5.9 Retrieving the Schwarzschild Metric from Einstein Equations |
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the procedure and we add such a correction to the gravitational equation we reconstruct a new renormalized Lagrangian that now includes up to quartic terms in the field, namely:
L (h) = L (2)(h) + κ L (3)(h) + κ2 L (4)(h) |
(5.8.14) |
It is clear that, with increasing and rapidly divergent algebraic effort, the above procedure can be repeated infinitely many times, generating an hμν Lagrangian that is developed in power series of the coupling parameter κ and which contains an unlimited number of interaction vertices at n-legs:
∞
L tot(h) = L (2)(h) + κn L (n)(h) (5.8.15)
n=3
Do we have any idea of what the sum of such an infinity series might look like? Of course we do. Introducing the tensor
and its inverse gμν the Lagrangian L tot(h) turns out to be nothing else but the geometrical Einstein-Palatini Lagrangian:
L tot(h) = R[g] |
− det g |
(5.8.17) |
The lesson taught by Feynman with its Venusian history is that without knowing anything about differential geometry, curvature of space-time and tensorial calculus the correct gravitational action could nonetheless be discovered starting from the crucial observation that the gravitational field is what couples to the current of fourmomentum P μ, namely the stress-energy tensor. This is not so much surprising if we reflect that the momentum P μ is the generator of translations and if we pretend to transform translations into a local symmetry then we are just requiring that the theory we want to construct should be diffeomorphic or as Einstein formulated it generally covariant invariant. In simple words what is a diffeomorphism if not a local translation?
5.9Retrieving the Schwarzschild Metric from Einstein Equations
Having discussed Einstein equations in all their aspects it is now the proper moment to prove that the Schwarzschild metric we used in Chap. 4 to retrieve all Newtonian Physics plus corrections is indeed an exact solution of Einstein equations. As we are going to show, Schwarzschild metric is just a vacuum solution, namely it solves Einstein equations with vanishing stress-energy tensor, being Ricci-flat. In the sequel, we rediscuss Schwarzschild solution at two levels. In Chap. 6 of this
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5 Einstein Versus Yang-Mills Field Equations |
volume, considering the equations of stellar equilibrium we join the Schwarzschild metric describing the region of space-time external to a spherical star with the interior solution of Einstein equations driven by the stress-energy tensor of the fluid composing the star. In Chap. 2 of Volume 2 we study Schwarzschild space-time and its analytic extension beyond the horizon as the first spherical symmetric example of a black-hole. Finally in Chap. 3 of Volume 2 the Schwarzschild solution will be incorporated as a particular case of the general Kerr-Newman solution which corresponds to the most general stationary black-hole.
The Schwarzschild metric belongs to the following class of spherical symmetric, static metrics, whose coefficients can depend only on the radial coordinate r:
ds2 = − exp 2a(r) dt2 + exp 2b(r) dr2 + r2 dθ 2 + sin2 θ dφ2
We can easily recast such a metric in the vielbein formalism writing:
E0 |
= dt exp a(r) ; |
E1 |
= dr exp b(r) |
E2 |
= r dθ ; |
E3 |
= r dφ sin θ |
Calculating the exterior differential of the vielbein (5.9.2) we obtain:
dE0 = a e−bE1 E0 |
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= 0 |
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dE |
2 |
= |
e−b |
E |
1 |
E |
2 |
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r |
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e−b |
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3 |
cos θ 1 |
2 |
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3 |
dE |
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E |
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+ sin θ |
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E |
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r |
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r |
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On the other hand the vanishing torsion equation takes the following explicit form:
dE0 − ω01 E1 − ω02 E2 − ω03 E3 = 0
dE1 − ω01 E0 − ω12 E2 − ω13 E3 = 0
(5.9.4)
dE2 − ω02 E0 + ω12 E1 − ω23 E3 = 0
dE3 − ω03 E3 + ω13 E1 + ω23 E2 = 0
so that, combining (5.9.4) with (5.9.3) we determine the following unique solution for the Levi Civita spin connection ωab :
ω01 |
= − ˙ |
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ω02 |
= |
ω03 |
= |
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ae−bE0 |
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0 |
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ω12 |
= |
e−b |
E2; |
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ω13 |
= |
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e−b |
E3 |
(5.9.5) |
r |
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r |
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cos θ 1 |
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ω23 |
= |
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E3 |
; |
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sin θ |
r |
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where the dot denotes the derivative with respect to the parameter r. Inserting this result in the definition of the curvature two-form we obtain:
5.9 Retrieving the Schwarzschild Metric from Einstein Equations |
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R01 |
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ab |
a2 |
exp 2b |
E0 |
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E1 |
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[− ] |
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R02 |
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a˙ |
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exp |
2b E0 |
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E2 |
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= r |
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[− |
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R03 |
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a˙ |
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exp |
2b E0 |
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E3 |
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= r |
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[− |
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12 |
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b |
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1 |
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2 |
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R |
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˙ |
exp[−2b]E |
E |
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r |
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13 |
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b |
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1 |
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3 |
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R |
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˙ |
exp[−2b]E |
E |
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r |
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R23 |
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1 − exp[−2b] |
E2 |
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E3 |
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r2 |
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From this result we easily read off the component of the Riemann tensor and we can calculate the Einstein tensor which has the following form:
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1 |
− |
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1 |
− |
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b |
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= r2 |
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r2 |
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r |
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G00 |
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e−2b |
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2 |
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˙ |
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= − r2 |
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r2 + |
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r |
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G11 |
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1 |
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e−2b |
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2 |
a˙ |
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b |
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− ˙ ˙ |
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= |
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= |
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r |
+ ¨ + |
˙ |
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G22 |
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G33 |
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e−2b |
a˙ − ˙ |
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a |
(a)2 |
ab |
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Gba |
= 0 |
otherwise |
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(5.9.15) |
In the vacuum we have to set all the components of Gab to zero. Summing the first two equations we obtain:
Hence the sum of the two functions a(r) and b(r) is a constant. Yet at infinity, namely when r → ∞ the considered metric should approach the Minkowski metric, namely both a(r) and b(r) should tend to zero. This means that the integration constant is zero and we have
Replacing (5.9.17) into the vanishing condition for G22 as given in (5.9.14) we get:
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+ 2 |
˙ |
− |
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b |
= |
d |
− |
− |
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˙ |
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0 |
b |
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b2 |
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2 |
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exp |
2b(r) r |
1 |
(5.9.18) |
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rdr
The last equation is immediately integrated yielding:
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= |
− |
2 m |
−1 |
(5.9.19) |
exp 2b(r) |
1 |
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5 Einstein Versus Yang-Mills Field Equations |
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= |
− |
r |
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(5.9.20) |
exp 2a(r) |
1 |
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2 m |
where m is an integration constant. This solution uniquely fixed by the boundary conditions at infinity is just the Schwarzschild metric.
References
1.Lord, E.A.: A theorem on stress-energy tensors. J. Math. Phys. 17, 37 (1976)
2.Einstein, A.: Die Feldgleichungen der Gravitation. in: Sitzungsberichte der Preussischen Akademie der Wissenschaften zu Berlin, pp. 844–847 (1915)
3.Einstein, A.: Die Grundlage der allgemeinen Relativitätstheorie. Ann. Phys. 49 (1916)
