The Underlying Mechanisms of Time Dilation and Doppler Effect in Curved Space-Time

Transcription

1 The Underlying Mechanisms of Time Dilation and Doppler Effect in Curved Space-Time enliang Li,,*, Hailiang Zhang 3, Perry Ping Shum 3, Qi Jie Wang,4 School of Electrical & Electronic Engineering, Nanyang Technological University, Singapore, Optoelectronics Research Centre, University of Southampton, Southampton SO7 J, United Kingdom 3 COFT, School of EEE, Nanyang Technological University, 50 Nanyang venue, Singapore School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore, * libenliang73@gmail.com In this paper, we investigate the time dilation and Doppler effect in curved space-time from the perspective of quantum field theory (QFT). coordinate transformation eeping the local metric values unchanged between an original coordinate ( tx, ) and another new coordinate system ( t, x ) is introduced. We demonstrate that the mathematical forms of physics formulas in QFT are the same in these two coordinate systems. s applications of the coordinate transformation between these two coordinate systems, the time dilation and Doppler effect with an arbitrary time-dependent relative velocity in curved space-time are analyzed. For Minowsi space-time, the time dilation and Doppler effect agree with the cloc hypothesis. For curved space-time, we show that even if the emitted wave has a narrow frequency range, the Doppler effect will, in general, broaden the frequency spectrum and, at the meantime, shift the frequencies values. t last, the Doppler-shifted frequency value will be given under certain conditions and approximations. These new findings will deepen our understanding on Doppler effect in curved space-time and may also provide theoretical guidance in future astronomical observations. I. Introduction The quantum field theory (QFT) in curved space-time is to investigate the behavior of quantum particles with fixed space-time metric. Since some operators in QFT (such as Dirac factor in QED and covariant derivative ) are metric dependent, as a result, the mathematical expressions of waves and the formulas describing the interactions between particles will carry the metric dependence. In our previous wor [], we argued that the formation of atomic structures can be theoretically deduced from the QFT. Therefore, the atomic spectra will carry the information about the local metric, and the gaps between the energy levels of the atom will vary with the position of the atom in curved space-time. In order to analyze the energy levels of the atoms located at different positions in curved spacetime, one can study the local curvature inside of the atoms to obtain the shifted energy spectrum, which has been a research topic under investigation using perturbation theory [] [7]. In modern cosmology, the electromagnetic signals coming from moving astronomical objects deliver the exhaustive physical information about the intriguing phenomena going on in the universe. Providing a solid theoretical instruction to analyze these signals is crucial in leading to new and important discoveries [8] [0]. s a diagnostic tool, the Doppler shift measurement is one of the most important spectroscopic measurements made in astronomy. chieving a complete understanding of the Doppler effect in the curved space-time becomes essential in studying the numerous astrophysical phenomena and some progress has been made by many researchers in the past [] [8]. For the Doppler effect in curved space-time, one needs to solve the wave equations of QFT to examine the evolution of waves during the

2 propagations. esides, since atoms usually are used as emitters or receivers of waves, the interactions between the propagating waves with the atoms whose energy levels might be shifted due to the motion in curved space-time need to be well understood. ecause of these reasons, the combination of general relativity with QFT to investigate the Doppler effect becomes a necessity, and some new features that have not been considered in previous wors may arise from it. In this paper, we investigate the influence of space-time curvature on two atoms located far away from each other such that the metric value inside the atoms can be taen as constants. In order to analyze the behavior of these atoms, we introduce a new coordinate system ( t, x ) which eeps the local metric the same as that in the original coordinate ( tx., ) s a result, the quantum states of the atoms in two different coordinate systems tae the same mathematical expressions. Meanwhile, the time dilation and Doppler effect with an arbitrary relative velocity in curved space-time will be studied. This paper is organized as follows. In section II, we will introduce the coordinate transformation maintaining the local metric values. Meanwhile, the significance of this coordinate transformation is discussed in this section. In sections III, IV and V, we will study the time dilation and Doppler effect in Minowsi, Schwarzschild and FRW space-times, respectively. The results show that the time dilation in these space-times agree with the cloc hypothesis while the Doppler effect in curved space-times displays some new features which have not been investigated by researchers before. For strong gravitational fields such as in the vicinity of blac holes, the Doppler effect can shift the value of wave frequency and, at the same time, broaden the spectrum of the emitted signals. Under certain conditions and approximations, the Dopplershifted frequencies are given in Schwarzschild and FRW space-times. These findings can provide a test for QFT in curved space-time and meanwhile may lead to important new modern astrophysical discoveries. In this paper, natural units in which G c will be used. II. Coordinate transformation maintaining local metric value In this section, we will discuss a coordinate transformation that can maintain the local metric value. y eeping the local metric value, we show that the physical formulas of local QFT remain the same in the transformed coordinate, the reason for doing such coordinate transformation is also discussed in this section. For QFT in curved space-time ( tx,, ) we write the two-dimensional equation of motion for massless scalar field ( tx, ) as where stands for the covariant derivatives and g ( t, x) g ( t, x) ( t, x) 0 (.) is the metric in ( tx, ) coordinate system. Now if we use another set of coordinate ( t, x ), we get the field equation as g ( t, x) ( t, x) 0 (.)

3 where x x g ( t, x) g ( t, x) (.3) x x x 0 0 and with the defined symbol x t( x t) and x x( x x). Therefore, x the operation g ( t, x) in coordinate ( t, x ) will become equivalent with g ( t, x) in coordinate ( tx., ) Thus, we can see that the scalar field solution ( tx, ) in Eq. (.) is also a solution for Eq. (.) in coordinate ( t, x ), that is, g ( t, x) ( t, x) 0. (.4) Therefore, we have ( t, x) ( t, x) (the value of the scalar function does not change). However, the mathematical structure of the solution ( t, x ) may be different with ( tx, ), this is because g ( t, x ) may have a different mathematical structure compared with g ( t, x). For the coordinate transformations that can maintain the mathematical structure (or mathematical form) of the metric g ( t, x), the metric g ( t, x ) in coordinate ( t, x ) remains unchanged such that g ( T, X ) g ( T, X ) [ g ( T, X ) is the metric in ( t, x ) coordinate with tt, x X and g ( T, X ) is the metric in ( tx, ) coordinate with t T, x X ] for any space-time value ( T, X ). In this way, Eq. (.4) becomes g ( t, x) ( t, x) 0. (.5) We also now from Eq. (.) that ( t, x ) is a solution of Eq. (.5), i.e., we have ( t, x) ( t, x). (.6) Therefore, the mathematical form of the scalar function ( tx, ) remains unchanged. Thus, we can describe the same field using the same mathematical expression in different coordinate i( t x) systems, the only difference is the value of the parameters. In fact, for the plane wave e as a scalar function which is a solution of Eq. (.) with Minowsi metric, the solution in ( t, x ) coordinate that maintains the mathematical form of Minowsi metric becomes i( t x ) e i( t x) which has the same mathematical structure as e, that is, the only difference between two solutions is the values of the four parameters ( t, x,, ). In practice, solving QFT equations involving curved metric g ( t, x) extended in entire space-time would be highly difficult, especially when several fields are interacting such as QED and QCD in curved space-time. However, for local QFT phenomena (such as quantum states of atoms which only extends within nanometers and the interactions of atoms with bacground fields), we do not expect the space-time metric far way would have any influence on such QFT phenomena at this spot. ased on this observation, next we are going to discuss 3

4 the coordinate transformation that maintains the local metric value, after that we will show how time dilation arises from such coordinate transformations. In order to discuss the time dilation phenomena, we need to now how the cloc wors. The atomic cloc is a time-eeping device by counting the number of cycles of the radiated i t t wave (for wave e, the number of cycles completed is n ) produced by transitions between the first excited state and the ground state of an atom. Currently people use atom Caesium33 and the transition frequency is defined as N Hertz where N , or equivalently to say that one second is defined as N cycles of the radiation produced by the atomic transition. This atomic cloc comprises two major parts: atoms and the radiation field produced by the transitions between the first excited state and the ground state, atoms and the radiation field are all confined within the cloc device. oth the atomic transition and the evolution of the radiation field are physical phenomena that can be described by QFT. Therefore, from now on, we will treat the ticing of atomic cloc as local field phenomena (local means within the size of the cloc device) which can be described by QFT. Now suppose there are two identical atomic clocs and, which are carried by lice and ob, respectively. lice will set the coordinate ( tx, ) to describe the physical phenomena which includes atomic transitions and evolutions of quantum fields inside of cloc, liewise, ob sets the other coordinate ( t, x ). Measured by lice, ob is located at x () t with a velocity () dx () t v t. Given all the above conditions, now we can mae a Motiondt Hypothesis (MH): The motion of a cloc (may be accelerated) does not exert influence on the QFT physical phenomena. In MH, the QFT physical phenomena include quantum states of atoms and the radiation fields generated by the transitions of atoms inside of the cloc. Later on we will see that MH agrees with the cloc hypothesis in special relativity, and MH is indeed a generalization of one of the postulates in special relativity: physics are the same in different inertial reference frames. nd we just generalized this postulate to acceleration frames with a time-dependent relative velocity () dx () t v t. However, such generalization does not include classical physics, we dt only refer to the microscopic physical phenomena arising from QFT. Moreover, MH is not only about the clocs, it is equivalent to state that if we confine an atom in some type of a box, the motion of the box does not affect the state of the atom and the interactions between the atoms with the bacground field. s a matter of fact, we would lie to regard it as an approximation rather than a hypothesis since a super large acceleration of a box may even tear the atoms inside apart and this acceleration will clearly cause effects on the quantum states of atoms as well as the radiation field inside of the box. However, since the cloc hypothesis has been verified with high precision by several experiments under a certain range 4

5 of accelerations [9] [3], as a convention, we still use the word hypothesis rather than approximation. Next we are going to discuss the physical meaning of MH in details. Fig. Setches of ox and ox, each one contains one hydrogen atom. (a) ox is at rest. (b) ox is moving with a velocity v () t measured in lice s ( tx, ) coordinate. If ob sets another coordinate ( t, x ) that maintains the local metric where ox occupies, then he can describe the quantum states of the atom using the same formula that lice applies to describe the states of the atom. s shown in Fig., suppose there are two identical hydrogen atoms and confined in two identical boxes and, respectively. ox is carried by lice who sets coordinate ( tx, ) and ox is carried by ob who sets coordinate ( t, x ). In lice s ( tx, ) coordinate the Hamiltonian of the hydrogen atom is H ( x ), the ground and the first excited states are (, ) tx and (, ) tx, respectively. Similarly, the atom measured by ob has H ( x ), (, ) t x and (, ) t x. Here each of the two boxes can refer to an atomic cloc device. We can denote g ( t, x) as the space-time metric measured in lice s ( tx, ) coordinate of the location where ox occupies. Now let us consider the case that g ( t, x) is independent of time t, and the box has a very small size (it can be comparable with the size of atoms) such that the metric can be approximated as a constant within the size of the box, that is, we can define g g ( t, x) where g is a constant within the size of ox measured in ( tx, ) coordinate (the local space-time curvature within the box is ignored under this approximation). Now if lice wants to describe the states of atoms and the interactions between atoms with the bacground fields inside of ox, she needs to start from the local Lagrangian density of QED in curved space-time written as [4] L ( t, x)[ m0] ( t, x) e0j ( t, x) ( t, x) F ( t, x) F ( t, x) (.7) 4 5

6 where notation in,,, F and j just denote that these quantities are defined related with ox. We also have g, j ( t, x) [, ], F ( t, x) [ ( t, x) ( t, x)] and (it can be replaced by since the matric is approximated as a constant within the size of ox ) denotes the covariant derivative. ll these quantities depend on the local metric value g. From Eq. (.7), the local fields ( tx, ) and ( t, x ) confined in ox obey the equations obtained from Euler-Lagrange equations as F ( t, x) e j ( t, x), (.8a) ( m ) ( t, x) e ( t, x) ( t, x). (.8b) Indeed, the states of atoms and the interactions between atoms with bacground fields in ox can be theoretically derived from Eq. (.7) (here we only consider QED phenomena) with the local metric g. Therefore, lice can obtain the atomic transition frequency between states (, ) tx and (, ) tx as. Now let ob who is carrying ox starts to move with a velocity v () t measured by lice as shown in Fig., we can also as ourselves how to derive the states of atom measured by ob. Therefore, if ob wants to describe the QED phenomena inside of ox, he can set a coordinate system ( t, x ) and the metric where ox occupies can be written as g ( x ). Similarly, he can apply the approximation as g g ( x ) where g is a constant within the size of ox measured in the coordinate ( t, x ). It is worth noting that g is not the metric value measured by lice using coordinate ( tx., ) Then the Lagrangian density that ob uses to describe the QED phenomena inside of ox can be written as L ( t, x)[ m0] ( t, x) e0j ( t, x) ( t, x) F ( t, x) F ( t, x) (.9) 4 where g, F ( t, x) [ ( t, x) ( t, x)], j ( t, x) [, ], and denotes the covariant derivative in ( t, x ) coordinate. Compare Eq. (.9) with Eq. (.7), we notice that these two Lagrangians have the same mathematical structure, in other words, there is no additional effect that is caused by the motion or acceleration of ox added into Eq. (.9). This is because we assumed that the acceleration has no influence on the QFT physical phenomena as stated by MH. For a constant velocity v, we all agree that the Lagrangian in ox is Eq. (.9) by the one of the postulates in special relativity, here we just generalize this postulate to any non-inertial frames (this generalization only applies to QFT physics), later on we will see such generalization will produce the cloc hypothesis. We notice that the QED physics inside of 6

7 ox is completely determined by the local metric g measured by ob, and all the differences between the QED physics in two boxes are caused by the difference between the values of g and g. The field solutions inside of ox can also be given as (, ) (, ), (.0a) F t x e0 j t x ( m ) ( t, x) e ( t, x) ( t, x). (.0b) 0 0 In general, if g g, the physics formulas in ox (such as, j and ) will have different mathematical forms compared with that in ox. s a result, the energy levels of the hydrogen atom may become different in the two boxes, that is where measured in ob s ( t, x ) coordinate is the transition frequency between states (, ) t x and (, ) t x of atom. In order to now the physics formulas in ox without repeating all the calculations done by lice who has obtained physics formulas in ox, ob can set his coordinate ( t, x ) to satisfy g g. (.) In other words, ob can set his coordinate ( t, x ) such that the metric value of ox measured in ( t, x ) coordinate equals to the metric value of ox measured in ( tx, ) coordinate, this can be done by adjusting the value of ( t, x ) (please see the sections IV and V as examples). Therefore, we get, then Eq. (.9) will become the same as Eq. (.7), and the field solutions of Eq. (.0) will have the same mathematical forms compared with the field solutions of Eq. (.8). That is, ( T, X ) ( T, X ) and ( T, X ) ( T, X ) where ( T, X ) are any given time-space values. s a result, all the physical formulas and equations in the two boxes deducted from QED physics will become the same. In ob s ( t, x ) coordinate, the Hamiltonian of the atom is H H, the time-independent ground and the first excited states are and, respectively. The energy levels of the two atoms are also the same, i.e.,. Therefore, the period T / of cloc measured in ob s ( t, x ) coordinate will equal to the period T / of cloc measured in lice s ( tx, ) coordinate. s a result, the time displayed by cloc (recall that the atomic cloc counts the number of cycles completed by the radiation wave) is n t T dn dt and the time displayed by cloc is n t / T. ecause of T T, we have, thus dn dt the relation between t with t indeed shows the correct relation between the displayed-time (which are n and n ) of the two clocs. In order to mae the above argument more clear, let us tae a loo at an example. For Fig. in Minowsi space-time, the metric is given as / 7

8 In ox, the QED Lagrangian density of Eq. (.7) becomes g 0 ( t, x). (.) 0 L ( t, x)[ i ( ie0 ) m0] ( t, x) F ( t, x) F ( t, x). (.3) 4 The Hamiltonian of the hydrogen atom can be deducted from QED physics as [] and the energy levels are H m e x (.4) 4 me n. (.5) n The ground state of hydrogen atom is ( x)exp( it ). If ob who is moving with an arbitrary velocity v() t [ v( t) ] sets a coordinate system ( t, x ) that does not change the Minowsi metric, i.e. g ( t, x) g ( t, x), then he will use the below Lagrangian density to describe the QED phenomena inside of ox as L ( t, x)[ i ( ie0 ) m0] ( t, x) F ( t, x) F ( t, x). (.6) 4 Therefore, the two Lagrangians have the same mathematical structure, the only difference between them is the value of space-time parameters, and the relationship between the values of ( t, x ) and ( tx, ) can be deducted from Eq. (.) (we will do this in following sections). The Hamiltonian of the hydrogen atom written in ( t, x ) coordinate can be deducted from Eq. (.6) as H m e x. (.7) Comparing Eq. (.7) with Eq. (.4), we can see that the atom has a ground state expressed by ( x)exp( it ). In other words, the expression of the quantum states of the atom can be obtained from the corresponding quantum states of the atom by replacing ( tx, ) with ( t, x ). nd the energy levels n of atom are exactly the same as t x by Eq. (.5). Therefore, the frequency measured in coordinate (, ) n expressed of the photon emitted by the atomic transition between the first excited state and the ground state of atom equals to the frequency measured in coordinate ( tx, ) of the emitted photon caused by the same transition of the atom. This is a result of MH and is also a very important conclusion which also holds in curved space-time. Meanwhile, note that in order to mae, this 8

9 must be measured in coordinate ( t, x ) which maintains the local metric value expressed by Eq. (.). In the case of g g, the mathematical form of states of atom will not be the same as states of atom, thus the energy levels measured in the corresponding coordinate will not be the same either. In this paper, the purpose of all the above discussions from Eq. (.7) to Eq. (.7) is to lay a solid theoretical bacground for the reason why we need to find a coordinate ( t, x ) to satisfy g g. In the following sections, we will show how we can use this coordinate ( t, x ) to study the time dilation and Doppler effect in curved space-time. efore that, in order to mae our discussions more clearly, we can loo at another example. There are two identical atomic clocs (carried by lice) and (carried by ob) in Minowsi space-time, ob is moving with a constant speed v measured by lice using ( tx, ) coordinate. Suppose the atomic transition in cloc generates a plane wave i( t x) e measured by lice in coordinate ( tx, ) and the atomic transition in cloc generates another plane wave i( t x ) e measured by ob in coordinate ( t, x ). We now that for a constant speed v the Lorentz transformation is given as t ( t v x), (.8a) x ( x v t) (.8b) which maintains the local metric value with / v, i.e., Eq. (.8) satisfy Eq. (.). s discussed before, the quantum states measured in coordinate ( t, x ) of the atoms in cloc are the same as the quantum states measured in coordinate ( tx, ) of the atoms in cloc. i( t x ) Therefore, the frequency in e ( ) is equal to in i t x e, i.e.,. Thus, the two clocs have the same period T / since cloc is not moving measured in coordinate ( t, x ) and cloc is not moving in coordinate ( tx, ), i.e., dx dx 0. Recall our discussions on how the atomic cloc wors earlier, we can see the number of cycles of radiated wave in cloc is n t/ T while the number of cycles in cloc is n t / T. t last we can calculate the number of seconds that have been recorded by the clocs, which can be given by n / N (displayed by cloc ) and n/ N (displayed by cloc ) with N Due to the fact that T has the same value in cloc and cloc, Eq. (.8a) indeed shows the correct time relation displayed by the two clocs. In the above discussions, we can see that the crucial step is. t this stage, let us see what will happen if we do not maintain the metric value. ssuming the speed v is still a constant, we rewrite the transformation as t ( t v x), (.9a) x ( x v t) (.9b) 9

10 instead of Eq. (.8). We just rescaled the value of the coordinate parameters ( t, x ) by a factor, i.e., we multiply one more factor measured in ( t, x ) becomes e i( t x ) e i( t x ) v in front of ( t, x) / in ( t, x ), i.e., e e i( t x ) i( tx ). Therefore,. Thus, we get / /, the wave frequency radiated by atomic transitions in cloc measured in ( t, x ) is not the same as that in cloc measured in ( tx,, ) therefore, the periods will not equal. Then Eq. (.9a) becomes incorrect since it cannot give the correct time relation shown by the two clocs. Indeed, we can do something more to give the correct time relation: the period T / T and the number of seconds that have passed is expressed as t / ( T N) t / ( TN ). Clearly, we do not want to mae troubles lie this, what we want is Eq. (.8a) since the time relation between the two clocs can be, straightforwardly, read out without any more calculations. Indeed, the reason why the Lorentz transformation is the correct transformation is that it maintains the metric value, i.e., it satisfies Eq. (.). This is just one example to show why we do not want to change the metric by rescaling, otherwise, other mode parameters (such as, ) need to change accordingly, and the physical meanings of the value of parameters, such as ( t, x,, ), do not remain the same as what they stand for in other coordinates. That is, if t represents the physical time displayed by the cloc, then the value of t given by Eq. (.9a) does not represent the physical time displayed by cloc and does not represent the frequency of the mode either. In conclusion, since the frequency of the moving cloc measured in coordinate ( tx, ) may not equal to the frequency of the stationary cloc measured in the same coordinate ( tx,, ) as a result, lice will see the cloc runs slower or faster than cloc. However, ob who carries cloc can set a new coordinate ( t, x ) satisfying Eq. (.) such that the frequency of cloc measured in coordinate ( t, x ) equals to of cloc measured in coordinate ( tx., ) In other words, the period of cloc measured in the coordinate ( t, x ) will equal to that of cloc measured in the coordinate ( tx., ) s a result, the relationship between t and t will be the correct relationship of the time displayed by the two clocs. Moreover, we already see that by setting a new coordinate ( t, x ), the mathematical forms of the physical formulas obtained in box can be maintained. Therefore, we do not have to repeat all the calculations to obtain the physical formulas in box, the only difference of the physical formulas between the two boxes is the value of space-time parameters ( t t, x x), and the time difference between t and t will be shown by the two clocs. This is an advantage of setting a new coordinate ( t, x ) that maintains the local metric value, soon it also will be shown as a powerful tool to study the Doppler effect in following sections III, IV, V. III. Time Dilation and Doppler Effect with Time-Dependent Velocity in Minowsi Space-Time The two dimensional Minowsi space-time has the metric 0

11 g 0 ( x) 0 (3.) 0 with the defined symbol x t and x x. Suppose that we have two identical atomic clocs and rest at x 0 which are carried by lice and ob, respectively. t time t 0, ob starts to move away from lice and he sets a new coordinate system ( t, x ) to measure the physical events observed by himself. Now we can write the coordinate transformation that maintains the metric value as dt cosh f ( x) dt sinh f ( x) dx, (3.a) dx sinh f ( x) dt cosh f ( x) dx (3.b) in which f( x ) can be any arbitrary functions of t and x. Similar with the Lorentz transformation, Eq. (3.) can be used to transform ( dt, dx ) between any two nearby physical events measured by lice into ( dt, dx ) between the same two physical events measured by ob. The velocity of ob v() t measured by lice can be obtained by demanding dx 0 in Eq. (3.b), we thus get the velocity of ob as ( ) dx v tanh ( ) t f x. (3.3) dt The first order partial derivatives can be obtained from Eq. (3.) as t x cosh f( x), (3.4a) t x t x sinh f( x). (3.4b) x t The metric g ( x ) in the ( t, x ) coordinate can be given by Eq. (.3). We can verify that g ( x) g ( x) shown as Eq. (3.), that is, the metric is unchanged after the transformation. ased on the argument from Eq. (.7) to Eq. (.7) for MH, since the value of the metric are the same at the spatial location where the two clocs occupy, the frequency of the emissions generated by the atomic transitions inside of the clocs will be the same and the two clocs will thus have the same period, therefore, the relationship between dt and dt shown by Eq. (3.a) is indeed the correct relationship for the displayed time of the two clocs. Note that different transformation function f( x ) in Eq. (3.) corresponds to different v () t, f( x ) for the Lorentz transformation with constant v can be wored out. The transformed coordinate x ( x) as functions of x can be obtained by integration of Eq. (3.) along a path, that is, path (3.5a) path t ( x ) dt cosh f ( x ) dt sinh f ( x ) dx,

12 path (3.5b) path x ( x ) dx sinh f ( x ) dt cosh f ( x ) dx. However, in order to get an integration-path-independent (IPI) function x ( x) (the integration value only depends on two end points of a path), the second order partial derivatives need to satisfy the following relationships y substituting Eq. (3.6) into Eq. (3.4), we get t t xt t x, (3.6a) x x xt t x. (3.6b) cosh f ( x) sinh f ( x), (3.7a) x t cosh f ( x) sinh f ( x). (3.7b) t x One solution can be given as f( x) in which is a constant. We can recognize that this solution corresponds to the Lorentz transformation with a constant velocity, the velocity of ob is v tanh. Therefore, the Lorentz transformation is well-nown as an isometry of Minowsi metric. We may also wonder what happens if the velocity of ob v tanh f ( x) is not a constant. Since for a general function f( x ) that does not satisfy Eq. (3.7), we cannot get the coordinates x ( x) as IPI functions of x. Nevertheless, we can still integrate along a certain path to obtain the integration-path-dependent (IPD) functions x ( x). Next we will loo at one example to illustrate this point and later on we will see how it is related to the twin paradox. s shown in Fig., the coordinate system ( tx, ) is set by lice who carries the cloc. t t 0, ob who carries the cloc starts to move with a constant positive acceleration d x a a ; at time t t (note that this t t is measured by cloc and at ), ob dt starts to decelerate with a a ; at t t, ob would start to move bac with a a ; finally, at t 3t ob starts to decelerate with a a and he meets lice at t 4t. s can be seen from Fig., during the time period 0t 4t, lice follows the path I and ob follows the path II, they meet at ( t 4 t, x 0).

13 4t 3t t Path I Path II t x Fig.. Setch of the two Paths measured in lice s coordinate ( tx, ). The red line indicates Path I that lice follows and the blue curve is Path II that ob travels, ob turns bac at t t and they meet at ( t 4 t, x 0). In order to calculate the time difference between the two clocs when they meet by using Eq. (3.5a), we can treat the ticing of cloc as physical events and all these events occur along path II. Therefore, we can integrate along path II to obtain the total time of the events measured by lice in coordinate ( tx,, ) and the total time of the events measured by ob is defined as t ( x) in coordinate ( t, x ), this t ( x) is also what the cloc shows when they II II meet. Note that because Eq. (3.) is the coordinate transformation of the same events observed by different observers, we cannot just perform the integrations along path I for cloc and along path II for cloc, and then compare the time difference between the two different paths. For lice in coordinate ( tx,, ) if we integrate along the path II, we get t dt, (3.8a) path II Note that x dx. (3.8b) path II dt is the total time of the events (ticing of cloc ) occurring along path II path II measured by lice. Since path I combined with path II forms a loop, we also now that in coordinate ( tx,, ) the integration is path-independent, that is, Note that dt 4 path II dt t path I, (3.9a) dx 0. path II dx (3.9b) path I dt in Eq. (3.9a) is the time that cloc shows when they meet. Therefore, path I cloc whose motion follows the path I can also be used to measure the total time of the 3

14 events (ticing of cloc ) occurring along path II. For coordinate ( t, x ), if we integrate along path II, according to Eq. (3.5), we obtain (see ppendix ) t ( x) cosh f ( x) dt sinh f ( x) dx arcsin a t t a t, II path II (3.0a) a x ( x II ) sinh f ( x ) dt cosh f ( x ) dx 0. (3.0b) path II Since t II( x) is the time shown by cloc when they meet and we can see that t ( ) 4 II x t, therefore, the travelling cloc runs slower. t this stage, we notice that the results of Eq. (3.8) and Eq. (3.0) are both integrations of path II observed by lice and ob, respectively. y symmetry, we can as ourselves: what if we treat the ticing of cloc as the physical events that we want to measure? Then we need to integrate along path I, Eq. (3.8) remain the same since the result is path-independent as shown by Eq. (3.9). For coordinate ( t, x ) set by ob, he obtains (see ppendix ) 4 t( x) cosh f ( x) dt sinh f ( x) dx arcsin a t, I path I (3.a) a xi ( x) sinh f ( x) dt cosh f ( x) dx 0. (3.b) path I We can see t( x) t ( x), this is as expected actually since we already discussed earlier that I II ( t, x ) will be IPD functions if v () t is not a constant. Therefore, cloc whose motion follows the path II cannot be used to measure the total time of the physical events (ticing of cloc ) occurring along path I since the time shown by cloc is t ( x) when they meet at t 4t. This fact can be used to explain the twin-paradox: we can only integrate along path II (followed by the moving twin) to calculate the time difference between the twins. Next we can tae a loo at another example. Two identical atomic clocs are carried by lice and ob, respectively. They are initially at rest and lice sets a coordinate system ( tx, ) while ob sets another coordinate ( t, x ) which maintains the local metric given by Eq. (3.). t t 0, ob starts to move with constant acceleration measured by lice s ( tx, ) coordinate satisfying f( x) In Eq. (3.), is the proper time measured by lice and The velocity of ob measured by lice can be expressed as II. (3.) t( ) sinh( ), (3.3a) x( ) cosh( ). (3.3b) 4

15 dx v( t) tanh( ) tanh[arcsinh( t)] dt t t. (3.4) From Eq. (3.a), the time difference between the two clocs can be given as dt [cosh f ( x) v( t)sinh f ( x)] dt dt v( t) dt (3.5) cosh which is the time dilation with time-dependent velocity in Minowsi space-time, this result is consistent with the cloc hypothesis. The scalar field ( tx, ) that satisfies Eq. (.) with Minowsi metric can be expressed as the superposition of plane waves (3.6) i( tx) i( tx) ( t, x) [ ae ae ] with. In ob s coordinate ( t, x ) that maintains the local metric value, his Lagrangian describing QED physics is given by Eq. (.5), therefore the modes can be obtained without any more calculations. s we argued in section II, this is a big advantage to maintain the local metric value, that is, the mathematical form for the physical formulas in coordinate ( t, x ) is exactly the same as that in coordinate ( tx,, ) what left for ob to do is just replacing ( tx, ) in Eq. (3.6) with ( t, x ) and obtain the modes in coordinate ( t, x ) as (3.7) i( t x ) i( t x ) ( t, x ) [ ae ae ] with. Now suppose that an atom carried by lice emits a plane wave ob can decompose this plane wave in his coordinate ( t, x ) as e i( t x) to ob, e [ e e ] (3.8) i( tx) i( t x ) i( t x ) in which and are parameters to be fixed. For a trial solution, we require 0, and 0 only if equals one certain value to be fixed, then we get i( t x) i( t x ) e e t x t x n. (3.9) Therefore, after an infinitesimal time interval dt we have In ob s co-moving frame ( t, x ), dx 0 and we obtain dt dx dt dx. (3.0) dt dx dt [ v( t)] v( t) [ v( t)] dt dt v ( t) v( t) (3.) in which dt is given by Eq. (3.5) and v() t is given by Eq. (3.4). Therefore, the Dopplershift with a time-dependent velocity is determined by the velocity at that instant. dt This 5

16 conclusion is obtained in accordance with the cloc hypothesis in special relativity. The relationship between the creation and annihilation operators in two coordinate systems can be given as a a, (3.a) a a (3.b) with v( t) v ( t). Therefore, for Eq. (3.6) and Eq. (3.7) we can also verify ( t, x) ( t, x) (3.3) which agrees with Eq. (.6). IV. Time Dilation and Doppler Effect with Time-Dependent Velocity in Schwarzschild Space-Time For simplicity, we will only consider the coordinate transformations in ( tr, ) directions for the Schwarzschild metric, therefore, the two dimensional Schwarzschild metric can be described by g M ( ) 0 r ( t, r) M 0 ( ) r. (4.) Similarly as Minowsi space-time, if we want to find another coordinate system ( t, r ) that maintains the metric form, the metric g ( t, r ) is given as M ( ) 0 r g ( t, r ) g ( t, r). (4.) M 0 ( ) r The metric transformation between different coordinate systems can be expressed by Therefore, we obtain g x x g. (4.3) x x g 00 g M M t M r r r t r t 0 0 ( ) ( )( ) ( ) ( ), M t t M r r g r t r r t r 0 ( ) ( ), (4.4a) (4.4b) 6

17 g M M t M r r r r r r ( ) ( )( ) ( ) ( ). (4.4c) Similarly as Minowsi space-time, if we want to find an IPI solution for ( t, r ) as functions of ( tr,, ) the second order partial derivatives need to satisfy the following equations t t rt t r, (4.5a) r r rt t r. (4.5b) In this section, we will not go further to find such IPI coordinate ( t, r ), but we assume that the coordinate ( t, r ) under study in this section is IPD. Meanwhile, we follow the convention that the coordinate system ( tr, ) is set at spatial infinity r. That is, if lice who carries cloc is rest at r, she can eep record of every physical event that occurs at location ( tr,, ) where t is the time displayed by cloc and r is the spatial distance from the gravitational source (note that r is not the distance from lice). Now we consider two observers lice who carries the atomic cloc and ob who carries an identical cloc. lice who sets the coordinate system ( tr, ) is rest at r and r is the location of cloc, ob is located at r r and this r is also measured in lice s coordinate system. ob himself sets another coordinate system ( t, r ). Note that in general, ob cannot find another IPI coordinate system given by Eq. (4.) even if he is rest at r r, this is because Eq. (4.5) do not have a general solution for the spatial translation from r dr to r r. t time t 0 and t 0, ob starts to move with velocity v() t. In order to dt compare the two clocs, based on MH and the discussions from Eq. (.7) to Eq. (.7), we must use the same metric value to describe cloc and cloc, that is, the components of the metric where cloc occupies measured by ob equals to the components of metric where cloc occupies measured by lice. Therefore, we have g ( t, r ) g ( t, r ) (4.6) where g ( t, r ) is given by Eq. (4.) with r replaced by r and (, ) g t r is the metric written in ob s coordinate ( t, r ), r is the spatial value of ob s location measured in his coordinate ( t, r ). Due to r, we get 0 g ( t, r ) g ( t, r) 0. (4.7) So the metric value at the position of cloc measured in lice s coordinate ( tr, ) equals to the metric value at the position of cloc measured in ob s coordinate ( t, r ). Therefore, 7

18 the frequency of the wave emitted by the atom inside of cloc measured in ob s coordinate ( t, r ) equals to the transition frequency of cloc measured by lice s coordinate ( tr., ) s a result, the period of cloc measured in the coordinate ( t, r ) will equal to that of cloc measured in the coordinate ( tr,, ) thus the time displayed by cloc can be given by t in Eq. (4.8). For the ticing event of cloc, according to Eq. (4.7), Eq. (4.4) can be rewritten as M t r r () t t t [ ] ( ) ( ), t t r r 0, t r t r M t r r () t r r [ ] ( ) ( ). (4.8a) (4.8b) (4.8c) In practice, ob does not have to specify the metric values at other locations with r r, since Eq. (4.7) is all the information we need to now about the local experiment performed at location r. However, if ob wants to set the full coordinate system ( t, r ) that extends to the entire space-time with the metric Eq. (4.), he needs to demand r r based on Eq. (4.6) and Eq. (4.), that is, r. For convenience only, let us just assume that ob himself indeed sets such coordinate system with the metric given by Eq. (4.) (note that such coordinate system would be IPD). In Eq. (4.8), r () t is the spatial location of cloc measured by lice and it depends on t since ob is moving with velocity can obtain the time relation between the two clocs as (see ppendix ) () dr () t v t. We dt dt M [ ] v( t) r () t M r () t dt. (4.9) Upon integration along the path that cloc travels, we get t and t as the times displayed by cloc and cloc, respectively. s we can see, the time difference is determined by the location and the velocity of ob measured by lice at that instant, this result agrees with the cloc hypothesis. 8

19 Fig. 3 Setch of the Doppler effect near the gravitational source. lice who is at spatial infinity sets a coordinate system ( tr, ) and ob who is located at r () t measured by lice sets a coordinate system ( t, r ). lice sends a wave with frequency measured in coordinate ( tr, ) to ob, the frequency of this wave becomes measured in ob s ( t, r ) coordinate. The frequency change from to is due to the switching of the reference frames from ( tr, ) to ( t, r ). The mode solution of Eq. (.) with Schwarzschild metric can be obtained as [5] [7] it, l, m, l l, m, l, m, l l, m it (4.0),, lm ( x) [ a R ( r) Y (, ) e a R ( r) Y (, ) e ] in which Ylm, (, ) is the essel function with mode number lm,, R, l () r is a function of r that satisfies a second order differential equation and a,, lm is the annihilation operator for mode,, lm. Since we are dealing with two-dimensional coordinate transformation in this paper, we will drop symbols lm, and Y, (, ) for all subsequent discussions. Then Eq. (4.0) can be written as lm. (4.) (, ) [ ( ) i t it t r a R r e a R ( r) e ] ccording to ob s coordinate ( t, r ) with the metric given by Eq. (4.), Eq. (4.) can be written as. (4.) it it ( t, r ) [ a R ( r ) e ar ( r ) e ] i t s shown by Fig. 3, lice sends a wave R () r e with frequency to ob, then ob i t decomposes this wave R () r e into the modes with positive and negative frequencies in coordinate ( t, r ) as i t i t i t R ( r) e [ R ( r ) e R ( r ) e ] (4.3) in which and are parameters to be fixed. When the wave reaches ob at location r () t, this location in ob s ( t, r ) coordinate is r. Since r, the metric becomes 9

20 i t Minowsi shown by Eq. (4.7), and thus the wave R ( r ) e in Eq. (4.3) becomes with. Note that since the wave is traveling with negative velocity, we write i( t r ) e the plane wave as i( t r ) e instead of i( t r ) e. Therefore, we have it i( tr ) i( tr ) ( ) [ ] R r e e e. (4.4) fter an infinitesimal time interval dt, along ob s co-moving frame dr 0, we have R r v dt e e e. (4.5) i ( tdt ) i( tdtr ) i( tdtr ) ( ) [ ] Expanding Eq. (4.5) to first order, we obtain it dr () r dt e v i e R r i e e (4.6) dt it i( tr ) i( tr ) rr ( ) [ ] dr in which dt is given by Eq. (4.9). Note that unlie Eq. (3.8) in Minowsi space-time, dt in Eq. (4.4) may not, in general, all vanish, it means that a mode with only positive frequency measured by lice may become modes involving negative frequencies measured ir de by ob. For Minowsi space-time, always vanish since the plane wave is a ir e dr dr ( r ) pure imaginary number ; on the contrary, in Eq. (4.6) is not always an R ( r ) dr imaginary number, thus we cannot ascertain to get a real solution if all equal to zero, usually this phenomena is interpreted as particle-creation in curved space-time [8] [30]. In this paper, we will not go further to solve Eq. (4.4) and Eq. (4.6) to obtain and, instead, we will loo at some simpler cases. First let us loo at a case with v 0, Eq. (4.6) becomes it i( tr ) i( tr ) ( ) [ ] dt ir r e i e e. (4.7) dt We apply a trial solution with 0 (all the vanish) and is non-zero only at one certain to be fixed, thus combine Eq. (4.7) with Eq. (4.4), we have M r (4.8) which is the gravitational blue-shift measured by ob who is rest at r. In this case, the relationship between a in Eq. (4.) and a in Eq. (4.) can be given as 0

21 a a (4.9) i t with given by Eq. (4.8). Note that the frequency of the wave R () r e does not change during the propagation measured in lice s coordinate ( tr, ), the frequency shift given by Eq. (4.8) is due to the switching of the reference frame from ( tr, ) to ( t, r ), this interpretation agrees with Einstein s equivalence principle: the changing of the photon frequency in the gravitational field is equivalent with the changing frequency in accelerating reference frames [3]. One may find that such interpretation contradicts with the energyconservation argument of photons (the energy of the photon together with the gravitational potential is conserved, since the potential energy is altered at different locations, the photon energy will be shifted accordingly). However, this energy-conservation argument is purely classical and it originates from the observation of macroscopic objects falling in the gravitational field. We can clarify this issue here by comparing the behavior of quantum particles with the behavior of classical charged objects moving in Coulomb field. In classical physics, a negatively charged object accelerates due to the attraction of a positively charged object, and its inetic energy gains with the decreasing distance from the positively charged object. However, in quantum physical model of the Hydrogen atom with an electron interacting with nucleus by Coulomb force, for a given eigenstate of electron with fixed energy, different spatial values (the distance from the nucleus) of electron correspond to the same energy. Similarly, Eq. (4.0) represents the eigenstates (or eigen-functions) of Eq. (.) with given quantum numbers (,, lm) and these quantum numbers do not change during the wave propagation measured in the same coordinate system. For the second simpler case with v 0, suppose that the gravitational field is wea and i r we can tae the flat-limit approximation R ( r ) e, then apply a trial solution with 0 and is non-zero only at a certain to be fixed, after substituting i r R ( r ) e into Eq. (4.4) and Eq. (4.6) we get dt [ v( t)] dt (4.0) in which dt is given by Eq. (4.9). Note that Eq. (4.0) is valid under the condition that the dt strength of the gravitational field is wea. For a strong gravitational field (such as near the blac-holes), ob will measure a range of spectrum given by Eq. (4.4) instead of a single frequency given by Eq. (4.0). ssuming lice carries an atom and ob carries an identical atom. lice s atom emits a i t wave R () r e which has the frequency measured in lice s coordinate ( tr, ) to ob. nd the frequency of the wave measured by ob is expressed as Eq. (4.0). ccording to MH, we now that the energy levels of ob s atom measured in coordinate ( t, r ) are exactly the same as lice s atom measured in coordinate ( tr,, ) where the two coordinates are related by Eq. (4.8). Therefore, ob s atom will emit or absorb a wave with the same frequency

22 measured in ( t, r ) coordinate. In order to mae ob s atom absorb this wave sent by lice, we require in Eq. (4.0), then the velocity v can be obtained as q q q q v q ( ) (4.) M in which q r absorption peas.. Therefore, the velocity has two solutions which correspond to two V. Time Dilation and Doppler Effect with Time-Dependent Velocity in FRW Space- Time The FRW (Friedmann-Robertson-Waler) metric in two dimensions can be written as 0 g ( t, r) e 0 Ht r (5.) in which H represents the Hubble constant measured in coordinate ( tr, ) and can tae values as (open universe), 0 (flat universe) or (closed universe), respectively. Similarly, in coordinate ( t, r ) we have the metric g ( t, r ) written in the same form as 0 g ( t, r) Ht e 0 r (5.) in which H represents the Hubble constant measured in coordinate ( t, r ). t this stage, we do not assume H H for any coordinate system ( t, r ). ccording to Eq. (.3) the metric transformations between the two coordinates can be given as g g t e r ( ) ( ), t r t Ht 00 0, t r r t r Ht t t e r r 0 g0 g e t e r r r r r Ht H t ( ) ( ) (5.3a) (5.3b) (5.3c) and Eq. (5.3) has an IPI solution which is given in ppendix C. Now we consider two observers in flat universe ( 0 ), lice who carries the atomic cloc sets the coordinate ( tr, ), and ob who carries an identical cloc sets the

23 dr coordinate ( t, r ). t time t 0 and t 0, ob starts to move with velocity v() t. dt Next we can apply the argument below MH, that is, the local metric value for the cloc measured by ob equals to the local metric value for cloc measured by lice, then ob s coordinate system ( t, r ) needs to satisfy g ( t, r) g ( t, r ) (5.4) in which g ( t, r ) measured by ob is the local metric where cloc locates and g ( t, r ) measured by lice is the local metric where cloc locates. gain, for local experiments performed at r, ob does not need to specify metric values at other locations, the relationship of Eq. (5.4) is all we need to now. For convenience in the following discussions, assuming that ob sets a coordinate system given by Eq. (5.) which extends throughout the entire space-time, note that in this scenario, ( t, r ) will be IPD functions of ( tr, ) for nonzero velocity v () t. Therefore, we have e flat universe, Eq. (5.3) become H t Ht e and for the ticing events of cloc in t t r t Ht ( ) e ( ), t t r r t r t r Ht 0 e, e t r ( ) e ( ). r r Ht H t (5.5a) (5.5b) (5.5c) For the first order differential relations, we have If we treat the ticing of cloc as physical events, we have t t dt dt dr, (5.6a) t r r r dr dt dr. (5.6b) t r r r r r dr ( v ) dt 0 v 0. (5.7) t r t r Combine Eq. (5.7) with Eq. (5.5), we can obtain the time dilation in expanding flat universe as (see ppendix C) dt e v t dt (5.8) Ht ( ) in which t is given by cloc and t is given by cloc. For the mode solution of Eq. (.) with FRW metric in flat universe, we obtain the equation 3

24 Ht 3 H e ( ). (5.9) t t x y z We can write the mode solution of Eq. (5.9) in ( tx, ) coordinate as Then the mode is given by. (5.0) ( t, x) [ a f ( t, x) a f ( t, x)] f t x Ve e h t in which 3Ht / i x (, ) ( ) ( ) 3Ht Ve can be regarded as the physical volume of the cube on which the periodic boundary condition is imposed. We mae the adiabatic approximation for h () t as [9], [3] t / ( ) ( ) exp[ ( ) ] h t t i t dt (5.) in which the frequency is given by Ht () t e. Then f ( t, x ) in Eq. (5.0) can be given by / () t f ( t, x) ( V ) exp[ i x i Ht]. (5.) H Similarly, in ob s coordinate ( t, x ) with the metric given by Eq. (5.), we get ( t, x ) [ a f ( t, x ) a f ( t, x )]. (5.3) The wave f ( t, x ) can be decomposed into modes with positive and negative frequencies in ( t, x ) coordinate as f ( t, x) [ (, ) (, )] f t x f t x. (5.4) Now suppose ob is moving in x direction and lice sends a wave f ( t, x ) to ob with y 0, when the wave reaches ob at the location x () t measured by lice, we have z f[ t, x( t)] [ f( t, x ) f( t, x )] (5.5) in which x is the spatial value of ob measured in coordinate ( t, x ). fter an infinitesimal time interval dt, we get f[ t dt, x( t) v( t) dt] [ f( t dt, x ) f( t dt, x )]. (5.6) Expanding Eq. (5.6) to first order, we obtain f f f f f[ t, x( t)] [ v( t)] dt [ ( f dt) ( f dt)] t x t t (5.7) 4