In this paper, general relativity theory (GRT) is used to describe the geometry of spacetime inside a black hole. The reader is assumed to be familiar with calculus and special relativity (SRT). We start with a review of the parts of SRT needed to understand GRT.
Special Relativity Theory
Special relativity concerns itself with inertial frames of reference. Two observers who are in uniform, straight-line motion relative to one another are said to be in inertial frames of reference. Such observers are called inertial observers. If two observers are accelerating relative to one another, they are not inertial observers. Einstein developed SRT first, by ignoring acceleration. Later on, he was able to generalize his theory to include acceleration and develop GRT.
SRT is based on two principles. First, the principle of relativity: the laws of physics are the same in all inertial frames of reference. This means that there is no absolute rest frame. Second, Einstein's light postulate: the speed of light in vacuum is constant, and has the same value in all inertial frames of reference. All experimental measurements to date have in fact found the speed of light in empty space to be constant, regardless of the (uniform) motions of the measuring devices. The speed of light, about 300,000 km/sec, is represented by the letter c, which stands for celeritas (latin for "swift").
In order for the speed of light to be constant in different inertial frames which are in motion relative to one another, Einstein realized that space and time cannot be absolute. For example, if observer A is in motion relative to observer B, then, from B's point of view, A’s units of measuring space must be shorter than B’s (in A's direction of motion). Also, according to B, A’s units of measuring time must be longer (slower) than B’s. Relative to B, space for A is contracted in his direction of motion and time is lengthened or slowed down (dilated). (Two handy mnemonics are: "moving sticks are shortened" and "moving clocks run slow.") The sizes of these relative distortions of space and time are precisely what is needed so that when A measures the speed of a passing light ray, he will always obtain the same value that B does.
These distortions of space and time can be quantified by the Lorentz transformation. Consider two observers O and O´, where O´ moves with constant velocity v along the positive x axis of a Cartesian coordinate system, and the positions of O and O´ coincide with the origin at time t = 0. In this configuration (known as the standard configuration), the Lorentz transformation is given by
t´ = g(t – vx), x´ = g(x – vt), y´ = y, z´ = z, (1)
where g (gamma) is given by
and c = 1. Here I have geometrized the units, by multiplying time in seconds by the speed of light in meters per second, so that the unit of time is meters. One meter of time is equal to 1/300,000,000 of a second, since the speed of light is 300,000,000 m/sec. In geometrized units, the speed of light is 1, and dimensionless. In these units, light travels one meter of space in one meter of time. In relativity, space and time are treated much alike. The use of geometrized units emphasizes this similarity. Also, geometrized units are more convenient in GRT.
Although Einstein found that space and time can vary in different inertial frames, his math teacher Minkowski found a way to measure space and time together in such a way that the measurement does not vary. This discovery was of fundamental importance to the subsequent development of GRT. Minkowski replaced the Euclidean distance with the spacetime interval. The frame-invariant spacetime interval Ds between two events is defined by
In order to prove that the spacetime interval is invariant under the Lorentz transformation, we start with
and replace t´, x´, etc. using the Lorentz transformation, as given above (1). After some algebraic manipulations, we find that
which shows that the interval is frame-invariant in SRT.
What does the frame-invariance of the spacetime interval really mean? Minkowski said: "From henceforth, space by itself, and time by itself, have vanished into the merest shadows and only a kind of blend of the two exists in its own right." Consider a three-dimensional object, say a stick. It casts a two-dimensional shadow against a wall. If we turn the stick, the length of the shadow changes, even though the stick itself remains the same length. In an analogous fashion, we can think about a four-dimensional spacetime "object." All inertial observers agree that this object has the same "length" (interval) in spacetime. However, different observers see different lengths for the three-dimensional "shadow" of the object in space.
In Euclidean geometry, we use the Pythagorean theorem to measure distance (Dd2 = Dx2 + Dy2 + Dz2). In Newtonian mechanics the Pythagorean theorem is frame-invariant, but not in SRT. Minkowski’s discovery allows us to use the spacetime interval in place of the theorem of Pythagoras in order to measure “distance” in the non-Euclidean geometry of SRT.
Using the spacetime interval, we can classify the separation between two events (the “distance”) in one of three ways:
1. If Ds2 < 0, the events are said to have timelike separation. In this case, there can exist one inertial observer who experiences both events. We say that the events lie on the world line of this observer. The observer is traveling at a speed less than c, the speed of light.
For any two events with timelike separation, the separation in space is less than the separation in time. To see this, consider the following:
2. If Ds2
= 0, the events are said to have null or lightlike separation.
In this case the events lie on the world line of a light ray. The
separation in space and the separation in time are equal (in geometrized
units).
3. If Ds2
> 0, the events are said to have spacelike separation. In this
case there exists an inertial frame in which the events are simultaneous
(separated in space but not in time).
In SRT, the paths of material particles are restricted to timelike world lines, and the paths of photons are restricted to null or lightlike world lines. Spacelike world lines are excluded. (Spacelike world lines correspond to paths that are faster than the speed of light, or that go backward in time.) See figure 1. In this diagram, the origin O represents the present for some observer. The observer's future lies somewhere between the two lightlike world lines, with t > 0. The observer's past lies somewhere between these two world lines, with t < 0. If we were to add a dimension by adding a y axis perpendicular to the x and t axes, the lightlike world lines would form two cones meeting at the origin. These are called "light cones."
When two events have timelike separation (Ds2 < 0), we define the proper time interval between the events to be Dt (delta tau), where
The proper time, also known as the wristwatch time, is the time measured by an observer who experiences both events. Since the proper time for lightlike world lines is zero, we can say that photons do not experience the passage of time.
Let’s consider the spacetime interval of a photon traveling in the x direction. For this photon, Dy and Dz are zero. Setting Ds2 = 0, we obtain
0 = -Dt2
+ Dx2
or 
so that
which is the result to be expected. In all inertial frames of reference, the speed of light is constant and equal to one in geometrized units.
As another example, let’s consider the case of the twin sisters, Mary and Jane. Jane decides to travel to the nearest star, Alpha Centauri. She has at her disposal a very fast space ship, one that can travel almost as fast as light. Her sister Mary is somewhat of a homebody, however, and prefers to stay on Earth. Let’s compare the proper time of both sisters, using Mary’s reference frame for the calculations. We ignore acceleration in this example, for which we would need to use GRT. Also, instead of meters, we use years for t and light-years for x.
Mary says good-bye to Jane, and waits for 12 years. After 12 years, Jane returns from her journey to Alpha Centauri, 4.5 light-years away. Mary calculates her own proper time interval as follows:
Since she stayed at home, she used Dx = 0 in her calculation.
Mary is surprised to find that Jane has not aged as much as her, however. She is not so surprised after calculating Jane’s proper time interval. Using Dx = 9 light-years, she obtains:
See figure 2. Again, this calculation is only approximately correct,
since no adjustment was made for the time Jane spent accelerating and decelerating.
However, this example illustrates an interesting fact: the longest
path through Minkowski spacetime is actually the one that involves no movement
through space, only time. This is because we subtract the
spatial components of the path in order to compute the interval.
General Relativity Theory
SRT does not take into account acceleration. In 1907, Einstein had what he called “the happiest thought of my life” that motion in an accelerating frame cannot be distinguished from motion in a uniform gravitational field. To put it another way, Einstein assumed that gravitational acceleration is equivalent to inertial acceleration (the acceleration due to inertia). Using this “principle of equivalence,” Einstein was able to extend the principle of relativity from inertial reference frames to all reference frames. He did this by showing that the same laws of physics that describe acceleration could also be used to describe motion due to gravity.
General relativity is Einstein’s theory of gravity. It is founded on two core principles: (1) The principle of relativity: the laws of physics are the same in all frames of reference, and (2) The principle of equivalence: accelerated motion and motion in a uniform gravitational field are equivalent.
GRT is a geometric theory. Gravity is not treated as a force. Instead, according to GRT, gravitational acceleration is caused by the warping of space and time. The warping or curvature of spacetime is caused by the presence of matter, energy and pressure.
Free-falling particles in a gravitational field follow geodesics, which are curves of extremal length. They are either the longest or the shortest possible paths between two points in spacetime. According to GRT, free-falling particles take the longest possible paths through spacetime, so they follow geodesics. We perceive free-falling particles in a gravitational field to be accelerating because they are following geodesics through curved spacetime.
Light rays also follow geodesics, but these geodesics cannot be defined
in terms of length, since all lightlike paths have the same length (spacetime
interval), namely zero. Instead, we define these geodesics as the
straightest
possible paths through curved spacetime. This definition is made
precise through the techniques of differential geometry, the generalized
non-Euclidean geometry developed in the 19th century by the great mathematicians
Gauss and Riemann, and utilized by Einstein in GRT.
In differential geometry, a curved surface is divided up into an infinite
number of infinitesimally small pieces. Each piece is measured by
the “metric” or “line element” ds, a differential which gives the distance
between two points that are infinitesimally close together on the surface.
See figure 3. The techniques of differential geometry can be used
to find an equation for the metric that is valid anywhere on the curved
surface. Unlike conventional geometry, with differential geometry
distances can be determined on a curved surface without reference to anything
outside of the surface. For example, if we consider a sphere as a
two-dimensional curved surface, the circumference of the sphere can be
determined without knowing its radius, which lies outside the surface of
the sphere.
The techniques of differential geometry will work in general, for any
continuous curved surface whatsoever, in any number of dimensions.
Just what Einstein needed for his theory of gravity.
In general, in the four dimensions of spacetime, the equation for the metric assumes the following form:
ds2 = g11 dt2 + g22 dx2 + g33 dy2 + g44 dz2
+ 2 g12 dt dx + 2 g13 dt dy + 2 g14 dt dz + 2 g23 dx dy + 2 g24 dx dz + 2 g34 dy dz. (3)
The ten coefficients gab are, in general, functions of t, x, y and z. In three-dimensional Euclidean space, we set
g22 = g33 = g44 = 1,
and the other coefficients equal to zero. We then obtain the conventional Pythagorean formula
ds2 = dx2 + dy2 + dz2,
which measures the distance between two infinitesimally close points in Euclidean space. To find the distance Ds between two points a finite distance apart, we sum up the differentials by integrating the metric along the geodesic connecting the two points. Of course, in Euclidean space we know the geodesic is just the straight line connecting the two points, and we don’t have to use integration to find the distance. However, the technique of taking the line integral of the metric along the geodesic between two points to find the distance between them will work in any geometry whatsoever, Euclidean or non-Euclidean, flat or curved.
The general formula for the metric given above (3) can be called the generalized Pythagorean theorem for four-dimensional spacetime. In the flat spacetime of SRT, we set
g11 = -1, g22 = g33 = g44 = 1,
and the other coefficients equal to zero. We then obtain the Minkowski metric
ds2 = -dt2 + dx2 + dy2 + dz2. (4)
By integrating this metric along the world line connecting two events in flat spacetime, we can find the interval Ds between the events. Again, there are easier ways to find the interval in flat spacetime, but in the curved spacetime of GRT, taking the line integral of the metric is the only way that will work, in general.
When acceleration is present, either gravitational or centrifugal (these are equivalent, according to Einstein), the formula for the metric becomes more complex than the ones used for Euclidean or Minkowski space. We will see an example of such a metric shortly.
According to GRT, the metric ds is frame-invariant. We can use it to find the interval between two events in any frame of reference, no matter what motion or acceleration may be present. In order to change from one reference frame to another, however, the coefficients gab must be transformed (using a transformation formula). The value of ds is not changed by this transformation, even though the coefficients of the metric do change.
In addition to differential geometry, another mathematical tool used in GRT is tensor analysis. Tensors are real-valued, multi-dimensional functions of vectors. They are used in GRT because they are frame-invariant. In other words, a tensor will give the same real number regardless of the reference frame in which the vector components are calculated. This property makes tensors a useful short-hand for representing the frame-invariant metrics of GRT.
In GRT, a tensor representing the curvature of spacetime is set equal to a tensor representing the stress-energy content of spacetime (the matter, energy and pressure present). This is the way that GRT mathematically models the concept that matter, energy and pressure cause the curvature of spacetime, and therefore gravity. This tensor equation can be written as
where G and T are the tensors representing the curvature and the stress-energy content of spacetime, respectively. This equation is the most important result of GRT, and can be used to construct mathematical models of the universe, as well as of stars and black holes. The expansion of this tensor equation gives the “Einstein field equations,” a system of ten non-linear partial differential equations. These cannot be solved in general. Instead, initial conditions and simplifying assumptions, such as spherical symmetry, are used to obtain a simpler, solvable set of equations. When these equations are solved, the result is the metric tensor, which contains the ten coefficients gab of the metric (3). The resulting metric describes the geometry of the spacetime being modeled (for example a star, a black hole, or the universe). (With simplifying assumptions, some of the ten coefficients will be zero.)
In the simplest case, that of flat spacetime (no acceleration or gravitational
field), the metric tensor expands to give the Minkowski metric (4).
When the methods of GRT are applied to a curved spacetime geometry (such
as we find inside or near a star), the result is a more complex metric.
In general, the proper time interval between two infinitesimally close
events on a timelike world line can be obtained from the metric by finding
dt = |ds|.
The proper time interval Dt between two events P and Q that lie on a timelike world line W is defined to be the line integral of dt along W from P to Q. See figure 4.
The proper time Dt from P to Q is equal to the time measured on a clock that moves along W from P to Q. (This is known as the Clock Hypothesis.)
Our experience of physical reality tells us that time has a direction.
We assume that it is not possible for an observer to travel along the world
line W from Q to P (backward in time).
The Schwarzschild Metric
Schwarzschild found a solution to the Einstein field equations that describes the curvature of spacetime caused by a non-rotating black hole. The Schwarzschild metric was derived using the reference frame in which the black hole is stationary. This means that the coefficients of the Schwarzschild metric are valid only for observers who are “at rest” or motionless relative to the black hole. "At rest" means that the spatial coordinates of these observers do not change over time.
The Schwarzschild solution applies only to non-rotating black holes. A solution for rotating black holes was found by Kerr. As the Kerr solution is more complex, this paper will only consider the Schwarzschild solution. (Most of the results given here are valid for the Kerr solution also.)
Consider a black hole of mass M (in geometrized units). For r > 2M, the Schwarzschild metric is given in spherical coordinates (t, r, q, j) by
There are many interesting and surprising features of the geometry described by this metric. First I mention that r = 2M is called the event horizon, for reasons to be explained shortly. The event horizon can be thought of as the boundary that divides the inside from the outside of the black hole.
As r approaches 2M, the coefficient of dt2 approaches zero, and the coefficient of dr2 approaches infinity. For a long time, it was not known whether the infinite “singularity” at r = 2M was a real, physical singularity which could not be passed through by an infalling observer or just a coordinate singularity. (A familiar example of a coordinate singularity is the one at r = 0 in polar coordinates, where q can take on any value and is therefore indeterminate. A transformation to Cartesian (rectangular) coordinates eliminates this singularity.) In general, a coordinate singularity indicates a mathematical problem, not a real, physical problem, and can be eliminated by a coordinate transformation.
Although the singularity at r = 2M was long suspected to be a coordinate singularity, this was not proved until the late 1950s, when a coordinate transformation was found that eliminated the singularity. Additional coordinate transformations have been discovered since. These will not be considered here, as they are mathematically complex.
By setting ds2 =
0 in the above Schwarzschild metric, we can study the behavior of a light
ray near the event horizon, as seen by an observer at rest at “infinity.”
Why do we consider an observer who is at rest at “infinity?” As r
approaches infinity, the coefficient of dt2
in the Schwarzschild metric approaches –1. The spatial coordinates
r, q and
j for a
resting observer are constant over time, so the differentials dr, dq
and dj are zero. Therefore, dt
= |ds| 
dt, and so the proper
time Dt of the observer resting at "infinity"
is approximately equal to the coordinate time Dt
of the Schwarzschild metric. Hence, any statements we make concerning
t or dt also apply to the far away observer's wristwatch time t
or dt.
Returning to the light ray near the event horizon, we can simplify the analysis by assuming the ray travels along a radial null geodesic, either directly toward the black hole or directly away from it. Then the q and j coordinates of the light ray do not change. This implies that dq and dj are zero, so we can eliminate the right-hand term in the Schwarzschild metric. Then
so that
or 
Now as r approaches 2M, dt/dr approaches infinity. This is a time dilation effect. Any message sent via light signal from near the event horizon (r = 2M) to an observer far from the black hole will be stretched out. The closer the emitter of the light signal is to the event horizon, the more stretched out the message will appear to the far away observer. The frequency of the light signal decreases, or redshifts, because lower frequency light carries less information per unit of time (the far away observer’s wristwatch time). The closer to the event horizon the light signal is sent from, the greater is the redshift observed from far away. When the emitter is very close to the event horizon, the observed redshift is so great that the light signal disappears altogether. For this reason, the event horizon is sometimes called the infinite redshift horizon.
Taking the reciprocal, we see that dr/dt approaches 0 as r approaches 2M. Since we solved the Schwarzschild metric for a radial lightlike geodesic (by setting ds2 = 0), dr/dt corresponds to the speed of light. Hence the speed of light approaches 0 as r approaches 2M, relative to an observer far from the black hole. At r = 2M, an outward directed light ray is frozen in time and space, and never reaches an observer at r > 2M.
Since we said that the speed of light is constant in SRT, one might ask how the speed of light can vary in GRT. The answer is that gravitational fields change the geometry of spacetime, and the speed of light is fundamentally tied to the nature of the spacetime geometry the light is passing through.
According to GRT, gravity is what we observe as the bending, stretching and compressing of spacetime caused by matter, energy and pressure. Light rays follow geodesics through this bent, stretched or compressed spacetime. The warping of spacetime warps the paths of the light rays.
Relative to an observer at rest far from a black hole, space is compressed (contracted) near the event horizon and time is stretched out (dilated). Each meter of space is shorter compared to the space far from a black hole, and each meter of time is longer. Relative to the far away observer, a light ray, traveling one meter of space per one meter of time, travels a short distance in a long time. To the far away observer, the light ray has slowed down.
Although the speed of light can vary in GRT, it is still always the case that material objects cannot attain or exceed this speed.
Inside the Black Hole
Now let’s consider the Schwarzschild solution for 0 < r < 2M (inside a black hole). A small but very important change must be made to the metric for this case. When r > 2M, the coefficient (1 – 2M/r) is positive. However, for 0 < r < 2M, this coefficient is negative. In order to work with positive coefficients for this case, we use –(1 – 2M/r) = (2M/r – 1). The metric then becomes
Notice how the minus sign has moved from the t coordinate to the r coordinate. This means that inside the event horizon, r is the timelike coordinate, not t. In GRT, the paths of material particles are restricted to timelike world lines. Recall the discussion of timelike separation earlier in this paper (2). It is the coordinate with the minus sign that determines the meaning of “timelike.” According to GRT, inside a black hole, time is defined by the r coordinate, not the t coordinate. It follows that the inevitability of moving forward in time becomes, inside the black hole, the inevitability of moving toward r = 0. This swapping of space and time occurs at r = 2M. Thus, r = 2M marks a boundary, the point where space and time change roles. For the observer inside this boundary, the inevitability of moving forward in time means that he must always move inward toward the center of the black hole at r = 0. All timelike and lightlike world lines inside r = 2M must move toward decreasing r and end at r = 0 (the end of time!) Because it is not possible for any particle or photon inside r = 2M to take a path where r remains constant or increases, the boundary r = 2M is called the event horizon of the black hole. No observer inside the event horizon can communicate with any observer outside the event horizon.
Earlier, we showed that the speed of light approaches zero near the event horizon, relative to an observer far from the black hole. This means that the outside observer can never see an infalling observer reach or cross the event horizon, because any light radiating from the infalling observer slows down and redshifts, with the redshift approaching infinity as the infalling observer nears the event horizon. Does this mean that the infalling observer does not actually reach or cross the event horizon? No. The infalling observer does in fact cross the event horizon. Remember that the singularity at r = 2M (the event horizon) was shown to be a coordinate singularity, not a real, physical singularity. Using transformed coordinates, it can be shown that the infalling observer passes from r > 2M to r = 0 in a finite amount of time (his proper time, or the interval along his world line).
Furthermore, it can be shown that the maximum amount of time from r = 2M to r = 0 for an observer who has fallen through the event horizon, even if he has at his disposal a rocket of unlimited power, is given by
where M is the geometrized mass used in the Schwarzschild metric. M is related to the Newtonian mass m by
M = Gm/c2,
where G is the gravitational constant in S.I. (standard international) units.
Let’s look at a real-life example. Astronomers believe that there is a supermassive black hole at the center of our galaxy, with an estimated mass of about 3.7 million solar masses. The tidal force near the event horizon of such a large black hole is weak. (The tidal force, or tidal acceleration gradient, is the difference in the gravitational acceleration between two points in a non-uniform gravitational field. The smaller the black hole, the larger this gradient is near the event horizon, because the curvature of spacetime is greater. An astronaut approaching a stellar black hole of a few solar masses would be torn apart by the tidal force before reaching the event horizon.) Thus, it is possible that an astronaut, if well protected from radiation, could survive to cross the event horizon of this supermassive black hole and continue inward. Let’s calculate the maximum time this astronaut could avoid reaching the center of the black hole. (For simplicity, we assume the black hole is not rotating, so that the above formula can be used.)
Our intrepid astronaut has less than a minute to explore the black hole! The Schwarzschild radius of this black hole is
r = 2M = 2 × 3,700,000 × 1,477 m = 11 million km.
The Center of the Black Hole
What happens at r = 0? In the Schwarzschild metric, the expressions 2M/r approach infinity as r approaches 0. This is a real, physical singularity, not a coordinate singularity. All the mass of a Schwarzschild black hole is concentrated at r = 0, a point of infinite density, where space and time come to an end. The presence of real singularities in solutions of the Einstein field equations suggests that GRT is an incomplete theory of gravity.
One of the assumptions underlying GRT is that spacetime is continuous. This can be seen in the usage of differential geometry as the mathematical basis of GRT. Differentials measure infinitesimally small distances, which only makes sense if spacetime is continuous. It may be that spacetime is actually discrete, with a smallest fundamental unit. Physicists are exploring this idea as they work on new theories of quantum gravity. If this idea is correct, then the differential equations of GRT are only approximations of reality. These approximations are valid in the large, even down to the atomic level, but at some level (the Planck length, the smallest possible length), they break down.
If spacetime is discrete, not in reality a continuum, then the differential equation is not the appropriate mathematical tool to use to study the center of the black hole. Fortunately, mathematicians have developed a more appropriate tool, the difference equation, which is the discrete analog of the differential equation. Difference equations are of the form Dy/Dt, analogous to dy/dt. Computers are used to solve these equations. Starting from an initial point in space and time (the initial state of the system), time is incremented (advanced) by a tiny bit Dt, then the computer finds the new state the system has evolved to. The time variable t is then incremented by another Dt and again the computer finds the new state the system has evolved to. If Dt is one trillionth of a second, the computer must repeat this process 10 trillion times in order to find the state the system has evolved to after 10 seconds. The branch of mathematics that studies these evolving systems using difference equations is called discrete dynamical systems. In order to develop a successful theory of quantum gravity, it is my belief that a new branch of geometry based on difference equations must be used. This new area of math, called "difference geometry," is currently under development. Once we have a viable theory of quantum gravity, we should be able to understand what really happens at the center of the black hole.
Real black holes almost certainly rotate. Such black holes can be modeled by the Kerr metric, a more complex metric whose detailed analysis is outside the scope of this paper. The interested reader should check out Taylor and Wheeler's fine book, Exploring Black Holes: Introduction to General Relativity. The Kerr black hole does not have a point singularity at r = 0. Instead, the singularity has a ring structure, and can be avoided by most trajectories. In theory, a particle that avoids the ring singularity can pass through r = 0 to a region where r < 0. It is unlikely that such a region actually exists. It is more likely that GRT does not correctly describe the nature of real black holes at r = 0. A future theory of quantum gravity should provide us with a more realistic description of the center of the rotating black hole.
Physical Interpretation of the Event Horizon
We found earlier that the Schwarzschild metric has a coordinate singularity at the event horizon, where the coordinate time becomes infinite. Recall that the coordinate time is approximately equal to the far away observer's proper time. However, a calculation using transformed coordinates shows that the infalling observer falls right through the event horizon in a finite amount of time (the infalling observer's proper time). How can we interpret solutions in which the proper time of one observer approaches infinity yet the proper time of another observer is finite?
The best physical interpretation is that, although we can never actually see someone fall through the event horizon (due to the infinite redshift), he really does. As the free-falling observer passes across the event horizon, any inward directed photons emitted by him continue inward toward the center of the black hole. Any outward directed photons emitted by him at the instant he passes across the event horizon are forever frozen there. So, the outside observer cannot detect any of these photons, whether directed inward or outward.
Consider two observers far from the black hole. Suppose they synchronize their watches, then one of them remains far from the black hole while the other descends slowly (at first) toward the event horizon. Then the time on the watch of the descending observer as he reaches and falls through the event horizon will be approximately equal to the time on the watch of the far away observer as she sees her companion disappear very near the event horizon.
Life Inside the Black Hole
Some have speculated that our universe might exist inside a gigantic
black hole. Let’s explore this idea further, in order to gain more
insight into what the interior of a black hole is really like (and for
the fun of it). If our universe is really inside a giant black hole,
one might ask where the event horizon is. Is there any path we can
take that will bring us closer to the event horizon? According to
GRT, if our universe is inside a black hole, every point in our universe
is moving closer to the center of this black hole, and away from the event
horizon. There is no (spatial) direction that will bring us closer
to the event horizon. As it is difficult to visualize a four-dimensional
curved surface (one needs to think in five dimensions to be able to do
this), subtracting a dimension or two makes it easier. Imagine a
giant sphere, and a point on the interior surface of this sphere. This
point detaches from the inner surface and moves toward the center, at the
same time expanding into a disk. This expanding disk represents our
universe expanding in space as it moves through time. In this model
let's suppose that our universe formed on the event horizon of the giant
black hole, represented by the surface of the sphere. We suppose
that the Big Bang occurred at the event horizon of the black hole.
See figure 5. The expanding disk is a two-dimensional representation
of the three spatial dimensions of our universe. (We could label
these spatial dimensions
q,
j
and t. Note that the disk represents an unimaginable curved 3-D hypersurface.)
Every point in our universe (the disk) is moving away from the inner surface
of the sphere (the event horizon) toward the center of the sphere (the
singularity of the giant black hole). The dimension through which
this disk is moving is a timelike dimension (let’s label it r). For
every point on the disk (our universe at a point in time), the event horizon
lies in the past and the singularity of the black hole lies, unseen, in
the future. All timelike and lightlike world lines in our universe
lead from the event horizon to the singularity of the black hole.
To travel to the event horizon would be to travel backward in time.
Therefore, there is no path we can take that will bring us closer to the
event horizon.
In this imaginary model, the only point of the spacetime of our universe that is connected to the event horizon of the giant black hole is the point in space and time at which the Big Bang occurred. With a powerful enough telescope, one can, in theory, look in any direction and see the Big Bang (or at least 380,000 years after the Big Bang, when the universe became transparent). One can look around in any direction and see the Big Bang, yet one cannot travel toward it, because it lies in the past. This is the way of things inside any black hole. Even a super-powerful rocket cannot prevail against the gentle timelike acceleration toward the singularity at the center of a black hole.
Could we detect the presence of this gigantic black hole? If we
had sensitive enough instruments, it should be possible to detect the tidal
acceleration gradient, at least over astronomical distances. This
might be mistaken for a slight variation in the strength of gravity over
very large distances. Also, the expansion of the universe should
eventually slow down and reverse, as the universe moves closer to the central
singularity of the black hole. In this model, our universe would
eventually shrink down to a single point (the Big Crunch).
Addendum: Toward a Viable Theory of Quantum Gravity
For several decades now, physicists have been struggling to find a way to unite general relativity (GRT) and quantum mechanics (QM). This started with Einstein himself, who spent the last decades of his life working on his "unified field theory." He was not successful. In the last few decades, a great deal of effort has been expended by many theoretical physicists toward this goal. Still, we do not have a viable theory of quantum gravity. It has proven to be a very difficult problem, indeed. Why? I believe the most fundamental problem is that these two great pillars of modern physics (QM and GRT) rest on different philosophical bases. Why is this important? Because different philosophical bases mean different mathematical bases. The philosophy and therefore the math are in conflict.
GRT is fundamentally a deterministic theory of reality. It is founded on the principle of cause and effect, that is, that every effect in the universe is determined by a cause. This principle is reflected in the deterministic mathematics used to develop GRT, namely differential geometry and tensor analysis. On the other hand, QM (or at least the conventional or "Copenhagen" interpretation of QM) is fundamentally a probabilistic theory of reality. The conventional interpretation of QM asserts that the behavior of matter and energy at the subatomic or quantum level is fundamentally random. It is not possible to predict the path of a single electron, for example, no matter how much information about its initial state is available to us. Only the collective behavior of large numbers of electrons can be predicted, using statistical methods. Probability and statistics form the mathematical basis of QM.
The deterministic nature of GRT is fundamentally incompatible with the stochastic (random) nature of QM. Einstein himself recognized this conflict between the two theories. His famous statement that "God does not play dice" reflected his recognition of this conflict (and also his bias toward determinism).
Any attempt to unify QM and GRT must address this fundamental conflict in the philosophy and therefore the mathematics employed. How can this conflict be resolved? There is a way. Mathematicians have recently developed a field of math known as chaos theory or chaotic dynamical systems. The fundamental premise of chaos theory is that many systems, such as the Earth's weather, are very sensitive to the initial states they start out in (at a given point in time). The slightest change in the initial state of a chaotic system can lead to a dramatic change in its subsequent evolution. A simple example of this can be seen in attempting to balance a pencil on its pointy end. The slightest change in the initial state (how you have the pencil balanced) can lead to a dramatic change in the subsequent evolution of the trajectory (in which direction the pencil falls). If we do not know the initial state of a chaotic system precisely enough, we cannot predict its subsequent evolution. Instead, the behavior appears to be random and we must employ probability or statistics if we are to make useful predictions. For example, the toss of a die appears to be random. In truth, however, it is chaotic. According to chaos theory, if we knew the initial state of the system (the die and its surrounding environment) precisely enough, we could predict the roll of the die. Chaos theory is deterministic in nature. It explains the appearance of randomness in certain systems by showing that the evolution of these systems is very sensitive to their initial states.
It may be that the behavior of subatomic particles is fundamentally chaotic rather than random in nature. If so, their chaotic behavior leads to the appearance of randomness. We cannot predict the trajectory of a subatomic particle, not because its trajectory is fundamentally random, but because we do not know its initial state precisely enough. The particle's path may be determined, yet appears random and unpredictable to us because of our ignorance of its initial state. There is in fact an interpretation of QM that is deterministic and based on chaos theory rather than probability theory. This is the Bohm interpretation. The Bohm interpretation of QM has been widely ignored by physicists because its predictions do not vary much from the conventional interpretation. The important point to be made here, however, is that a chaotic interpretation of QM can employ the same mathematical basis as GRT.
If spacetime is in fact quantized and discrete, not continuous, then
the mathematical basis of GRT can be changed from differential geometry
to difference geometry, as mentioned above (in the section entitled "The
Center of the Black Hole"). Difference equations (the discrete analogs
of differential equations) form the basis of difference geometry.
Also, difference equations form the basis of chaos theory. Thus,
GRT and QM can be unified by a single philosophy (a deterministic view
of nature) and a single mathematical basis (difference equations).
The result could be a viable theory of quantum gravity.
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