Inside the Black Hole

Peter D. Alison

December 11, 2009

Einstein’s theory of general relativity revolutionized the field of physics in theearly twentieth century. It was explained many of the inconsistencies in Newton’stheory of gravity that had plagued scientists for years, such as the precession of Mer-cury’s orbit, gravitational redshift, and stellar aberration. Its elegance and aestheticswere something at which to marvel. However, general relativity also predicted theexistence of black holes, mass densely packed into a small enough volume where theescape the velocity exceeds the speed of light (this escape velocity can also be derivedclassically, vesc =

√2GM/r). In general relativity terms, a black hole occurs when

a star runs out of thermonuclear fuel and proceeds to an end state of ongoing grav-itational collapse, despite pressures from the Pauli exclusion principle. Black holeshave never been fully understood in that they cannot be directly examined becauseof their gravitational pull on light. Properties of black holes are examined indirectlythrough their interactions with visible matter. At first black holes only existed intheory, through the predictions of general relativity; yet, Einstein did not believethat black holes could form because of the infinities that resulted at the singularity.But still the question remains, what happens inside a black hole? Though there isstill much unknown, what we know now can take us into the black hole and explainwhat happens.

1 Defining the Boundaries

It is important to describe the boundaries of a black hole in order to accuratelydefine where exactly the inside of the black begins. Our discussion will limit us tojust Schwarzschild black holes, where the Schwarzschild geometry is the metric at

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Figure 1: Schwarzschild metric’s r term Note: the term becomes negative whenr < 2M

work. The metric in spherical space-time coordinates is

ds2 = −(1− 2GM

c2r)dt2 + (1− 2GM

c2r)−1 dr2 + r2(dθ2 + sin2 θdφ2) (1)

where G is the gravitational constant, c is the speed of light, and M is the mass of theblack hole. In this coordinate system the black hole is taken to be stationary at theorigin and non-rotating. In this metric there are two singularities that occur; first wehave the physical singularity at r = 0 and a coordinate singularity at r = 2GM/c2.At first it was thought that something incredible occurred at this radius, but thissingularity only occurred because of the choice of coordinates. If one switches toEddington-Finkelstein coordinates with the transformation and to geometrized units(G = 1, c = 1)

t = v − r − 2M log| r2M− 1| (2)

we obtain the new metric

ds2 = −(1− 2M

r)dv2 + 2dvdr + r2(dθ2 + sin2 θ dφ2), (3)

with a smooth transition all the way to r = 0. However the radius, r = 2M , is aquantity of importance, the Schwarzschild radius. This is the point of no return for ablack hole. Though nothing particularly special happens at this point, this is wherethe escape velocity matches the speed of light. This is analogous to swimming in thewater near a waterfall. As one nears the waterfall, the water moves faster and faster.At some point, the water would be moving as fast as maximum swimming speed; this

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Figure 2: Two-dimensional slice of Schwarzschild geometry. The event horizon iswhere the surface ends.

is the point of no return for a waterfall. In terms of general relativity, r = 2M is thepoint where all light cones, the set of possible paths for a particle at a certain point,point inward towards the singularity, the only way to go is in. This radius, also calledthe event horizon, that shall define the boundary of the black hole, the differencebetween inside and out. If we switch back to Eddington-Finkelstein coordinates andsolve for the null geodesics (paths of light rays) we find the inside the event horizoningoing and outgoing light rays both move towards r = 0, the singularity. A betterway to picture this curvature of space-time is to take a two-dimensional slice of theSchwarzschild geometry (constant t and θ), a spacelike surface. If a light ray isexactly at the event horizon moving outward, it will appear to be stationary to anobserver also on the event horizon. As a result of this surface no information canemerge from the black hole, hence black holes are completely black.

2 The Descent

The journey to the event horizon is characterized by the radial distance from the blackhole. We shall start with the first point of interest, the innermost stable circular orbitat r = 6M = 3rs, where a particle will remain in orbit despite perturbations in themotion because we are in a well of the effective potential,

Veff = −Mr

+l2

2r2− Ml2

r3, (4)

where r = 6M is the local minimum of this potential, otherwise known as a stableequilibrium point. As we travel closer we reach the innermost unstable circular

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Figure 3: Graph of the effective potential with the unstable and stable orbit distances

orbit, r = 3M . To maintain an orbit at this radius the speed required is the speedof light; only light can orbit this close to the black hole. Inside the IUCO, all freefall orbits will fall into the black hole. In order to prevent going into the black holethis close would require an immense amount of acceleration. At this point it is goodto discuss the tidal forces of a black hole. The tidal force, or acceleration gradient,is the difference in the gravitational acceleration between two points in space in anon-uniform gravitational field. Tidal forces are actually inversely dependent on themass of the black hole at the event horizon. For the supermassive black hole atthe center of our Milky Way galaxy with a mass of 3.7 million solar masses andSchwarzschild radius of 11 million km, the tidal force would be less than one g. For amore conventional black holes of 30 solar masses the tidal force would be one milliong at the Schwarzschild radius; needless to say you would be ripped to shreds at theleast. Of course this vast difference in tidal forces could be explained by the simplefact the event horizon for the smaller black hole is a mere 90000 km and the inversesquare law dominates at these differences.

r30

rSM

=88620

10929800000= 8.108× 10−6, (

r30

rSM

)2 = 6.574× 10−11 (5)

Since we are at the event horizon we know that it would take a finite amount of toreach the singularity. We can calculate this quantity for our two black holes usingthe equation:

dr

dτ= −

√2M

r→ ∆τ =

∫dτ =

∫dr

dτ

dr= −r

3

√2r

M. (6)

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Plugging in our values for the mass we obtain the times it takes to reach the singu-larity

τ30 = 10−5s and τSM = 57s (7)

Though a finite amount of time would elapse to travel to the singularity, to an ob-server far away it would take an infinite amount of time to observe a particle travelingto the event horizon. This is caused by the strong redshifting light undergoes whennear the event horizon. As the particle approaches the event horizon the redshiftingbecome infinite and communication slows down to nothing. Light that is redshiftedto the far away observer would appear to have slowed down. Light emitted at acertain frequency would arrived at the observer’s position less frequently and withless energy. Now that we are at the event horizon there is no turning back.

3 Beyond the horizon

Now that the event horizon is behind us, the geodesics described by the null lightcones are both moving toward the singularity. If light cannot escape, no informationcan emerge from the black hole. Inside the event horizon the r-term changes sign toa negative while the t-term changes to a positive. Now time is spacelike while r istimelike and have in effect switched places. r must now decrease to the singularity,

(1− 2M

r)−1dr2 > 0 and − (1− 2M

r)dt2 < 0 (8)

and now all world lines cannot stay on paths of constant or increasing r. We can takea look at the null world lines using Eddington-Finkelstein coordinates and assumingpurely radial motion for simplicity,

0 = −(1− 2M

r)dv2 + 2dvd (9)

and we have the equations

dv = 0 and − (1− 2M

r)dv + 2dr = 0 (10)

which admit the solutions

v = const. and v − 2(r + 2M log| r2M− 1|) = const. (11)

which are ingoing when r > 2M . In fact, r decreases fast enough to reach nearlight speeds on the way to the singularity. If one were to enter the event horizon

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Figure 4: Light cones only can only point towards decreasing r inside the horizon.

feet first, he would not be able to see his feet, because light rays cannot move frompoints inside the horizon to points out of it. When the observer is completely insidethe event horizon, images are distorted in such they appear wrapped around him.The Schwarzschild surface, the sphere made by the horizon, appears to be split intotwo distorted surfaces. Due to relativistic beaming, as the observer’s speed increasesas he goes in, the images are concentrated into a thin line. When the singularity isclose, an observer’s presence is enough to cause violent oscillations in the tidal forces,even to the point of creating particle-antiparticle pairs out of the vacuum. When thesingularity is finally reached, we are at a point of infinite curvature and zero volumewhere all the mass of the black hole is concentrated. Einstein’s laws, unfortunately,break down at this point. Physics is a science where there is no tolerance for physicalinfinities, and yet we seem to have a point in space-time that have infinite densityand infinite curvature.

There was one possible thought of how rectify this singularity when this wasdiscovered to be predicted by general relativity, to directly incorporate quantummechanics into the theory. When this was first attempted many decades ago, aworse outcome resulted. Instead of a single infinity, there was a infinite sum ofinfinities. New theories are under development, yet none so far have proven effectiveor experimentally verifiable. There are quantities (length, time, energy, and density)

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arranged from the universal constants centered on Planck’s constant that show wheregeneral relativity apparently breaks down and quantum effects are relevant. Theyare

lPl =

√G~c3

= 1.62× 10−33 cm, (12)

tPl =

√G~c5

= 5.39× 10−44 s, (13)

EPl =

√~c5G

= 1.22× 1019 GeV, (14)

ρPl =c5

~G2= 5.16× 1093 g/cm3. (15)

It seems the only place conditions for the Planck scales to occur today is in a blackhole. Though general relativity beautifully explains gravitation, it is still a classicaltheory in which spacetime is assumed to be continuous. General relativity makes useof differential geometry (continuous spacetime) which would now have to be replacedwith discrete difference geometry (discrete spacetime).

4 References

J. Haldenwang, Spacetime Geometry Inside a Black Hole, 1-15 (2008)A.J.S Hamilton Falling to the singularity of the Black Hole, http://casa.colorado.edu/ ajsh/singularity.html., 1-3 (2006)J. Van Der Pool and Stephen Cooter, Who’s Afraid of the Big Black Hole?, BBCHorizon (2009)J.B. Hartle Gravity: An Introduction to Einstein’s General Relativity, Addison Wes-ley, 1-280 (2003)

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