Redshift Drift in Uniformly Accelerated Reference Frame*

Chinese Physics C Vol. 44, No. 7 (2020) 075103

Redshift drift in uniformly accelerated reference frame*

1,2 1,2;1) Zhe Chang(常哲) Qing-Hua Zhu(朱庆华) 1Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China 2University of Chinese Academy of Sciences, Beijing 100049, China

Abstract: We construct an alternative uniformly accelerated reference frame based on a 3+1 formalism in adapted coordinates. In this frame, a time-dependent redshift drift exists between co-moving observers, which differs from that in Rindler coordinates. This phenomenon can be tested in a laboratory and improve our understanding of non-inertial frames. Keywords: non-inertial frame, uniform acceleration, redshift drift, Unruh effect DOI: 10.1088/1674-1137/44/7/075103

1 Introduction ates, they do not present a redshift drift. This seems not consistent with the observations from inertial observers. Huang [12] suggested that the redshifts without a drift in Owing to the special relativity, inertial frames are Rindler coordinates should be attributed to the norms of well tested and understood. The principle of relativity in- four-accelerations, which are not the same for all co-mov- dicates that physical equations remain the same in all in- ing observers. Besides, Minser, Thorne, and Wheeler ertial frames. For non-inertial frames, there is a general (MTW) [13] derived Møller coordinates with the hypo- principle of relativity; namely, the physical equations remain the same in arbitrary reference frames. This prin- thesis of locality. This indicates that Rindler coordinates ciple should involve all non-inertial reference frames. are in fact local frames. The redshift drift might be a However, even uniformly accelerated reference frames higher order effect. All these considerations motivated us are not yet understood well [1]. Moreover, different uni- to construct an alternative uniformly accelerated refer- formly accelerated reference frames are set up from dif- ence frame that is different from Rindler coordinates and ferent points of views [2–6]. the local frame [1, 5, 13–16]. Propagation of light in non-inertial frames provides a In this study, we investigate an adapted coordinate in way for testing relativity in non-inertial reference frames. which all co-moving observers have the same norms of The Sagnac effect states that in a rotating reference four-accelerations. Explicit metric and coordinate trans- frame, counter-propagating rays that propagate around a formations are obtained. Moreover, the redshifts for co- closed path would take different time intervals [7, 8]. moving observers in the new uniformly accelerated refer- This can be described by the Born metric, known as a re- ence frames are calculated. Using the new proposed uni- lativistic effect [9–11]. Likewise, does a similar effect ex- formly accelerated frames, we investigate a possible Un- ist in uniformly accelerated frames? As we know in the ruh effect and show a non-thermal distribution perceived view of inertial observers, accelerated detectors would by the uniformly accelerated observers in Minkowski va- observe a time-dependent redshift of light from a co- cuum. moving source. Could the redshift drift be observed in This paper is organized as follows. In section 2, we uniformly accelerated reference frames as a relativistic review the redshift between co-moving objects in a non- effect? relativistic approximation and in Rindler coordinates. We In order to answer these questions, a metric of uni- find that there is a redshift drift in the non-relativistic ap- formly accelerated reference frames should be given ex- proximation, whereas this is not in the Rindler coordin- plicitly. Rindler coordinates [3], also named as Møller co- ates. In section 3, we present the construction of a uni- ordinates or Lass coordinates [2], are generally used in formly accelerated reference frame as well as its features. uniformly accelerated reference frames. As rigid coordin- In section 4, we provide explicit metrics of the acceler-

Received 27 November 2019, Revised 13 April 2020, Published online 15 June 2020 * Supported by National Natural Science Foundation of China (11675182, 11690022) 1) E-mail: [email protected] ©2020 Chinese Physical Society and the Institute of High Energy Physics of the Chinese Academy of Sciences and the Institute of Modern Physics of the Chinese Academy of Sciences and IOP Publishing Ltd

075103-1 Chinese Physics C Vol. 44, No. 7 (2020) 075103 ated frames. The redshift drift and the possible Unruh effect in the accelerated frames are studied. Finally, conclusions and discussions are summarized in section 5. Throughout, we use the convention that c = 1.

2 Redshift drift and uniformly accelerated reference frames

We suppose that two light sources A and B and a detector are fixed on a carrier. Light source A is located at a distance L on the right of the detector, whereas light

source B is located on the left of the detector. The schem- Fig. 1. (color online) Schematic of redshift drift in uni- atic is shown in Figure 1(a). We know that no redshift is formly accelerated reference frames. Light sources A and B observed by the detector when the carrier undergoes an and detector are fixed on carrier. Panel (a): No redshift is inertial motion. This would be different when the carrier observed by detector when carrier undergoes inertial mo- undergoes a non-inertial motion (see Figures 1(b) and tion. Panel (b): When carrier moves to right with constant

(c)). acceleration a0, detector will observe redshift from source B In this section, we firstly review the redshift in a non- and blueshift from source A. Panel (c): Observed redshift relativistic approximation and in Rindler coordinates. and blueshift will drift with time, if carrier remains in uni- 2.1 Non-relativistic redshift for accelerated observers formly accelerated motion. As time is absolute in non-relativistic kinematics, the With time, the blueshift would become lower. The red- frequency of light is universal in different reference shift and blueshift drifts are illustrated in Figure 1 from frames. This indicates that a redshift calculated in a labor- process (b) to (c). In non-relativistic kinematics, this atory reference frame is equal to that calculated in the ref- should be understood as a Doppler effect, because there is erence frame for a moving detector. a difference in the velocities when the ray is emitted and We consider that light source B is assigned to the left received. However, in the reference frames for the carrier, of the detector at a distance of L. The carrier undergoes a the difference in the velocities might not be perceived. uniformly accelerated motion to the right. This is shown The redshift drift in the accelerated frames should be un- in Figure 1(b). The source emits a photon at t', and the derstood as a relativistic effect. We will show this in sec- detector observes the photon at t. In the non-relativistic tion 4. The situation is similar to the understanding of the approximation that at ≪ 1 and L ≪ 1/a, the processes can expansion of the universe. ≪ = be formulated as In most cases where at c 1, the redshift and the blueshift would reduce to the most common version, − ′ − 1 2 − ′2 = , aL (t t ) a(t t ) L (1) namely, z = ∓ . 2 c2 where a is the acceleration of the carrier. There is a dif- 2.2 Redshift in Rindler coordinate ference in the time intervals of the emitted and received photons. Using Eq. (1), we have the ratio of the time in- In relativity, a uniformly accelerated motion is de- tervals, scribed by a constant norm of a four-acceleration for a √ ∆ − − 2 − worldline. In the t–x diagram, a uniformly accelerated t = 1 at = (1 at) 2aL. motion is a hyperbolic motion, as the trajectory of the ∆ ′ − ′ − (2) t 1 at 1 at uniformly accelerated motion is a hyperbola, which can From Eq. (2), the redshift, z−, is given by be of the form, ∆t aL ≡ − = + O 2 . 2 2 1 z− ′ 1 ((aL) ) (3) x − t = . (5) ∆t (1 − at)2 a It shows that the redshift is time dependent and will The hyperbolic motion can be described by the following equations: increase with time. Likewise, we consider light source A  located on the right of the detector. It will observe a blue- du0  = au1, shift from the source, which is given by  dτ  (6) aL du1 z+ ≈ − . (4)  = au0, (1 + at)2 dτ This shows that the blueshift is also time dependent. where u0 ≡ dt/dτ, u1 ≡ dx/dτ, and τ is the proper time.

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Using the normalization condition of uµ, one can find that coordinates and inertial frames is given by µ/ τ  the norm of a four-acceleration du d is a constant a. A  t = f (X)sinh(aT),  solution of Eq. (6) can be obtained,  x = f (X)cosh(aT), {  0 = τ ,  = , (14) u cosh(a )  y Y (7)  = . u1 = sinh(aτ). z Z Under a specific initial condition, the parametrized where f (X) could be understood as different rulers of 1 trajectory of a uniformly accelerated motion can be of the space. If f (X) = + X , they are the so-called Møller co- a form, 1  ordinates. Moreover, if f (X) = eaX , they are the so-  1 a t = sinh(aτ),  a called Lass coordinates [2]. In general, the metrics are of  (8)  1 the form, x = cosh(aτ). ′ a ds2 = −a2 f 2dT 2 + ( f )2dX2 + dY2 + dZ2. (15) Another point of view for a uniformly accelerated However, it should be noted that the norms of four- motion in relativity is from electrodynamics [16]. The accelerations are not the same for different co-moving equations of motion for a charged particle are of the form, observers. They depend on coordinate X of the observers. µ 1 du q µ ν = + = Fν u , (9) For example, if f X, the norm of the four-accelera- dτ m a µ tion is given by where Fν is the electromagnetic tensor and m and q are √ µ ν 1 a the static mass and charge of a particle, respectively. We gµνa a = = . (16) f (X) 1 + aX consider a uniform electric field in the direction of x-axis. For simplicity, we ignore other spatial coordinates. The If we wish to construct a uniformly accelerated frame potential is given by Aµ = (E0 x,0), where E0 is the based on a picture of a charged particle in a uniform elec- strength of the electric field. From the potential, the electric field, the norms of four-accelerations should be con- tromagnetic tensor is of the form, stant for all the observers located at arbitrary positions. ( ) This led Huang and Guo [4] to construct new types of µ µσ E0 Fν = η Fσν = . (10) uniformly accelerated frames; in these new frames, the E 0 norms of accelerations for co-moving observers are the Eq. (9) can be rewritten as same constant, a. Another understanding of Eq. (16) was   0 given by MTW [13], who derived Møller coordinates du E0q  = u1,  dτ m with the hypothesis of locality. At the location where  (11) ≪ −1  1 X a , the norms of four-accelerations for different co- du = E0q 0. τ u moving observers are nearly the same constants. This in- d m dicates that Rindler coordinates are in fact local frames. For charged particles, the equations of motion are Observables in Rindler coordinates are redshifts shown to be the same as those for a hyperbolic motion between co-moving observers [17], which can be given by √ = E0q with acceleration a . + m = gTT (X) = 1 aX − ≈ − ′ , A Rindler coordinate might be the most commonly z ′ ′ 1 a(X X ) (17) gTT (X ) 1 + aX used uniformly accelerated frame. The metric is given by where X and X' are the fixed positions of the detector and ds2 = −X2dT 2 + dX2 + dY2 + dZ2. (12) the source, respectively. The redshift is time-independent The coordinate transformation between inertial and is different from that calculated in the non-relativist- frames and Rindler coordinates is of the form,  ic approximation, Eq. (3).  t = X sinh(aT), Huang [12] suggested that this difference is origin-   x = X cosh(aT), ated from Eq. (16) that norms of 4-accelerations are not  (13)  y = Y, the same constant for all co-moving observers. For their  z = Z. new kind of uniformly accelerated reference frames with the same accelerations for co-moving observers [4], the From Eq. (8), the coordinate transformation suggests redshift was shown to be time-dependent. T ∼ τ. Namely, the coordinate time of a uniformly accelerated frame has a similar status of proper time for co- moving observers. Form this point of view, there are oth- 3 Uniformly accelerated reference frame er coordinates that may be regarded as uniformly accelerated frames. The general transformation between Rindler The construction of the alternative uniformly acceler-

075103-3 Chinese Physics C Vol. 44, No. 7 (2020) 075103 ated references was based on a 3+1 formalism in adapted The transformation for X at present is arbitrary, which coordinates, which is also different from the uniformly can be written as accelerated reference frames suggested by Huang [4]. dX = ∂0Xdt + ∂1Xdx. (22) For simplicity, the accelerations of the frames are set For coordinates Y and Z, the transformations are re- along the x direction. A coordinate transformation spectively given by dY = dy and dZ = dz. From Eqs. (20) between the uniformly accelerated frames and the iner- and (22), we obtain the inverse transformations for dt and tial frames is expected in the form,  dx,  = , ,  ( )  T T(t x) 1   N u  X = X(t, x), dt = ∂ XdT + dX ,  (18)  0∂ + 1∂ 1  Y = y,  u 1X u 0X N   ( ) (23) Z = z,  N u0 dx = −∂ XdT + dX . 0∂ + 1∂ 0 where T, X, Y, and Z are the coordinates of the uniformly u 1X u 0X N accelerated frames, and t, x, y, and z are the coordinates With the transformations, one can obtain the metric of of the inertial frames. For an accelerated frame, we ex- the uniformly accelerated reference frames, pect that a type of principle of relativity should be satis- ds2 = − dt2 + dx2 + dy2 + dz2 fied. 2 ( N 2 2 2 Axiom 1 The co-moving observers in the uniformly = (−(∂1X) + (∂0X) )dT 0∂ + 1∂ 2 accelerated frames undergo uniformly accelerated mo- (u 1X u 0X) ) tions with respect to the inertial frames. 1 2 + dX2 − (u0∂ X + u1∂ X)dTdX Axiom 2 The co-moving observers in the inertial N2 N 0 1 frames undergo uniformly accelerated motions with re- + dY2 + dZ2. (24) spect to the uniformly accelerated frames. A uniformly accelerated motion in the axioms is for- From axiom 1, accelerated observer u should be a co- mulated as Eq. (6). We would use axiom 1 for construct- moving observer of the uniformly accelerated frames, ing the uniformly accelerated reference frames. This sug- namely, gests that the uniformly accelerated observers defined in dX u0∂ X + u1∂ X = = 0. (25) the inertial frames should move attached to the acceler- 0 1 dτ ated reference frames. In the following, we verify axiom γ = 1 2 by the fact that the geodesics in uniformly accelerated Moreover, if we set 0 1 , by making u ∂1X + u ∂0X frames can be formulated as uniformly accelerated mo- use of Eq. (25), we can rewrite ∂0X and ∂1X as tions.  1  u ∂0X = − , 3.1 Construction of uniformly accelerated reference  γ  (26)  0 frame ∂ = u . 1X γ In the 3+1 formalism, four-velocities u of accelerated observers are normal vectors of a space-like hypersur- Using Eqs. (23), (24), and (26), we know that the face ΣT , which is formulated as metric is reduced to µ 2 2 2 2 2 2 2 uµdx = −NdT, (19) ds = −N dT + γ dX + dY + dZ . (27) where N is the so-called lapse function and N > 0. A uni- In the adapted coordinates, condition axiom 1 guaran- formly accelerated frame is adapted to u. Moreover, we tees that the metric is always diagonal. This is different set that the u is along the direction of the x-axis, namely, from the uniformly accelerated frames suggested by u2 = u3 = 0 for simplicity. T is the coordinate time of the Huang [4]. uniformly accelerated frames. The transformation for dT One may wonder how much this is related to the 3+1 can be written as formalism. In general, a metric in the 3+1 formalism can u0 u1 be written as dT = dt − dx. (20) 2 2 2 i i j j N N ds = −N dT + γi j(β dT + dX )(β dT + dX ), (28) 0 1 i u u where γi j is the reduced metric and β is the so-called shift This suggests that ∂0T = and ∂1T = . N as an in- N N function. Owing to axiom 1, βi is shown to vanish. As we tegrating factor ensuring that differential form dT is in- X X dX Y Z tegrable. Because d2 = 0, Eq. (20) leads to know β = Nu = N = 0 and u = u = 0, this leads to ( ) ( ) dτ 0 1 βi = 0. We can rewrite the metric in Eq. (28) as ∂ u + ∂ u = . 1 0 0 (21) 2 2 2 i j N N ds = −N dT + γi jdX dX . (29)

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Comparing it with Eq. (27), one can find that the re- coordinate time T. This indicates that the Rindler metric duced metric, γi j , in the accelerated frames is of the cannot be included in our uniformly accelerated frames. form, Secondly, N = 1 is not permitted. This means that co-    γ2  ordinate time T of the uniformly accelerated frames is not   γ =  . the proper time for co-moving observers. This could be i j  1  (30) 1 understood by analogy. In the Schwarzschild space-time, one might not require that the coordinate time for a co- γ As also functions as an integrating factor, we prefer moving observer be a proper time, because there is grav- γ γ to i j in our derivation. ity. In the uniformly accelerated frames also this is the In the metric in Eq. (27), there are two unknown case, because there is a fiction force. fields, g00 and g11, which depend on the choice of the From Eq. (36), we may prove that the metric (Eq. four-velocity, u. Here, we consider uniformly accelerated (27)) is a solution of vacuum Einstein equations. We can reference frames. Namely, u is described by Eq. (6), start to check this by calculations of the connection,  which can be rewritten as  ∂  T T N 1 Γ = , du = 1 ∂ 1 = 0.  TT N τ T u au (31)  d N  ∂X N ΓT = = aγ, By making use of Eq. (26), we rewrite Eq. (21) in  TX N  terms of the coordinates, (T, X,Y,Z),  γ∂T γ ( ) ( ) ΓT = ,  XX 2 u0 u1 1 1 a  N ∂ + ∂ = ∂ + = 0. (32)  2 1 0 γ X  N∂X N aN (38) N N N N ΓX = = ,  TT γ2 γ 2 =  As d X 0, it leads to an equation as follows:  ( ) ( ) ( )  ∂T γ 1 0 ΓX = , u u 1 1 1  TX ∂ + ∂ = ∂ + ∂ u1 = 0. (33)  γ 1 γ 0 γ T γ γ2 X  N u0  ∂ γ ΓX = X , We rearrange Eqs. (31), (32), and (33) as  XX γ    ∂ = γ, others = 0.  X N aN ∂ = ,  T G aN (34) Non-trivial components of the Ricci tensor are given  1 ∂XG = ∂T γ, by N  ( ( ) ( ))  ∂ ∂ γ 1 1 µ  1 2 X N 2 T where we set ∂G ≡ ∂u . As u uµ = −1, it leads to RTT = Nγ ∂X − Nγ ∂T , u  γ3 γ N = 1 0  ( ( ) ( )) (39) G arcsinh(u ). Associating it with Eq. (7), we find  ∂ ∂ γ  1 2 X N 2 T G = aτ. (35) RXX = −γN ∂X + γN ∂T . N3 γ N Then, Eq. (34) can be rewritten in a natural manner, From Eq. (36), one has ∂ = γ, ( )  X N aN    ∂T γ ∂ τ = N, ∂ = a2Nγ,  T (36)  T   N ∂ τ = 1 ∂ γ.  ( ) (40) X T  ∂ N aN ∂ X = 2 γ. X γ a N The solutions of Eq. (36) provide an explicit metric of the uniformly accelerated reference frames. The expres- This causes the Ricci tensor to be zero, γ sion of the coordinate transformation depends on N, , Rµν = 0. (41) and proper time τ. From Eqs. (20), (26), and (35), the co- Namely, the metric of the uniformly accelerated ordinate transformation, which takes the form of Eq. (18), frames is a solution of vacuum Einstein equations. The can be derived from  checking process seems trivial, as the Einstein equation  cosh(aτ) sinh(aτ) dT = dt − dx, always allows coordinate transformation as a gauge sym-  N N  metry.  sinh(aτ) cosh(aτ)  = − + , dX γ dt γ dx (37) 3.2 Inertial motion in uniformly accelerated frame   = , dY dy In subsection A, we have constructed the metric of dZ = dz. the uniformly accelerated frames with axiom 1. The ax- Besides the diagonal, there are other features of the iom presents an equivalent description for uniformly ac- metric from Eq. (36). Firstly, the metric must depend on celerated motions in different reference frames. On the

075103-5 Chinese Physics C Vol. 44, No. 7 (2020) 075103 other side, an inertial motion also requires an equivalent vector bases of the frames, and κ, τ, and b are the so- description. This suggests that the inertial motion should called curvature and torsions of the worldline of the ob- be formulated as a uniformly accelerated motion in the servers in the Lorentz manifold, respectively. view of a uniformly accelerated frame. Our uniformly accelerated frames can be expressed in We consider the inertial motion in the uniformly ac- terms of the Frenet –Serret frames, in the tetrad formal- celerated frame at the T–X plane, ism. Firstly, we rewrite coordinate bases ∂µ as tetrads ea,      = µ∂  dT   cosh(aτ) sinh(aτ)  dt  which are formulated as ea ea µ. In the uniformly accel-    −    τ   N N  τ  erated frames, the tetrads can be given by  d 0  =   d 0 , (42)   dX   sinh(aτ) cosh(aτ)  dx      −    = 1 ∂ , τ γ γ τ e0 T d 0 d 0  N ( )  dt dx  1 , τ e = ∂X, (46) where is a constant velocity vector, 0 is the  1 γ dτ0 dτ0   = ∂ , proper time of co-moving observers in the inertial frame, e2 Y τ  which is distinguished from proper time for the uni- e3 = ∂Z. formly accelerated observers. From Eqs. (36) and (42), One can find that κ = a, τ = b = 0 for our uniformly ac- one can obtain  ( ) celerated frames, namely,        d dT = − γ dX ,  e   0 a  e   N a  0    0  dτ dτ0 dτ0      D  e1  =  a 0  e1 .  ( ) (43)      (47)  dτ  e2   0  e2   d γ dX = − dT .  aN e3 0 e3 dτ dτ0 dτ0 dT dX The curvature of the moving frames is exactly the If we set υT = N and υX = γ , then Eq. (43) reduces constant acceleration, a, in our uniformly accelerated dτ0 dτ0 to frames, whereas Rindler coordinates cannot be described  in terms of the Frenet-Serret frames with tetrads. dυT  = −aυX,  dτ 3.3.2 Kinematical quantities  (44) The congruence of co-moving observers u indicates a  υX d = − υT . deformation of the space –time. The difference between τ a d the worldlines of co-moving observers can be described From Eq. (44), the inertial motion in the view of the in terms of deviation vector χµ, which satisfies uniformly accelerated frames can be formulated as a hy- µ ν µ ν µ [u,χ] = u ∇νχ − χ ∇νχ = 0 . (48) perbolic motion with a reverse acceleration. Using Eq. µ µ µ ν dT dX For the spatial distance of χ , namely, χ˜ = γν χ , the (42), one can verify that and satisfy geodesic evolution of χ˜µ indicates the spatial deformation of the dτ0 dτ0 reference frames, equations in the accelerated frames. All these indicate ( ) µ that axiom 2 is verified. In addition, the redefinitions of D˜ χ˜ µ 1 µ µ µ = χ θγν + σν + wν , (49) the covariant velocities(υT ,υX) are insightful. In curvilin- dτ 2 ear coordinates, this is exactly the standard definition of a D˜ ∗ vector, where N,γ are the so-called Lamé coefficients. where = γ ∇u is the spatial covariant derivation with dτ µ µ Moreover, in the tetrad formalism, one may find respect to u. θ,σν, and wν are the so-called kinematical a a µ υ = eµu . quantities and named after the expansion scalar, shear tensor, and rotation tensor, respectively. 3.3 Features of uniformly accelerated reference frames In the accelerated frames, co-moving observers u are 3.3.1 Frenet–Serret frames given by µ The Frenet –Serret frames describe the evolution of uµdX = (−N,0,0,0) . (50) the frames along the worldline of the observers. It can be The covariant derivative of observers u can be de- generally written as      composed into acceleration, expansion scalar, shear, and  e   κ  e  rotation tensor,  0    0  D  e   κ τ  e   1  =   1 , 1 τ    −τ   (45) ∇νuµ = −uνaµ + θγµν + σµν + wµν, (51) d  e2   b  e2  3 e −b e 3 3 where, D µ µ where is the covariant derivative, eµ represents the a = ∇uu , (52) dτ

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µ θ = ∇µu , (53) ing special in the directions of the Y- and Z-axes, we consider two-dimensional metrics for simplicity. 1 1 σµν = (∇µuν + ∇νuµ + uµ∇uuν + uν∇uuµ) − θγµν, (54) 2 3 4.1 Hyperbolic metric and redshift drift 1 σ ρ The components of the metric turn to be hyperbolic wµν = γµ γν (∇σuρ − ∇ρuσ). (55) 2 triangle functions, when the uniformly accelerated ob- From Eqs. (29) and (52), the accelerations of co-mov- server, u, is just a function of coordinate time t, namely, µ a = ing observers are given by aµ = δ . The norms of the ac- u u(t). Associating it with Eq. (36), we get the metric as 1 γ dT 2 dX2 celerations are s2 = − + . (59) √ d 2 2 µ ν sinh (−a(T + X)) tanh (−a(T + X)) |a| ≡ gµνa a = a . (56) As N > 0, it leads to −a(T + X) > 0. Transformation This shows that the accelerations of any co-moving from the inertial frames to the accelerated frames is of the observers in our uniformly accelerated frames are the form, same constant acceleration, a, which is different from that   1 in Rindler coordinates (Eq. (16)). t = ,  asinh(−a(X + T)) We present the expansion scalar, shear tensor, and ro-  (60)  1 tation tensor in the following: x = X + .  atanh(−a(X + T)) wµν = 0,   ∂ γ From Eq. (36), we can obtain proper time τ for the θ = T ,  co-moving observers in the accelerated frames,  Nγ ( )       (57) 1 1 1   0  τ = arcsinh = arcsinh(at). (61)    a sinh(−a(T + X)) a σ = 1θ 2 .  µν  −   3  1  Eqs. (59) and (61) lead to sinh(aτ) = at = N > 0.  − 1 If a > 0, the accessible region of space–time is that with The vanished rotation tensor indicates that co-mov- τ,t ⩾ 0. Namely, the reference frames undergo uniformly ing observers u are Eulerian observers. There is a simul- accelerated motions from t = 0. The coordinate lines of taneous hypersurface ΣT orthogonal to all the co-moving the uniformly accelerated frames in the t–x plane are observers. presented in Figure 2. From Eqs. (48), (49), and (57), the evolution of the The metric in Eq. (59) can describe the reference spatial deviation vector is given by frames of the carrier in Figure 1, so that we can recon-   sider the redshift drift in the uniformly accelerated refer-  0    ence frames. The reference frames move with constant D˜ χ˜  θ  =  χ˜ . (58) acceleration a to the right, and light source B on the left dτ  0  0 of the detector co-moves with the carrier. The source emits light that is along a null curve. By utilizing the met- It shows that there is a spatial deformation between ric in Eq. (59), we obtain the equation of trajectories of the co-moving worldlines in the direction of the X-axis. the light, In the uniformly accelerated frames, it can be understood as a non-inertial effect. The fiction force can affect the deformation of the space. Evolutions of these kinematical quantities are described by the Raychaudhuri equations. In our uniformly accelerated frames, the equations can be deduced from Eq. (36). It did not lead to any constraints for construct- ing the uniformly accelerated frames.

4 Explicit solutions

Now, we try to obtain the solutions of Eq. (36). Moreover, using these solutions, we calculate the red- Fig. 2. Coordinate lines of uniformly accelerated frames in shift drift between co-moving objects and the possible t–x diagram. In case of a > 0, accessible region for acceler- Unruh effect in the accelerated frames. As there is noth- ated frames is that with t > 0.

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1 1 dX = 0 . (62) sinh(−a(T + X)) tanh(−a(T + X)) dT In this case, only forward-propagating light reaches the detector. Namely, the ′′−′′ in Eq. (62) is required to be chosen. Then, the trajectories are obtained ( ) ( ) a a tanh − (T + X) = −a(X − X′) + tanh − (T ′ + X′) , (63) 2 2 where the detector and the source are fixed at spatial coordinates X and X', respectively, where X > X′. From Eq. (63), it takes different time intervals, when two light sig- nals are emitted and received. The ratio can be given by ( ) 2 −a + Fig. 3. Redshift (left panel) and blueshift (right panel) ∆T cosh (T X) = ( 2 ), between co-moving objects in uniformly accelerated refer- ∆ ′ a (64) T cosh2 − (T ′ + X′) ence frames as functions of proper time. 2 where ∆T ′ and ∆T are the emitted and received time in- Table 1. Redshift z− and blueshift z+ between co-moving objects in tervals, respectively. Using Eqs. (59), (61), (63), and (64), uniformly accelerated reference frames calculated with different ap- we can derive the redshift [18] observed by the detector proaches. in the uniformly accelerated reference frames, z− z+ √ , ∆ Møller coordinate [17] aL − aL ∆τ gTT (T X) T c2 c2 z− ≡ − 1 = √ − 1 ( )− ( )− ∆τ′ ′ ′ ′ Non-relativity aL − at 2 − aL + at 2 g (T , X )∆T 2 1 c 2 1 c ( TT ) c c aL aτ aL − aτ a ′ ′ Huang's [12] e c − e c − + 2 c2 tanh (T X ) c ( ) 2 τ aL 1 = ( ) − aL a − τ 1 Hyperbolic metric 2 cosh 2 a a c c c cosh( c ) tanh − (T + X) 2 =a(X − X′)cosh(aτ) . (65) metric is close to that calculated in Rindler coordinates. It shows that the redshift drifts with the proper time of In the non-relativity case, it turns to be meaningless when ≳ τ the detector. There is an additional factor cosh(aτ) com- at c. In the relativistic case, there is no limitation, as a pared to the result in Rindler coordinates, Eq. (17). With is not a three-velocity. time, the redshift would become higher and finally tend 4.2 Conformally flat metric and Unruh effect to infinity. Similarly, we can consider light source A on the right By making use of Eq. (36) and constraint N = γ, we of the detector in Figure 1. The result shows that a blue- get a conformally flat metric, shift, namely, z < 0, is observed by the detector, 1 ds2 = (−dT 2 + dX2). (67) a(X − X′) a2(T + X)2 z+ = − . (66) cosh(aτ) As N > 0, it also leads to −a(T + X) > 0. Transforma- In this case, X < X′, and the blueshift would become tion between the inertial frames and the accelerated lower with time until it vanishes. frames is given by  ( ) These results, Eqs. (65) and (66), are consistent with  1 1 t = − + T − X , Huang's [12] and those in the non-relativistic case, qualit-  2 a2(T + X) atively. Namely, there is a redshift drift in the uniformly  ( ) (68)  1 1 accelerated reference frames. Further, we compare all x = − − T + X . these results in detail, which are summarized in Table 1. 2 a2(T + X) We recover the speed of light, c, in the formulations, and Proper time τ for co-moving observers can be ex- ′ set |X − X| ≡ L for consistency. pressed in terms of the space–time coordinate, ( ) In Figure 3, the redshift and blueshift are presented as 1 1 1 functions of the proper time. In the non-relativistic case, τ = ln − = ln(a(t + x)). (69) a a(T + X) a time t is absolute. Therefore, we did not distinguish it with the proper time. The results of Møller coordinates The coordinate lines of the uniformly accelerated and the non-relativistic approximation case are contrast- frames in the t–x plane are presented in Figure 4. ing, which are independent of and sensitive to time, re- The Unruh effect states that in uniformly accelerated spectively. The redshift calculated with the hyperbolic frames (Rindler coordinates), the co-moving observers

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We focus on left-moving sectors ϕ+ of the field. Dif- ferent sectors ϕ− and ϕ+ would not interact with each other [22]. We quantize ϕ+ in the form of ∫ ∞ 1 dk † ϕˆ = √ √ {ˆ −ik(T+X) + ˆ ik(T+X)}, + bke bke (76) 2π 0 2k

where bˆk is a ladder operator satisfying canonical com- munication relations, ˆ , ˆ† = δ − ′ , = , [bk bk′ ] (k k ) others 0 (77) and

bˆk|0A⟩ = 0, (78)

where |0A⟩ is a vacuum state in the uniformly accelerated reference frames. The mode function is read from the field operator in Eq. (76),

1 −ik(T+X) gk(T, X) = √ e . (79) 4πk

On the other side, we know the field operator of a Fig. 4. Coordinate lines of uniformly accelerated frames in left-moving sector in a flat space–time, t–x diagram. In case of a > 0, accessible region for acceler- ∫ 1 ∞ dp ated frames is that with t + x > 0. ϕ = { −ip(t+x) + † ip(t+x)}, ˆ+ √ √ aˆ pe aˆ pe (80) 2π 0 2p would perceive a thermal distribution of the Minkowski where aˆ is a ladder operator. The canonical communica- vacuum [19–21]. The temperature of the distribution is p tion relations are given by proportional to the constant acceleration, a, in Rindler co- , † =δ − ′ , ordinates. In this subsection, we use the metric in Eq. (67) [ˆap aˆ p′ ] (p p ) to calculate the possible Unruh effect. For simplicity, we others =0 . (81) consider a massless Boson. Moreover, one has The Klein–Golden equation for a massless Boson is | ⟩ = , given by aˆ p 0M 0 (82) µ | ⟩ ∇µ∇ ϕ = 0, (70) where 0M is the Minkowski vacuum state. The mode function is of the form, where ∇µ is the covariant derivative. With our metric, the 1 −ip(t+x) equation of motion can be written as fp(t, x) = √ e . (83) 4πp µν 1 2 2 g ∇ν∇µϕ = − (∂ ϕ − ∂ ϕ) = 0 , (71) a2(T + X)2 T X The ladder operators in the accelerated and inertial frames are related to the so-called Bogolubov transforma- namely, tion, which is given by ∂2 ϕ − ∂2 ϕ = 0 . (72)  ∫ T X  † aˆ = dk{α bˆ + β∗ bˆ } , It is the same as the Klein–Golden equation in a flat  p kp k kp k  ∫ (84) space–time. The solution of the equation can be given by  ∫ ˆ = {α∗ − β∗ † } , bk dp kpaˆ p kpaˆ p 1 − k T−kX ϕ = √ d2k{δ(k2 − k2)ϕ˜(k ,k)e i( 0 )} . (73) π 0 0 2 where αkp and βkp are the so-called Bogolubov coeffi- It can be expanded as cients satisfying the relations, ∫ ∫ ∞  1 dk −i(|k|T−kX) −i(−|k|T−kX)  ∗ ∗ ′ ϕ = √ {ϕ˜ + }  dk{α α ′ − β β ′ } = δ(p − p ) , | | (e e )  kp kp kp kp 2π −∞ 2 k ∫ ∫ ∞  1 dk  ∗ ∗ ′ −ik(T−X) ik(T+X)  dp{α α ′ − β β ′ } = δ(k − k ) , = √ {ϕ˜ke + ϕ˜ke  kp k p kp k p π 2k ∫ 2 0  (85) −ik(T+X) ik(T−X)  ∗ ∗ + ϕ˜ + ϕ˜ } = ϕ + ϕ ,  {α β ′ − β α ′ } = , −ke −ke − + (74)  dk kp kp kp kp 0 ∫ where  ∫  ∗ ∗ ∗ ∗ ∞  dp{α β ′ − β α ′ } = 0 . 1 dk −ik(T+X) ik(T+X) kp k p kp k p ϕ+ = √ {ϕ˜−ke + ϕ˜ke }. (75) 2π 0 2k For the mode functions, orthogonal relations can be

075103-9 Chinese Physics C Vol. 44, No. 7 (2020) 075103   = , derived from the so-called Klein–Gordon inner product,∫  U constant  1 (94) µ ∗ ∗  V = . (ϕ,χ) ≡ i dΣ {ϕ ∇µχ − χ∇µϕ }. (86) 2λ Σ 2a One can find that Using Eq. (94), we obtain the volume elements of the { , = δ − ′ , , ∗ = , null hypersurface [26], ( fp fp) (p p ) ( fp fp ) 0 ′ ∗ (87) µ ν 1 (g ,g ′ ) = δ(k − k ) , (g ,g ′ ) = 0 . dΣ ≡ ϵµνζ ξ dλ = dV, (95) k k k k 2a2V2 In different coordinates, field operator ϕˆ+ remains the µ where ϵµν is the Levi–Civita tensor and ζ is an auxiliary same under the Bogolubov transformation. Thus, one can µ µ null vector satisfying ξ ζµ = −1 and ζ ζµ = 0. From Eq. derive the Bogolubov transformation for the mode func- (95), the directed surface element is obtained, tions, ∫ Σµ = −ξµ Σ = δµ . d d 1dV (96) g = dp{α f + β f ∗}. (88) k kp p kp p Here, we choose that a > 0. For the metric with N > 0, From Eq. (88), the Klein –Gordon inner product can it leads to V < 0. be used to calculate the Bogolubov coefficients, Form Eqs. (79), (83), (89), and (96), we can calculate { α = , , the Bogolubov coefficients, kp ( fp gk) ∫ ∗ (89) 0 βkp = −( f ,gk). k αkp =( fp,gk) = i dV What the accelerated observers perceive in the  −∞     1 1 1 1  Minkowski vacuum is formulated as the expectation × √ ipυ∂ √ −ikV− √ −ikV ∂ √ ipυ  e V e e V e  value of occupation number operators Nk of the acceler- 4πp 4πk 4πk 4πp ( ( ) ) ated observers for the Minkowski vacuum state, ∫ ∞ ∫ ∞ ∫ 1 + p p 1 + p i(kV 2 ) i(kV 2 ) = √ k dVe a V − d e a V ⟨ | | ⟩ = ⟨ |ˆ† ˆ | ⟩ = β β∗ . π a2 V 0M Nk 0M 0M bkbk 0M dp pk pk (90) 4 pk 0 0 ∫ ∞ √ { √ } 1 2 pk iη = η2 − a2 , It shows that the expectation value only involves the π d 1 e β a 1 Bogolubov coefficients, pk. (97) As in Refs. [22, 23], we use light-cone coordinates to and calculate the Bogolubov coefficients from Eq. (89) for a ∫ given null hypersurface. The null coordinates are usually 0 β = − ∗, = − closely related to radiation [24, 25]. Light-cone coordin- kp ( fp gk) i dV  −∞  ates of the inertial frames and the uniformly accelerated    1 1 1 1  frames are given by (u,v) = (t − x,t + x) and (U,V) = (T − X, × √ −ipυ∂ √ −ikV−√ −ikV ∂ √ −ipυ  e V e e V e  T + X), respectively. From Eq. (68) and the light-cone co- 4πp 4πk 4πk 4πp ( ∫ ∞ ( ) ∫ ∞ ) ordinates, the transformation between the uniformly ac- 1 p 1 − p − p i(kV 2 ) i(kV 2 ) = − √ d e a V + k dVe a V celerated frames and the inertial frames is obtained, 2 4π pk a 0 V 0 1 ∫ ∞ √ { √ } = , υ = − . (91) 1 kp η u U 2 2 2 i a2V = − d η + 1 e a . 2πa −∞ In the light-cone coordinates, the metric of the accel- (98) erated frames can be rewritten as ϵ With the tricks of η → ei ∞ and ϵ > 0 [22], the Bogol- 1 ds2 = −dudυ = dUdV. (92) ubov coefficients can be obtained explicitly, a2V2  √  1  pk  We choose the null hypersurface as α = K −2i , (99) kp πa 1 a2 Φ(U,V) ≡ U = cosntant. (93) and One can find that it is the only non-trivial null hyper-   √  surface for calculating the possible Unruh effect, other- 1   pk  β ≡ β = − 1 + iK 2  wise it would lead to pk 0. The normal vectors of the kp π 1 2 0 2 a a null hypersurface are null vectors, ξν = −∂νΦ = −δν. We   √   √       use λ to parametrize the integral curve (U(λ),V(λ)) of π   pk   pk  + L 2  − I 2  ξµ(λ). As the null vector, ξµ, is also tangent to the null hy- 2 1 a2 1 a2 ( )) persurface, λ also can be used to parametrize the null hy- 4 1 3 1 , a 1 persurface in the two-dimensional case. Thus, we can + G1 3 ,2 2 2 , (100) 2π 3,1 k2 p2 1 solve the integral curve, (U,V), as

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where K1 and I1 are the modified Bessel functions of the ted coordinates for uniformly accelerated observers u. second and first kind, respectively, L1 is the modified The norms of four-accelerations of co-moving observers 1,3 Struve function, and G3,1 is the generalized Meijer G are the same in the uniformly accelerated reference function. The above shows that the Bogolubov coeffi- frames. The inertial motion in the view of a uniformly ac- cients, αkp and βkp, are completely different from those in celerated observer can be formulated as a uniformly ac- Rindler coordinates [19, 22, 23]. This suggests that the celerated motion. Moreover, the space–time would be de- expectation value might not be the form of that calcu- formed by a non-inertial effect. We also presented expli- lated in Rindler coordinates. We finally obtain the expect- cit metrics and coordinate transformations. In contrast ation value in our uniformly accelerated frames as fol- with Rindler coordinates, the redshift for co-moving observers would drift with time, in our accelerated frames. lows: ∫ This is consistent with earlier results [12]. Besides, we ⟨ | ˆ | ⟩ = β β∗ 0M Nk 0M dp kp kp calculated the possible Unruh effect and showed a non- ∫ ∞ {( ∫ ∞ √ { √ }) thermal distribution for the Minkowski vacuum state per- 1 2 kp iη = − η2 + a2 ceived by the uniformly accelerated observers. dp π d 1 e 0 2 a −∞ From our approaches, the constraint equations in Eq. ( ∫ ∞ √ { √ })∗} 1 kp η′ ′2 2 2 i (36) for uniformly accelerated frames are under-determ- × − d η + 1 e a πa −∞ 2∫ ∫ ∫ ined. It results in that the metric of the reference frames is 1 ∞ ∞ ∞ non-unique. The degrees of freedom for different meas- = dχ dη dη′ π2 urements and synchronisation conventions are allowed. 8 k 0 −∞ −∞   ′  Firstly, Eq. (36) is invariant under transformations  χηη ′  Λ × √ √ iz(η−η ) ≡ , T → f (T) and X → g(X). This is owing to integrating  e  η2 + 1 η′2 + 1 k factor N and γ being not unique. It indicates that the time √ (101) and distance can be measured with different clocks and kp rulers. Secondly, there are different simultaneous hyper- where χ = 2 , and a2 surfaces for the explicit solutions in Section 4. It sug- ∫ ∫ ∫   ∞ ∞ ∞  ′  gests different synchronisation conventions. Moreover, it 1  χηη ′  Λ ≡ dχ dη dη′  √ √ eiz(η−η ) is rather interesting to explore what kinds of conventions π2  ′  8 0 −∞ −∞ η2 + 1 η 2 + 1 ∫ ∫ ∫ { } are physically operational. ∞ ∞ ∞ ′ ∞ 1 χηη ′ 1 After Huang [12] first suggested that a redshift drift < dχdηdη′ eiχ(η−η ) = lnχ . 2 ′ 2 8π 0 −∞ −∞ |η||η | 4π 0 can be observed in uniformly accelerated reference (102) frames, we also provided a similar prediction with differ- Constant Λ is divergent. We use the trick of η → eiϵ∞ ent approaches. The redshift of a co-moving object is dif- to obtain the last equal sign. ferent from that in Rindler frames, because the co-mov- Firstly, this shows that the distribution of Eq. (101) is ing objects of these reference frames are defined differ- independent of the acceleration for our accelerated frame. ently. As mentioned in Ref. [27], the concept of relative Secondly, the expectation value of the number operator velocities between non-inertial observers is usually am- for the Minkowski vacuum state is a non-thermal distri- biguous. Thus, it is non-trival to obtain a convention for bution of k. The uniformly accelerated observers cannot co-moving objects via introducing physical reference perceive temperature in the Minkowski vacuum. frames. Moreover, if a set-up of co-moving objects can be realized in experiments, as shown in Figure 1, we might expect that these results derived from different frames 5 Conclusions and discussions can be tested in the future.

In this study, we constructed new uniformly acceler- The authors wish to thank Prof. Chao-Guang Huang ated reference frames based on a 3+1 formalism in adap- for useful discussions.

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