2. Negative Energy in Nonrotating Black Hole
A nonrotating black hole of mass
M in Schwarzschild coordinates is described by the metric
where
is the gravitational radius of the black hole,
G is the gravitational constant, and
c is the light velocity. The geodesic complete space–time of the nonrotating black hole can be described in Kruskal–Szekeres coordinates,
, which, in region
, are connected with the Schwarzschild coordinate in
in the following way:
For
and
, the transformation from the Schwarzschild coordinate into the Kruskal–Szekeres coordinates has the form
Schwarzschild coordinates are singular at
. Regarding their connection with Kruskal–Szekeres coordinates for other
, see [
7], Section 31.5.
Any possible movement of physical bodies and particles must satisfy the condition
leading to
For
, the coordinate
is space and
is the time coordinate.
Geodesic equations in Schwarzschild coordinates in the plane
are [
8]
where
E is the energy of a moving particle (measured by the static observer in
r,
,
coordinates),
J is the conserved projection of the particle’s angular momentum on the axis orthogonal to the plane of motion,
m is the particle mass, and
is an affine parameter on the geodesic. For massive particles,
, where
is the proper time.
In an external region of the black hole (
) for any physical object, the time coordinate
t is always increasing and so the energy
E of the particle is positive (see (
6). Inside the horizon of the black (
), where
t is space-like (
), one has movement as in increasing as in decreasing
t. As is seen from the first formula in (
6) for a particle moving inside the horizon in the direction of a decrease in the coordinate
t, the energy
E of the particle will be positive, while, for increasing coordinate
t, the energy
E is negative. For constant
t inside the black hole,
due to formula (
6).
Surely,
t inside of the black hole is space-like and
E is proportional to the
t-component of the momentum. Inside a black hole, one can use other reference frames and corresponding energies [
9]. However, for the observer outside the black hole, the conserved
E along all trajectories of the free fall is equal (see formula (88.9) in [
10]) to
where
is the velocity measured by the observer at rest in the Schwarzschild coordinates. Thus, we can call
E inside the black hole, following [
7], the “energy at infinity”. For a discussion of other ways to determine the energy within the horizon and the movement of particles there, see, for example, the articles [
9,
11,
12,
13].
In
Figure 1, the trajectories for radial movement with positive, zero and negative energies in Kruskal–Szekeres coordinates are represented by red, green and blue lines.
As one can see from (
2) and (
3), the coordinate lines of constant
t in Kruskal–Szekeres coordinates are straight lines through the origin of the coordinates. In region II, coordinate
t decreases when moving from
to
(positive
E) and increases when moving from
to
(negative value of
E). Direct lines (
) correspond to constant
and therefore
.
Let us consider the problem of the back influence of falling particles on the metric of the black hole space–time. For macroscopic bodies with 4-velocity
, with the energy density
and pressure
p in space–time with metric
, the energy–momentum density tensor is [
10]
. The trace of the energy–momentum tensor
is invariant and it will be negative for
—in particular, for dust-like matter (
) with negative energy
. The back influence of falling particles with negative energy will be determined by the energy–momentum tensor in the right-hand side of the Einstein equations. The notion of the existence of particles with negative energies as it is known was used by S. Hawking to predict the Hawking effect for black holes [
14].
For a discussion of other ways to determine the energy within the horizon and the movement of particles there, see, for example, works
3. Negative and Zero Energies in Flat Space–Time
The geodesic line equations can be obtained for space–time with metric
from the Lagrangian
where
is the affine parameter on the geodesic [
8]. The energy of the particle
E is equal to the zero covariant component of the momentum
multiplied on the light velocity
Defining the affine parameter for the massive particle as
, where
is the proper time of the moving particle, one obtains
and the energy of the particle is
Using notation
for the translation in the time coordinate generator, one can write (
12) for the energy of the particle as
If the metric components do not depend on the time coordinate
, then
is the time-like Killing vector and the energy
E is conserved on the geodesic. For time-like vector
and a massive particle, one has [
15]
and the energy (
15) is positive. For space-like vector
, as it takes place in the ergosphere of a rotating black hole, the arbitrary positive and negative values are possible (see [
15], p. 325).
Note that in spite of the invariance of the scalar product (
15), the value (
14) of the energy depends on the choice of the reference frame. This occurs due to the fact that by changing the reference frame in which the physical measurements are made, the observer is changing vector
. The analysis of the situation in a rotating coordinate system in flat space–time is provided in [
3].
In Minkowski space–time in the Galilean coordinate system or any other coordinate system with
,
, (
), the energy (
12) is
and it is always positive in movement “forward” in time because in the future light cone, one has
.
Consider the coordinate system in which the metric of flat space–time has the form of the metric of the expanding homogeneous isotropic Universe—the Milne universe [
16]:
where
, and coordinate
is changing from 0 to
. In new coordinates
the interval (
18) takes the form of a Minkowski interval
This space–time with coordinate
,
corresponds to the future cone in coordinates
.
The radial distance between points
and
in metric (
18) is
. Taking
D as the radial coordinate [
17], one obtains the interval as
From the condition
, one obtains that if
D is larger than
, no physical object can be at rest in coordinates
. The value
corresponds to
and it plays the role of the static limit for the rotating black hole in Boyer–Lindquist [
18] coordinates.
The energy of the particle with mass
m in coordinates
is
From (
18), one obtains, for any physical object, the inequality
Thus, a particle can have negative energy only for
, i.e., out of the static limit, if
Note that the components of metric (
21) depend on time and the energy (
22) in general is not conserved on the geodesics. If the energy is zero, then the particle is moving noninertial according to the law
The trajectory of such movement for case
,
is represented by the curve in
Figure 2 in coordinates
T,
r (see (
19)).
In the event that the inertial movement trajectory in these coordinates is a straight line, the possible region of movement of particles with negative and zero energies in the reference frame is defined in the coordinate by conditions .
Velocities of movement in coordinates
and
satisfy condition [
19]
Thus, for
particles at rest in inertial frame
will have negative energies in the frame
. This region can be seen in
Figure 2 as the region above the blue line in the red district. Zero energy of the particle at rest in
coordinates is possible only if the blue line is defined by the root of equation
, i.e.,
.
Thus, one can see that for a specific choice of coordinates, one can obtain negative and zero energies for particles at rest in an inertial frame.
Note that for small distances (
), the metric (
21) becomes the metric of a comoving spherical coordinate system of Minkowski space–time
and the energy (
22) will be equal to the usual energy in the inertial system of flat space–time
because, for
, one has
.
The decay of the body on two bodies, one with negative energy and the other with the positive energy being larger than the energy of the initial decaying one, corresponds to the Penrose process. This process occurs outside the static limit on distance
. However, later, these two products of the decay move inside the static limit, and, during flight in the direction of the origin, where the metric is that of Minkowski space, they change their energies in such a manner that the result will be the same as in the inertial frame. In fact, due to (
22)
Here,
is the energy in the reference frame
, such that
,
, and
does not depend on time. Thus,
is conserved. At the point of decay, both energies
E and
are conserved. When body 2 with the positive energy arrives at the coordinate origin
, its energy
E (
30) will be equal to
and no growth in the energy will be observed.
Body 1, with the negative energy
E due to (
22) after decay, will have a negative value of velocity
larger (in absolute value) than that of body 2. This means that it will arrive at the origin before the arrival of body 2. Its energy in the origin of the coordinate frame will be also positive and the full energy of 1 and 2 will be equal to that of the decaying body. Thus, at the origin, one will not observe any effect that makes this situation similar to the situation for Kerr’s black hole.
In fact, for rotating black hole [
1], as in the case of rotating coordinate system [
4], the energy is conserved. In this case, when body 2 travels out of the ergosphere, far from it, body 1, with negative energy, moves further inside the horizon of the black hole or goes to infinity in case of rotating coordinates in Minkowski space. Thus, the energy of the body with positive energy, due to the conservation of the energy, will be always larger than that of the initial decaying body.
Note that the existence of states with negative energies for Milne’s metric leads to an effect similar to the Hawking effect [
14] for the Schwarzschild metric. Particle creation in quantum theory will occur and the detector of particles checks them (see Section 5.3 in [
20]). This will be the creation of virtual particles (see Section 9.8 in [
21]) so no change in the metric due to them can be observed.
4. Negative Energy in Gödel Universe
The metric of the Gödel cosmological model of the rotating Universe proposed in 1949 (see [
22] or [
23]) can be written as
where
is constant. Such a metric is the exact solution of Einstein’s equation with background matter as an ideal liquid without pressure and negative cosmological constant
where
. Here,
has the sense of the angular velocity of rotation of the vector of fluid of the background matter
.
Taking, instead of
, new coordinates
,
one writes the interval (
31) in the form [
22,
24]
where
,
,
and identifying
and
.
Now, consider the general case of space–time
with the interval
where
a is constant,
,
,
,
, and identifying
and
. Let us say
and
for
. For Gödel Universe
,
and
The metrical tensor is
where indexes
correspond to
. Note that for any
, the metrical tensor is not degenerate
. The degeneration for
in the Gödel Universe is coordinate degeneracy. The eigenvalues of the
tensor are
For
, one has
Note that although
is positive for
, the signature of
for all
is the standard
.
Possible movement of particles is defined by
, so, for the interval (
38), one has
It is important that for any coordinate system with interval (
38), the physical body for any values of
can be at rest, i.e., there is no static limit. However, in (
38), there is nondiagonal term
as in Kerr’s metric. However, differently from the case of rotating coordinate system [
3], there is the possibility of a change in the sign before
.
From (
44), one obtains
The solution of this inequality is the union of two intervals
Considering cases of different signs of
, one obtains the following sets of solutions of (
45):
These sets define light “cones” of the future and past for the metric (
38). The form of these cones in cases
,
and
is shown in
Figure 3 for the Gödel Universe with
Let us find limitations on possible values of the energy of particles moving in such a Universe. The coordinate
is dimensionless, so the “physical energy” of the particle is expressed through the time component of the momentum as
For the frame with coordinates (
38), the covariant
components are conserved, because the components of the metric depend only on
r. Thus, the conserved energy on the geodesic for the interval (
38) is
From (
47) and (
48), for the case of movement “forward” in time, i.e., in the future light cone, one obtains
so
It means that for a particle moving in the future cone in the Gödel Universe, the energy is not negative.
For movement “back in time”, the energy is limited from above by
and so it can be less than or equal to zero. However, such movement physically is inconsistent. The “time machine” effect in the Gödel Universe corresponds to continuous movement in the future cone. Thus, for
, where
closed loops (they are not geodesic lines)
,
, called Gödel cycles, from
to
, are closed time-like curves [
24]. Particles moving along such a cycle are moving “forward” in time but, due to the identification of values
different on
, it occurs in the past after the whole cycle. Its energy is positive due to (
53). Such a “time machine” is different from that moving in the past by the sign of the particle energy.