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15Physics of Complex Systems, 2022, vol. 3, no. 3 _www.physcomsys.ru
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UDC 524.8 EDN PYVSES
https://www.doi.org/10.33910/2687-153X-2022-3-3-117-121
The forces outside the static limit in the rotating frame
A. A. Grib™ V. D. Vertogradov1, I. E. Fedorov1
1 Herzen State Pedagogical University of Russia, 48 Moika Emb., Saint Petersburg 191186, Russia
Authors
Andrey A. Grib, ORCID: 0000-0002-6389-991X, e-mail: andrei grib@mail.ru
Vitalii D. Vertogradov, ORCID: 0000-0002-5096-7696, e-mail: vdvertogradov@gmail.com
Ivan E. Fedorov, e-mail: fedivan97@gmail.com
For citation: Grib, A. A., Vertogradov, V. D., Fedorov, I. E. (2022) The forces outside the static limit in the rotating frame. Physics of Complex Systems, 3 (3), 117-121. https://www.doi.org/10.33910/2687-153X-2022-3-3-117-121 EDN PYVSES
Received 15 March 2022; reviewed 25 March 2022; accepted 6 April 2022.
Funding: This study was supported by grant No. 22-22-00112 RSF. It was also part of the SAO RAS state assignment Fundamental Scientific Research.
Copyright: © A. A. Grib, V. D. Vertogradov, I. E. Fedorov (2022). Published by Herzen State Pedagogical University of Russia. Open access under CC BY-NC License 4.0.
Abstract. This paper focuses on the rotating frame of the Minkowski spacetime to describe the inertial forces outside the static limit. We consider the inertial forces inside the static limit to find the classical
analogue. Then, we find out the expressions for these forces outside the static limit where we cannot consider
the limit - -> 0. We show that, generaly, if the angular velocity of the object D is equal to the angular momentum c
a, then the acceleration is equal to zero. When a * D, we show that a - D is a decreasing function of r. Keywords: rotating frame, static limit, inertial force, geodesic, Minkowski spacetime
Introduction
The focus on rotating black holes led the authors of (Grib, Pavlov 2019) to conclude that ergosphere, geodesics with negative energies, and the Penrose effect (Grib, Pavlov 2011; Grib et al. 2014; Vertogradov 2015) are typical not only for black holes but also for the Minkowski spacetime in rotating coordinates. It was shown that the role of the static limit in the theory of rotating black holes is taken by the distance from the origin of rotating coordinates to the value of the radius where the linear velocity of rotation becomes equal to the speed of light. The region beyond this value plays the role of ergosphere. In this region all bodies must be moving. This corresponds to use of the rotating rods and presents the main difference from observations inside the limit where all bodies can be at rest. In this paper we consider the geodesic lines both inside and outside the static limit in rotating coordinates. This is done to initiate the discussion of the inertial forces appearing in these coordinates in the Minkowski spacetime. These forces inside the limit are well-known centrifugal and Coriolis forces. The analogues of these forces for the region outside the static limit are presented in this paper.
In the paper (Grib, Pavlov 2019) authors showed that the analogue of the Penrose effect outside the static limit can lead to observational effects inside this static limit. The study reported in this paper is the investigation of a new phenomenon—the inertial forces outside the static limit and their difference from those inside the static limit. It is necessary to take into account the relativistic corrections when
Theoretical physics. Cosmology
one considers the movement of the spacecrafts in this region. This can be achieved by using the rotating frame and observing the signals from the Earth.
In this paper Latin indices take values 0, 1, 2, 3 and Greek indices take values 1, 2, 3.
Inside the static limit
In this paper we consider the flat Minkowski spacetime in cylindrical coordinates {ct, r, z}:
ds2 = c2dt2 - dr2 - r2d(p'2 - dz2 . (1)
If we consider the rotating coordinates in the Minkowski spacetime
(p^ip-COt, (2)
then we obtain the following line element:
ds2 = c2dt2 - r2 (d(p - codtf - dr2 - dz2 . (3)
Here a is angular velocity.
c
One can note that the metric component g00 changes its sign at r = rsl = —. However, the line element
CO
(3) might still be timelike due to the positivity of the off-diagonal term g02dtd$= ar2dtd$. If we compare
the metric (3) with the Kerr metric, then the hypersurface r = rd plays a role of the static limit in the Kerr black hole.
d d
The Killing vector outside the static limit has a different form than inside the static limit i. e. +
Due to this fact, we have another expression for the proper time where the off-diagonal term is not equal to zero. Thus, the proper time will be imaginary for a body at rest. In this region and the geodesic equations will lead to new phenomenon—the inertial forces.
If we consider the observer on the Earth (motionless with everything rotating around it), then the object which is situated at r > rsl cannot be static for the observer. It must also rotate otherwise the line element (3) will be spacelike. Also, in this case the proper time will be imaginary: if we consider the connection between the proper t and coordinate t times when r = const., $ = const., z = const., then we obtain:
dr=y[g^dt, (4)
and we know that g00 < 0 at r > rsl.
So, if we want to calculate inertial forces for the objects inside the static limit using (4), we have to consider only the region 0 < r < rsl. In this region g00 < 0. From the Newtonian mechanics we know that acceleration is proportional to force. So, to calculate forces one should use the right hand-side of the geodesic equation:
du' i k l , ^
— = -r>V, (5)
dz
where ¡-u is the Christoffel symbol.
One should note that acceleration is the time derivative of the 4-velocity ul. From differential geometry we know that the derivative does not obey the tensor transformation law. So, when we define force we should use the covariant derivative instead of the usual one. We want to calculate the three-force, which requires the three covariant derivative instead of (5). Thus, we get (Landau, Lifshitz 1980):
ra _ „ a _ Mva „/!,/ (6)
f"=u?pup=^-+Y;supu6
where Yps is the purely three-Christoffel symbol. We can note (and show below) that the terms in the right hand-side (5) which are proportional to u0 correspond to the centrifugal force, while those which
are proportional to u0 ua correspond to the Coriolis force and those which are proportional to ua uP are part of the three covariant derivative and are not forces at all.
We need the Christoffel symbol of the metric (3) to calculate the geodesic equation (5). Using the well-know formula:
Fkl — ~SJ (Skj,l + 8jl,k &kl,j) »
(7)
one can obtain non-vanishing component of the Christoffel symbols:
r1
1 00 ~ 2 '
1 02 ~ > c
r1 =-r 1 22 ' '
1 12
r2 -- — 1 0\~
cr
(8)
To calculate the inertial forces in the region 0 < r < rsl, we can use another well-known formula (Landau, Lifshitz 1980):
Fa =
mc
dxa
In
2 2 \ CO r
+J1-
2 2 Í CO r
dxa dxp
c
(9)
where
Soa
2 2 ' COr
V =
cdxa
1-
2 2 CO r
(cdt-E,ßdxß )
(10)
Note that the first term in the square brackets corresponds to the centrifugal force and the second term depends on the velocity linearly and corresponds to the Coriolis force. Substituting (10) into (9) we obtain:
F =
mc
11-
2o r
2 2 2 vc -co r J
+
2 cor
dcp
yV cdt-Ç2dcp
F* =
2 cor
1-
dr
CO
2r2 Y cdt-Ç2dcp
1 -
ff»
V
y
(11)
(12)
Let us find out where the static limit rd is situated in the Solar system. For this purpose let us consider two cases:
1. The static observer is on the surface of the Earth and everything is rotating around it. In this case we should solve the following algebraic equation:
fflV2
1--— = goo=°. (13)
c
The period is equal to 24 h and the static limit is situated between the Uranus and the Neptune.
2. Everything is rotating around the Sun and the observer is on the Earth. In this case the period is equal to 1 year and rsl = 1.5 x 1015m . This static limit is situated inside the Oort's clouds.
General case
Here and in what follows we consider the system in units c = 1. In the general case the proper time t has the following form:
dz2 = (l-a2r2}dt2 +2cor2dtd(p-dr2 -dz2 -r2d(p2 . (14)
_ 1
One can see that the square of proper time (14) might be positive in the region fy — — < r < +co
CO
because of the off-diagonal term +2ar2dtd$. Hence we state that we can consider the region
d(p
dt
we assume that r = const. and r = const. for the observer and the connection between the proper and coordinate times is the following:
rsl < r < + ™ only in the case when the observer has non-vanishing angular velocity Q = However,
dz = dt^\-r2{Q-(o)2. (15)
To obtain force expression we should write down geodesic equations which, in the general case, are given by:
d2r
dz
-co2r
'dt^
dz
dt d(p ( d(p
\UL
-2 cor--— + r
dz dz
dz
(16)
d2(p _ 2 dr dcp co dt dr — — |- £ ^
dz r dz dz r dz dz
If we followed the definition of force from the previous section, then we would see that forces have finite values on the static limit but they are extremely large. No one sees these large values so we should redefine forces.
In the previous section we defined forces as three covariant derivative of the momentum and did not
count some terms in the right-hand side of the equation (16) because they were part of the three covariant
d
derivative. However, in the general case, outside the static limit, the Killing vector — is spacelike and as
dt
the result is unphysical. In the region r > rsl the observer, like in the ergoregion of a rotating black hole, has to move along both t and $ coordinates. As the result of such movement all the Christoffel symbols proportiaonal to (u0)2, u0u$ and (u$)2 are inertial forces. So substituting (15) into (16) one obtains:
d2r = r , si dz2 1 -r2((o-Q)2 ^ °)) ' d2(p _ 2vr
dz2 r-r\(o-Q)'
dr v =— . r dt
{(o-Q), (17)
From (17) one can see that the object has non-vanishing acceleration only if a - D * 0. The object does not have any acceleration if D = a because in this case it is equal to zero. If œ * D, then the acceleration
is a growing function of r and r = —-— diverges at the surface. This situation is the same like in the
co-Q
previous section with the angular velocity a being replaced by a - D. If the acceleration of an object is
a constant W, then the Q = ^SL is the decreasing function of r, i. e.:
dt
Q = . (18)
> r + Wr2
As is seen, with the proper time (15) we should redefine inertial forces. It leads us to the fact that if D = a, then the acceleration is absent. However, if the body has constant acceleration, it means that D * a but a - D tends to zero with growing r. Also, one should notice that D must always be positive otherwise all metric components become negative outside the static limit and the line element (16) would be spacelike in this region. However, in the region 0 < r < rd we can consider negative values of a.
Conclusion
In this paper we have considered forces inside and outside the static limit in the Minkowski spacetime for rotating coordinates. This frame has only two types of inertial forces, i. e., centrifugal and the Coriolis
forces. In the case of the static bodies one can consider forces only up to the radius r = rsl because in this
case the proper time and velocity is imaginary outside the static limit. Thus, to consider the inertial forces
in the region rsl < r < ™ one should consider the general proper time with non-vanishing angular velocity
M — ——. We have found out that with the proper time (15), the acceleration tends to zero if D ^ a. dt j
It should be also noted that if a * D, then we have another surface r —-. This surface shows that
co-Q
the line element outside it (3) is spacelike. That is, the case a * D is an analogue of the static observer with the angular momentum of spacetime a - D. It is worth mentioning that all forces which are considered in this article are fictitious ones.
Conflict of Interest
The authors declare that there is no conflict of interest, either existing or potential.
Author Contributions The authors have made an equal contribution to the preparation of the text.
References
Grib, A. A., Pavlov, Y. V. (2011) Particles with negative energies in black holes. International Journal of Modern
Physics D, 20 (05), 675-684. https://doi.org/10.1142/S0218271811019013 (In English) Grib, A. A., Pavlov, Y. V., Vertogradov, V. D. (2014) Geodesics with negative energy in the ergosphere of rotating black holes. Modern Physics Letters A, 29 (20), article 1450110. https://doi.org/10.1142/S0217732314501107 (In English)
Grib, A. A., Pavlov, Yu. V. (2019) Static limit and Penrose effect in rotating reference frames. Theoretical and
Mathematical Physics, 200 (2), 1117-1125. https://doi.org/10.1134/S004057791908004X (In English) Landau, L. D., Lifshitz, E. M. (1980) Course of theoretical physics series. Vol. 2. The classical theory of fields. 4th ed.
Oxford: Butterworth-Heinemann Publ., 444 p. (In English) Vertogradov, V. D. (2015) Geodesics for particles with negative energy in Kerr's metric. Gravitation and Cosmology, 21 (2), 171-174. https://doi.org/10.1134/S0202289315020115 (In English)
