Int J Theor PhysDOI 10.1007/s10773-014-2118-5

New Characterizations for Minimizing Energyof Biharmonic Particles in Heisenberg Spacetime

Talat Korpinar

Received: 18 January 2014 / Accepted: 24 March 2014© Springer Science+Business Media New York 2014

Abstract In this work, we study energy of timelike biharmonic particle in a new spacetimeHeisenberg spacetime H4

1. We give a geometrical description of energy of a Frenet vectorfields of timelike biharmonic particle in H4

1. Moreover, we obtain different cases for thisparticles.

Keywords Energy · Heisenberg spacetime · Biharmonic particle · Bienergy

1 Introduction

Electromagnetic fluids in dynamical, strongly curved spacetimes play a central role in manysystems of current interest in relativistic astrophysics. The presence of magnetic fields maydestroy differential rotation in nascent neutron stars, form jets and influence disk dynamicsaround black holes, affect collapse of massive rotating stars, etc. Many of these systemsare promising sources of gravitational radiation for detection by laser interferometers, [9,19–23].

For physical reasons, a spacetime continuum is mathematically defined as a four-dimensional, smooth, connected Lorentzian manifold. This means the smooth Lorentzmetric has signature . The metric determines the geometry of spacetime, as well as determin-ing the geodesics of particles and light beams. About each point (event) on this manifold,coordinate charts are used to represent observers in reference frames, [8, 9, 27].

Computing energy minimizing curves which are restricted to a surface and which fulfillinterpolation and/or approximation conditions is a basic problem of Geometric Computing.Its applications go far beyond the most obvious task of designing a curve on a surface inordinary Euclidean space — there are many nonlinear geometric settings of higher dimen-sion which have manifestations in the familiar case of three dimensions. An efficient methodfor variational curve design on surfaces, where neither the dimension of the surface, nor the

T. Korpinar (�)Department of Mathematics, Mus Alparslan University, 49250 Mus, Turkeye-mail: talatkorpinar@gmail.com

Int J Theor Phys

dimension of ambient space are restricted to two or three, provides a basic tool usable e.g.for energy minimizing rigid body motions, or the handling of obstacles, [10].

(Left) Counterpart to a C2 cubic spline curve on a surface the fourth derivative vectors(blue) are in direction of the surface normals (yellow). (Right) Curves minimizing E2 (blue),Et (red), and E1 (green) interpolating the same points on a surface, [10].

On the other hand, a smooth map φ : N −→ M is said to be biharmonic if it is a criticalpoint of the bienergy functional:

E2(φ) =∫N

1

2|T (φ)|2dvh,

where T (φ) := tr∇φdφ is the tension field of φ.The Euler–Lagrange equation of the bienergy is given by T2(φ) = 0. Here the section

T2(φ) is defined by

T2(φ) = −�φT (φ)+ trR(T (φ), dφ)dφ, (1.1)

and called the bitension field of φ. Non-harmonic biharmonic maps are called proper bihar-monic maps, [6, 7, 11]. In [13–15, 24–26], they characterized timelike biharmonic curvesin the Lorentzian Heisenberg group.

In this work, we study timelike biharmonic particle in a new spacetime Heisenbergspacetime H4

1. We give a geometrical description of energy of a Frenet vector fields oftimelike biharmonic particle in H4

1. Moreover, we obtain different cases for this particles.Additionally, we illustrate our results.

2 The Heisenberg SpaceTime H41

In Heisenberg spacetime H41, the metric gc is given by

gc = −c2dt2 + dx2 + dy2 + (dz− xdy)2, (2.1)

where c is light velocity in the vacuum.

Int J Theor Phys

The Lie algebra of Heisenberg space-time has a basis

e1 = ∂

∂x, e2 = ∂

∂y+ x

∂

∂z, e3 = ∂

∂z, e4 = 1

c

∂

∂t, (2.2)

for which we have the Lie products

[e1, e2] = e3, [e2, e3] = [e3, e1] = [e4, e1] = [e4, e2] = [e4, e3] = 0

with

gc(e1, e1) = gc(e2, e2) = gc(e3, e3) = 1, gc(e4, e4) = −1, (2.3)

where c is light velocity in the vacuum, [12, 17].

3 Biharmonic Particles in the Heisenberg Space-Time H41

A “particle” in special relativity means a curve γ with a timelike unitary tangent vector,[18].

The timelike curve γ is called timelike Frenet curve if there exist three smooth functionsk1, k2, k3 on γ and smooth nonnull frame field {T,N,B1,B2} along the curve γ. Also, thefunctions k1, k2 and k3 are called the first, the second, and the third curvature function onγ , respectively. For the timelike Frenet curve γ , the following Frenet formula is⎡

⎢⎢⎣∇TT∇TN∇TB1∇TB2

⎤⎥⎥⎦ =

⎡⎢⎢⎣

0 k1 0 0k1 0 k2 00 −k2 0 k30 0 −k3 0

⎤⎥⎥⎦

⎡⎢⎢⎣

TNB1B2

⎤⎥⎥⎦ . (3.1)

Here, due to characters of Frenet vectors of the timelike curve, T,N,B1 and B2 aremutually orthogonal vector fields satisfying equations

gc(T,T) = −1, gc(N,N) = gc(B1,B1) = gc(B2,B2) = 1. (3.2)

Theorem 3.1 ([16]) Let γ be a timelike biharmonic particle with constant slope in H41.

Then, the parametric equations of biharmonic particle are

x = 1

f

[c2 − cosh2 E

] 12

sin [fs + f0] + f1,

y = −1

f

[c2 − cosh2 E

] 12

cos [fs + f0] + f2,

z = sinhEs+ 1

2f

[c2 − cosh2 E

][fs + f0] − 1

4f

[c2 − cosh2 E

]sin 2 [fs + f0]

− f1

f

[c2 − cosh2 E

] 12

cos [fs + f0] + f3,

t = 1

cs + f4, (3.3)

where c is light velocity in the vacuum and f1, f2, f3, f4 are constants of integration and

f = k1

[c2 − cosh2 E] 12

+ sinhE .

Int J Theor Phys

4 Energy of Biharmonic Particles in the Heisenberg SpaceTime H41

In relativity, all of the energy that moves along with an object (that is, all the energy which ispresent in the object’s rest frame) contributes to the total mass of the body, which measureshow much it resists acceleration. Each potential and kinetic energy makes a proportionalcontribution to the mass, [2–4, 8, 27].

Definition 4.1 The energy of a differentiable map f : (M, g) → (N, h) betweenRiemannian manifolds is given by

Energy(f ) = 1

2

∫M

n∑a=1

h(df (ea), df (ea))v, (4.1)

where v is the canonical volume form in M and {ea} is a local basis of the tangent space[1, 5].

The energy of a unit vector field X is defined to be the energy of the section X : M →T 1M; where T 1M is the unit tangent bundle equipped with the restriction of the Sasakimetric on TM . Now let π : T 1M → M be the bundle projection and let T

(T 1M

) = V⊕Hdenote the vertical/horizontal splitting induced by the Levi-Civita connection. Further, wewrite TM =F⊕G where F denotes the line bundle generated by X; and G is the orthogonalcomplement [5].

Proposition 4.2 The connection map K : T (T 1M) → T 1M satisfies the followingconditions:

i) π ◦ K = π ◦ dπ and π ◦ K = π ◦ π , where π : T (T 1M) → T 1M is the tangentbundle projection;

ii) for ω ∈ TxM and a section ξ : M → T 1M; we have

K(dξ(ω)) = ∇ωξ, (4.2)

where ∇ is the Levi-Civita covariant derivative [5].

Definition 4.3 For η1, η2 ∈ Tξ (T1M), we define

gS(η1, η2) = g(dπ(η1), dπ(η2))+ g(K(η1),K(η2)). (4.3)

This gives a Riemannian metric on TM: Recall that gS is called the Sasaki metric. Themetric gS makes the projection π : T 1M → M a Riemannian submersion.

Case 1 Energy of tangent vector by using Sasaki metric is given by

Energy(T) = 1

2

(−1 +

[c2 − cosh2 E

](−f+ sinhE)2

)s.

Proof Assume that γ is a timelike biharmonic particle in H41. From (4.1) and (4.2) we can

write

E(T) = 1

2

∫ s

0gS(dT(T), dT(T))ds

Int J Theor Phys

Fig. 1 f1 = f2 = −f3 = −1,E= π4

Using Eq. (4.3) we have

gS(dT(T), dT(T)) = g(dπ(T(T)), dπ(T(T)))+g(K(T(T)),K(T(T)))

Since T is a section, we obtain

d(π) ◦ d(T) = d(π ◦ T) =d(idC) = idTC.

Then

K(T(T)) =[c2 − cosh2 E

] 12

sin [fs + f0] (−f+ sinhE)e1

+[c2 − cosh2 E

] 12

cos [fs + f0] (f− sinhE)e2.

So, we have

gS(dT(T), dT(T)) = −1 +[c2 − cosh2 E

](−f+ sinhE)2

Thus it is easy to obtain

Energy(T) = 1

2

(−1 + [c2 − cosh2 E](−f+ sinh E)2

)s.

The proof is completed.

Int J Theor Phys

Fig. 2 f1 = f2 = f3 = 1,E= π3

Thus, the following remark may be given;

Remark 1 If T have constant energy, then

c2 = 1

(−f+ sinhE)2+ cosh2 E .

Case 2 Energy of N by using Sasaki metric is given by

Energy(N) = 1

2

∫ s

0

([1

k1

[c2 − cosh2 E

] 12(f− sinhE) cos[fs + f0]

(−f+ 1

2sinhE

)]2

+[

1

k1

[c2 − cosh2 E

] 12(f− sinhE) sin[fs + f0]

(−f+ 1

2sinhE

)]2

+(

1

2k1

[c2 − cosh2 E

](−f+ sinhE)

)2

− 1

)ds,

Int J Theor Phys

Fig. 3 f1 = f2 = −f3 = 1,E= π6

where c is light velocity in the vacuum and

f = k1

[c2 − cosh2 E] 12

+ sinhE .

Proof Taking into account (4.1) and (4.3) we have

gS(dN(T), dN(T)) = g(dπ(N(T)), dπ(N(T)))+g(K(N(T)),K(N(T))).

Since T is a section, we obtain

d(π2) ◦ d(N) = d(π2 ◦ N) =d(idC) = idT C.

Thus we have

K(N(T)) =[

1

k1

[c2 − cosh2 E

] 12(f− sinhE) cos [fs + f0]

(−f+ 1

2sinhE

)]e1

Int J Theor Phys

+[

1

k1

[c2 − cosh2 E

] 12(f− sinhE) sin[fs + f0]

(−f+ 1

2sinhE

)]e2

+(

1

2k1

[c2 − cosh2 E

](−f+ sinhE)

)e3.

Finally, we obtain

Energy(N) = 1

2

∫ s

0

([1

k1

[c2 − cosh2 E

] 12(f− sinhE) cos[fs + f0]

(−f+ 1

2sinhE

)]2

+[

1

k1

[c2 − cosh2 E

] 12(f− sinhE) sin[fs + f0]

(−f+ 1

2sinhE

)]2

+(

1

2k1

[c2 − cosh2 E

](−f+ sinhE)

)2

− 1

)ds.

This completes the proof.

Remark 2 If N have constant energy, then([1

k1

[c2 − cosh2 E

] 12(f− sinhE) cos[fs + f0]

(−f+ 1

2sinhE

)]2

+[

1

k1

[c2 − cosh2 E

] 12(f− sinhE) sin[fs + f0]

(−f+ 1

2sinh E

)]2

+(

1

2k1

[c2 − cosh2 E

](−f+ sinhE)

)2

− 1

)= 0,

where c is light velocity in the vacuum and

f = k1

[c2 − cosh2 E] 12

+ sinhE .

Case 3 Energy of B1 by using Sasaki metric is given by

Energy(B1) = 1

2

∫ s

0

[[(− f

k2k1

[c2 − cosh2 E

] 12(f− sinh E) sin[fs + f0]

(−f+ 1

2sinh E

)

+ fk1

k2

[c2 − cosh2 E

] 12

sin [fs + f0] + 1

2

([c2 − cosh2 E

] 12

sin[fs + f0][

1

2k2k1

[c2 − cosh2 E

](−f+ sinh E)− k1

k2sinhE

]+

[1

k2k1

[c2 − cosh2 E

] 12

(f− sinh E) sinhE sin[fs + f0](−f+ 1

2sinh E

)− k1

k2

[c2 − cosh2 E

] 12

sinh E

sin[fs + f0]])+ k2

k1

[c2 − cosh2 E

] 12

sin [fs + f0] (−f+ sinh E))

+k2

k1

[c2 − cosh2 E

] 12

sin [fs + f0] (−f+ sinh E)]2

+[+

(f

k2k1

[c2 − cosh2 E

] 12(f− sinh E) cos[fs + f0]

(−f+ 1

2sinh E

)

Int J Theor Phys

− fk1

k2

[c2 − cosh2 E

] 12

cos [fs + f0] − 1

2

([c2 − cosh2 E

] 12

cos [fs + f0]

[1

2k2k1

[c2 − cosh2 E

](−f+ sinh E)− k1

k2sinh E

]+

[1

k2k1

[c2 − cosh2 E

] 12

(f− sinh E) sinhE cos[fs + f0](−f+ 1

2sinh E

)− k1

k2

[c2 − cosh2 E

] 12

sinh E

cos[fs + f0]])+ k2

k1

[c2 − cosh2 E

] 12

cos[fs + f0](f− sinh E))

+k2

k1

[c2 − cosh2 E

] 12

cos[fs + f0](f− sinh E)]2

+[

1

2

([1

k2k1

[c2 − cosh2 E

](f− sinh E) sin 2[fs + f0]

(−f+ 1

2sinh E

)

− k1

k2

[c2 − cosh2 E

]sin 2 [fs + f0]

])]2

− 1

]ds,

where c is light velocity in the vacuum and

f = k1

[c2 − cosh2 E] 12

+ sinhE .

Remark 3 If B1 have constant energy, then

[[(− f

k2k1

[c2 − cosh2 E

] 12(f− sinhE) sin [fs + f0]

(−f+ 1

2sinhE

)

+ fk1

k2

[c2 − cosh2 E

] 12

sin [fs + f0] + 1

2

([c2 − cosh2 E

] 12

sin [fs + f0]

[1

2k2k1

[c2 − cosh2 E

](−f+ sinhE)− k1

k2sinhE

]+

[1

k2k1

[c2 − cosh2 E

] 12

(f− sinhE) sinhE sin [fs + f0]

(−f+ 1

2sinhE

)− k1

k2

[c2 − cosh2 E

] 12

sinhE

sin[fs + f0]])

+k2

k1

[c2 − cosh2 E

] 12

sin [fs + f0] (−f+ sinhE)

+k2

k1

[c2 − cosh2 E

] 12

sin [fs + f0] (−f+ sinhE)]2

+[+

(f

k2k1

[c2 − cosh2 E

] 12(f− sinhE) cos [fs + f0]

(−f+ 1

2sinhE

)

− fk1

k2

[c2 − cosh2 E

] 12

cos[fs + f0] − 1

2

([c2 − cosh2 E

] 12

cos [fs + f0]

[1

2k2k1

[c2 − cosh2 E

](−f+ sinhE)− k1

k2sinhE

]+

[1

k2k1

[c2 − cosh2 E

] 12

(f− sinhE) sinhE cos [fs + f0]

(−f+ 1

2sinhE

)− k1

k2

[c2 − cosh2 E

] 12

sinhE

Int J Theor Phys

cos[fs + f0]])+ k2

k1

[c2 − cosh2 E

] 12

cos [fs + f0] (f− sinhE)

+k2

k1

[c2 − cosh2 E

] 12

cos [fs + f0] (f− sinhE)]2

+[

1

2

([1

k2k1

[c2 − cosh2 E

](f− sinhE) sin 2 [fs + f0]

(−f+ 1

2sinhE

)

−k1

k2

[c2 − cosh2 E

]sin 2[fs + f0]

])]2

= 1.

Case 4 Energy of B2 by using Sasaki metric is given by

Energy(B2) = 1

2

∫ s

0

[[1

k1

[c2 − cosh2 E

] 12(f− sinh E) cos [fs + f0]

(−f+ 1

2sinhE

)

−k1

[c2 − cosh2 E

] 12

cos [fs + f0]

]2

+[

1

k1

[c2 − cosh2 E

] 12(f− sinhE) sin [fs + f0]

(−f+ 1

2sinhE

)

−k1

[c2 − cosh2 E

] 12

sin [fs + f0]

]2

+[

1

2k1

[c2 − cosh2 E

](−f+ sinhE)− k1

k2sinhE

]2

−[k1

k2c

]2

− 1

]ds,

where c is light velocity in the vacuum and

f = k1

[c2 − cosh2 E] 12

+ sinhE .

Remark 4 If B2 have constant energy, then

[[1

k1

[c2 − cosh2 E

] 12(f− sinhE) cos [fs + f0]

(−f+ 1

2sinhE

)− k1

[c2 − cosh2 E

] 12

cos[fs + f0]]2

+[

1

k1

[c2 − cosh2 E

] 12(f− sinhE) sin [fs + f0]

(−f+ 1

2sinh E

)

−k1

[c2 − cosh2 E

] 12

sin[fs + f0]]2

+[

1

2k1

[c2 − cosh2 E

](−f+ sinh E)− k1

k2sinh E

]2

−[k1

k2c

]2

= 1.

Int J Theor Phys

5 Some Pictures

In this section we draw some pictures corresponding to different cases Figs. 1, 2 and 3.

Acknowledgments The authors would like to express their sincere gratitude to the referees for the valuablesuggestions to improve the paper.

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