ds2 = −(

1− rs

)c2dt2 +

1− rsr

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The integration constant can be identified by comparing theSchwarzschild metric with the weak field approximation

g00 = −(

1− 2GMrc2

)⇒ rs =

2GMc2 .

The length rs is referred to as the Schwarzschild radius.Example: for the sun, we have

M� = 1.99× 1030 kg ⇒ rs = 2.95 km.

The sun’s Schwarzschild radius lies inside the sun, whose radius isR� = 6.96× 105 km. There, the vacuum equations do not apply,hence r = rs has no physical meaning in that case.

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The integration constant can be identified by comparing theSchwarzschild metric with the weak field approximation

g00 = −(

1− 2GMrc2

)⇒ rs =

2GMc2 .

The length rs is referred to as the Schwarzschild radius.Example: for the sun, we have

M� = 1.99× 1030 kg ⇒ rs = 2.95 km.

The sun’s Schwarzschild radius lies inside the sun, whose radius isR� = 6.96× 105 km. There, the vacuum equations do not apply,hence r = rs has no physical meaning in that case.

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The vacuum field equations apply for r > R�, hence for

2.956.96× 105 = 4.24× 10−6 � 1.

We observe that in this case the Schwarzschild metric differs onlyslightly from the Minkowski metric, for which rs = 0.We know, however, that neutron stars collapse if their mass is largerthan the Chandrasekhar limit, which is ≈ 1.3 M�. Then R < rs, and the surface r = rs lies outside the collapsed starwithin the vacuum. The Schwarzschild surface r = rs is now physical. To an outside observer rs is in effect a boundary of space-time. Timelike or lightlike geodesics from events inside will never reachthe outside. This kind of object is referred to as a black hole.

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The vacuum field equations apply for r > R�, hence for

2.956.96× 105 = 4.24× 10−6 � 1.

We observe that in this case the Schwarzschild metric differs onlyslightly from the Minkowski metric, for which rs = 0.We know, however, that neutron stars collapse if their mass is largerthan the Chandrasekhar limit, which is ≈ 1.3 M�. Then R < rs, and the surface r = rs lies outside the collapsed starwithin the vacuum. The Schwarzschild surface r = rs is now physical. To an outside observer rs is in effect a boundary of space-time. Timelike or lightlike geodesics from events inside will never reachthe outside. This kind of object is referred to as a black hole.

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Problem of Einstein’s Theory

The observed angular velocity of stars in a galaxy as a function of thedistance from the galactic center is not consistent with the theoreticalprediction from Einstein’s theory:

In principle, there are two options:We may postulate the existence of additional, yet unobservedmatter (“dark matter”).We may devise a modified theory of General relativity.

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Black hole of 10 sun masses, seen from a distance of 600 km; milkyway in the background. Movie1: circulating a neutron starMovie2: passing black hole Movie3: approaching a black hole

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Summary

General Relativity is a purely classical theory that emerges fromthe postulated Einstein-Hilbert action.As any form of energy acts as a source of the spacetimedynamics, the gravitational field acts as its own source. The dynamics is non-linear, no superposition principle.On the elementary particle level, this means that gravitionsinteract — in contrast to photons that do not directly interact.A quantized theory of gravity based on the Einstein-Hilbert actionappears to be not workable.The Einstein theory has been experimentally verified to a highprecision.

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The observed motion of galaxies is not consistent with Einstein’sequations. Postulate of dark matter.As up to now all attempts failed to detect dark matter, it may notexist at all.There are also claims that dark matter would not be compatiblewith the actual state of the universe.The failure of setting up a quantum field theory of gravity and theshaky concept of “dark matter” justify alternative approachesbased on a modified Einstein-Hilbert action

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fundamentals of general relativity - gsiweb-docs.gsi.de/~struck/hp/wsem/genrel_talk.pdf ·...

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