The Schwarzschild solution describes spacetime under the influence of a massive, non-rotating, spherically symmetric object. It is considered by some to be one of the simplest and most useful solutions to the Einstein field equations .[citation needed]
Working in a coordinate chart with coordinates ( r , θ θ --> , ϕ ϕ --> , t ) {\displaystyle \left(r,\theta ,\phi ,t\right)} labelled 1 to 4 respectively, we begin with the metric in its most general form (10 independent components, each of which is a smooth function of 4 variables). The solution is assumed to be spherically symmetric, static and vacuum. For the purposes of this article, these assumptions may be stated as follows (see the relevant links for precise definitions):
The first simplification to be made is to diagonalise the metric. Under the coordinate transformation, ( r , θ θ --> , ϕ ϕ --> , t ) → → --> ( r , θ θ --> , ϕ ϕ --> , − − --> t ) {\displaystyle (r,\theta ,\phi ,t)\rightarrow (r,\theta ,\phi ,-t)} , all metric components should remain the same. The metric components g μ μ --> 4 {\displaystyle g_{\mu 4}} ( μ μ --> ≠ ≠ --> 4 {\displaystyle \mu \neq 4} ) change under this transformation as:
But, as we expect g μ μ --> 4 ′ = g μ μ --> 4 {\displaystyle g'_{\mu 4}=g_{\mu 4}} (metric components remain the same), this means that:
Similarly, the coordinate transformations ( r , θ θ --> , ϕ ϕ --> , t ) → → --> ( r , θ θ --> , − − --> ϕ ϕ --> , t ) {\displaystyle (r,\theta ,\phi ,t)\rightarrow (r,\theta ,-\phi ,t)} and ( r , θ θ --> , ϕ ϕ --> , t ) → → --> ( r , − − --> θ θ --> , ϕ ϕ --> , t ) {\displaystyle (r,\theta ,\phi ,t)\rightarrow (r,-\theta ,\phi ,t)} respectively give:
Putting all these together gives:
and hence the metric must be of the form:
where the four metric components are independent of the time coordinate t {\displaystyle t} (by the static assumption).
On each hypersurface of constant t {\displaystyle t} , constant θ θ --> {\displaystyle \theta } and constant ϕ ϕ --> {\displaystyle \phi } (i.e., on each radial line), g 11 {\displaystyle g_{11}} should only depend on r {\displaystyle r} (by spherical symmetry). Hence g 11 {\displaystyle g_{11}} is a function of a single variable:
A similar argument applied to g 44 {\displaystyle g_{44}} shows that:
On the hypersurfaces of constant t {\displaystyle t} and constant r {\displaystyle r} , it is required that the metric be that of a 2-sphere:
Choosing one of these hypersurfaces (the one with radius r 0 {\displaystyle r_{0}} , say), the metric components restricted to this hypersurface (which we denote by g ~ ~ --> 22 {\displaystyle {\tilde {g}}_{22}} and g ~ ~ --> 33 {\displaystyle {\tilde {g}}_{33}} ) should be unchanged under rotations through θ θ --> {\displaystyle \theta } and ϕ ϕ --> {\displaystyle \phi } (again, by spherical symmetry). Comparing the forms of the metric on this hypersurface gives:
which immediately yields:
But this is required to hold on each hypersurface; hence,
An alternative intuitive way to see that g 22 {\displaystyle g_{22}} and g 33 {\displaystyle g_{33}} must be the same as for a flat spacetime is that stretching or compressing an elastic material in a spherically symmetric manner (radially) will not change the angular distance between two points.
Thus, the metric can be put in the form:
with A {\displaystyle A} and B {\displaystyle B} as yet undetermined functions of r {\displaystyle r} . Note that if A {\displaystyle A} or B {\displaystyle B} is equal to zero at some point, the metric would be singular at that point.
Using the metric above, we find the Christoffel symbols, where the indices are ( 1 , 2 , 3 , 4 ) = ( r , θ θ --> , ϕ ϕ --> , t ) {\displaystyle (1,2,3,4)=(r,\theta ,\phi ,t)} . The sign ′ {\displaystyle '} denotes a total derivative of a function.
To determine A {\displaystyle A} and B {\displaystyle B} , the vacuum field equations are employed:
Hence:
where a comma is used to set off the index that is being used for the derivative. The Ricci curvature is diagonal in the given coordinates:
where the prime means the r derivative of the functions.
Only three of the field equations are nontrivial (the fourth equation is just sin 2 --> θ θ --> {\displaystyle \sin ^{2}\theta } times the third equation) and upon simplification become, respectively:
Subtracting the first and second equations produces:
where K {\displaystyle K} is a non-zero real constant. Substituting A ( r ) B ( r ) = K {\displaystyle A(r)B(r)\,=K} into the third equation and tidying up gives:
which has general solution:
for some non-zero real constant S {\displaystyle S} . Hence, the metric for a static, spherically symmetric vacuum solution is now of the form:
Note that the spacetime represented by the above metric is asymptotically flat, i.e. as r → → --> ∞ ∞ --> {\displaystyle r\rightarrow \infty } , the metric approaches that of the Minkowski metric and the spacetime manifold resembles that of Minkowski space.
The geodesics of the metric (obtained where d s {\displaystyle ds} is extremised) must, in some limit (e.g., toward infinite speed of light), agree with the solutions of Newtonian motion (e.g., obtained by Lagrange equations). (The metric must also limit to Minkowski space when the mass it represents vanishes.)
(where K E {\displaystyle KE} is the kinetic energy and P E g {\displaystyle PE_{g}} is the Potential Energy due to gravity) The constants K {\displaystyle K} and S {\displaystyle S} are fully determined by some variant of this approach; from the weak-field approximation one arrives at the result:
where G {\displaystyle G} is the gravitational constant, m {\displaystyle m} is the mass of the gravitational source and c {\displaystyle c} is the speed of light. It is found that:
So, the Schwarzschild metric may finally be written in the form:
Note that:
is the definition of the Schwarzschild radius for an object of mass m {\displaystyle m} , so the Schwarzschild metric may be rewritten in the alternative form:
which shows that the metric becomes singular approaching the event horizon (that is, r → → --> r s {\displaystyle r\rightarrow r_{s}} ). The metric singularity is not a physical one (although there is a real physical singularity at r = 0 {\displaystyle r=0} ), as can be shown by using a suitable coordinate transformation (e.g. the Kruskal–Szekeres coordinate system).
The Schwarzschild metric can also be derived using the known physics for a circular orbit and a temporarily stationary point mass.[1] Start with the metric with coefficients that are unknown coefficients of r {\displaystyle r} :
Now apply the Euler–Lagrange equation to the arc length integral J = ∫ ∫ --> τ τ --> 1 τ τ --> 2 − − --> ( d s / d τ τ --> ) 2 d τ τ --> . {\displaystyle {J=\int _{\tau _{1}}^{\tau _{2}}{\sqrt {-\left({\text{d}}s/{\text{d}}\tau \right)^{2}}}\,{\text{d}}\tau .}} Since d s / d τ τ --> {\displaystyle ds/d\tau } is constant, the integrand can be replaced with ( d s / d τ τ --> ) 2 , {\displaystyle ({\text{d}}s/{\text{d}}\tau )^{2},} because the E–L equation is exactly the same if the integrand is multiplied by any constant. Applying the E–L equation to J {\displaystyle J} with the modified integrand yields:
where dot denotes differentiation with respect to τ τ --> . {\displaystyle \tau .}
In a circular orbit r ˙ ˙ --> = r ¨ ¨ --> = 0 , {\displaystyle {\dot {r}}={\ddot {r}}=0,} so the first E–L equation above is equivalent to
Kepler's third law of motion is
In a circular orbit, the period T {\displaystyle T} equals 2 π π --> / ( d ϕ ϕ --> / d t ) , {\displaystyle 2\pi /(d\phi /dt),} implying
since the point mass m {\displaystyle m} is negligible compared to the mass of the central body M . {\displaystyle M.} So B ′ ( r ) = − − --> 2 G M / r 2 {\displaystyle B'(r)=-2GM/r^{2}} and integrating this yields B ( r ) = 2 G M / r + C , {\displaystyle B(r)=2GM/r+C,} where C {\displaystyle C} is an unknown constant of integration. C {\displaystyle C} can be determined by setting M = 0 , {\displaystyle M=0,} in which case the spacetime is flat and B ( r ) = − − --> c 2 . {\displaystyle B(r)=-c^{2}.} So C = − − --> c 2 {\displaystyle C=-c^{2}} and
When the point mass is temporarily stationary, r ˙ ˙ --> = 0 {\displaystyle {\dot {r}}=0} and ϕ ϕ --> ˙ ˙ --> = 0. {\displaystyle {\dot {\phi }}=0.} The original metric equation becomes t ˙ ˙ --> 2 = − − --> c 2 / B ( r ) {\displaystyle {\dot {t}}^{2}=-c^{2}/B(r)} and the first E–L equation above becomes A ( r ) = B ′ ( r ) t ˙ ˙ --> 2 / ( 2 r ¨ ¨ --> ) . {\displaystyle A(r)=B'(r){\dot {t}}^{2}/(2{\ddot {r}}).} When the point mass is temporarily stationary, r ¨ ¨ --> {\displaystyle {\ddot {r}}} is the acceleration of gravity, − − --> M G / r 2 . {\displaystyle -MG/r^{2}.} So
The original formulation of the metric uses anisotropic coordinates in which the velocity of light is not the same in the radial and transverse directions. Arthur Eddington gave alternative forms in isotropic coordinates.[2] For isotropic spherical coordinates r 1 {\displaystyle r_{1}} , θ θ --> {\displaystyle \theta } , ϕ ϕ --> {\displaystyle \phi } , coordinates θ θ --> {\displaystyle \theta } and ϕ ϕ --> {\displaystyle \phi } are unchanged, and then (provided r ≥ ≥ --> 2 G m c 2 {\displaystyle r\geq {\frac {2Gm}{c^{2}}}} )[3]
Then for isotropic rectangular coordinates x {\displaystyle x} , y {\displaystyle y} , z {\displaystyle z} ,
The metric then becomes, in isotropic rectangular coordinates:
In deriving the Schwarzschild metric, it was assumed that the metric was vacuum, spherically symmetric and static. The static assumption is unneeded, as Birkhoff's theorem states that any spherically symmetric vacuum solution of Einstein's field equations is stationary; the Schwarzschild solution thus follows. Birkhoff's theorem has the consequence that any pulsating star that remains spherically symmetric does not generate gravitational waves, as the region exterior to the star remains static.
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