relativity and time dilation pdf

Relativity And Time Dilation Pdf

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Time Dilation and Length Contraction

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In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript. At the intersection of quantum theory and relativity lies the possibility of a clock experiencing a superposition of proper times. We consider quantum clocks constructed from the internal degrees of relativistic particles that move through curved spacetime.

The probability that one clock reads a given proper time conditioned on another clock reading a different proper time is derived. From this conditional probability distribution, it is shown that when the center-of-mass of these clocks move in localized momentum wave packets they observe classical time dilation.

We then illustrate a quantum correction to the time dilation observed by a clock moving in a superposition of localized momentum wave packets that has the potential to be observed in experiment. The time of an event taking place at a point is then defined as the time shown on the clock simultaneous with the event. Bridgman highlighted the significance of this definition of time 2 :.

Such descriptions of quantum clocks have been developed in the context of quantum metrology 4 , 5 , 6 , 7. In this regard, time observables are identified with positive-operator valued measures POVMs that transform covariantly with respect to the group of time translations acting on the employed clock system 8 , 9. This covariance property ensures that these time observables give the optimal estimate of the time experienced by the clock; that is, they saturate the Cramer—Rao bound Given that clocks are ultimately quantum systems, they too are subject to the superposition principle.

In a relativistic context, this leads to the possibility of clocks experiencing a superposition of proper times. Such scenarios have been investigated in the context of relativistic clock interferometry 18 , in which two branches of a matter-wave interferometer experience different proper times on account of either special or general relativistic time dilation 19 , 20 , 21 , 22 , 23 , 24 , 25 , Such a setup leads to a signature of matter experiencing a superposition of proper times through a decrease in interferometric visibility.

Other work has focused on quantum variants of the twin-paradox 27 , 28 and exhibiting nonclassical effects in relativistic scenarios 29 , 30 , 31 , 32 , 33 , 34 , 35 , We introduce a proper time observable defined as a covariant POVM on the internal degrees of freedom of a relativistic particle moving through curved spacetime. This allows us to consider two relativistic quantum clocks, A and B , and construct the probability that A reads a particular proper time conditioned on B reading a different proper time.

To compute this probability distribution we extend the Page—Wootters approach 37 , 38 to relational quantum dynamics to the case of a relativistic particle with internal degrees of freedom. We then consider two clocks prepared in localized momentum wave packets and demonstrate that they observe on average classical time dilation in accordance with special relativity.

We then illustrate a quantum time dilation effect that occurs when one clock moves in a superposition of two localized momentum wave packets: On average, the proper time of a clock moving in a coherent superposition of momenta is distinct from that of the corresponding classical mixture, see Fig.

We describe the average quantum correction to the classical time dilation observed by such a superposed clock. Two clocks are depicted as moving in Minkowski space. Clock A experiences a quantum contribution to the time dilation it observes relative to clock B due to its nonclassical state of motion.

In adhering to the operational view espoused earlier, we employ the Page—Wootters formulation of quantum dynamics in which time enters like any other quantum observable. This Hilbert space is defined as the Cauchy completion of the set of solutions to the constraint equation:.

What makes T C a time observable is that the effect operators transform covariantly with respect to the group generated by H C 5 , 6 , 7 , 9. It then follows from Eqs. When the relativistic particle has positive energy, the physical state satisfies Eq. With this identification, the dynamics implied by the Page—Wootters formalism is in agreement with previous descriptions of a relativistic particle with internal degrees of freedom 19 , 31 , We now make precise how the internal degrees of freedom of the relativistic particle introduced in the previous section constitute a clock by introducing a proper time observable.

We define a clock to be the quadruple:. The physical significance of the covariance condition in Eq. The following two physical properties of such a time observable follow:. Taking this notion of a clock and applying it to the relativistic particle model introduced in the previous section, we may construct a proper time observable that transforms covariantly with respect to the internal clock Hamiltonian H clock of the particle.

For an unbiased estimator, like the proper time observable T clock introduced in the previous section, the Helstrom—Holevo lower bound 4 , 5 places the fundamental limit on the variance of the proper time measured by the clock.

From Eq. This inequality gives the ultimate bound on the precision of any measurement of proper time. It turns out that covariant observables satisfying Eq.

Such a proper time observable T clock is optimal in the sense that it maximizes the so-called Fisher information 10 , which quantifies how well two slightly different values of proper time can be distinguished given a particular quantum measurement. For the effect operatrors E clock and the fiducial state in Eq. Let us point out a connection between our above construction of a proper time observable and quantum speed limits.

That is. We remark that in this construction both proper time and mass are treated as genuine quantum observables; the former as a covariant POVM T clock and the latter as a self-adjoint operator M clock. Such a formulation of proper time and mass in the regime of relativistic quantum mechanics has been argued as necessary by Greenberger 41 , To evaluate this probability distribution note that the clock states defined below Eq.

We employ such clocks for their mathematical simplicity in illustrating the quantum time dilation effect, however we stress that for any covariant time observable, on account of Eq. By substituting Eq. It follows that the observed average time dilation between two such clocks is. Therefore, upon comparison with Eq. It is natural to now ask: Does a quantum contribution to the time dilation observed by these clocks arise if the center-of-mass of one of the clocks moves in a superposition of momenta?

Using Eq. This is expected given that in these cases the center-of-mass of the clock particle is no longer a superposition of momentum wave packets; see Eq. It is seen that quantum time dilation can be either positive or negative, corresponding to increasing or decreasing the total time dilation experienced by the clock compared to an equivalent clock moving in a classical mixture of the same momenta wave packets. We note that classical special relativistic time dilation has been observed with atomic clocks moving at these velocities 44 , 45 , and perhaps the momentum superposition can be prepared by a momentum beam splitter realized using coherent momentum exchange between atoms and light 46 , We note that the required coherence time is comparable to coherence times of the superpositions created in the experiments of Kasevich et al.

Concretely, one might imagine observing quantum time dilation in a spectroscopic experiment using the width of an emission line, which is inversely proportional to the lifetime of the associated excited state, as a quantum clock.

Indeed, it has recently been shown that the lifetime of an excited hydrogen-like atom moving in a superposition of relativistic momenta experiences quantum time dilation in accordance with Eq.

Alternatively, Bushev et al. Similar remarks apply to the ion trap atomic clock discussed in the ref. We considered the internal degrees of freedom of relativistic particles to function as clocks that track their proper time.

It was shown that the Helstrom—Holevo lower bound 4 , 5 implies a time-energy uncertainty relation between the proper time read by such a clock and a measurement of its energy. From this relation, we derived an uncertainty relation between proper time and mass, which provided the ultimate bound on the precision of any measurement of proper time.

This yielded a consistent treatment of mass and proper time as quantum observables related by an uncertainty relation, resolving past issues with such an approach 41 , The approach adopted here differs in that we construct a proper time observable T clock in terms of a covariant POVM rather than a self-adjoint operator.

We then specialized to two such clock particles moving through Minkowski space and evaluated the leading-order relativistic correction to this conditional probability distribution. It was shown that on average these quantum clocks measure a time dilation consistent with special relativity when the state of their center-of-mass is localized in momentum space.

However, when the state of their center-of-mass is in a superposition of such localized momentum states, we demonstrated that a quantum time dilation effect occurs. We exhibited how this quantum time dilation depends on the parameters defining the momentum superposition and gave an order of magnitude estimate for the size of this effect.

We conclude that quantum time dilation may be observable with present day technology, but note that the experimental feasibility of observing this effect remains to be explored. It should be noted that the conditional probability distribution in Eq. It thus remains to investigate the effect of other nonclassical features of the clock particles such as shared entanglement among the clocks and spatial superpositions.

In regard to the latter, it will be interesting to recover previous relativistic time dilation effects in quantum systems related to particles prepared in spatial superpositions and each branch in the superposition experiencing a different proper time due to gravitational time dilation 18 , 19 , 20 , 34 , We emphasize that the quantum time dilation effect described here differs from these results in that it is a consequence of a momentum superposition rather than gravitational time dilation.

Nonetheless, it will be interesting to examine such gravitational time dilation effects in the framework developed above and make connections with previous literature on quantum aspects of the equivalence principle 51 , 52 , We also note that while we exhibited the quantum time dilation effect for a specific clock model, based on the preceding analysis in terms covariant time observables it is expected that any clock will witness quantum time dilation.

Given this, it will be fruitful to examine our results in relation to other models of quantum clocks that have been considered 3 , 32 , 54 , 55 , 56 , 57 and establish whether quantum time dilation is universal, affecting all clocks in the same way, like its classical counterpart.

Another avenue of exploration is the construction of relativistic quantum reference frames from the relativistic clock particles considered here 15 , 58 , 59 , 60 , In particular, one might define relational coordinates with respect to a reference particle and examine the corresponding relational quantum theory and the possibility of changing between different reference frames 62 , 63 , 64 , 65 , 66 , 67 , 68 , Related is the perspective-neutral interpretation of the Hamiltonian constraint in terms of which a formalism for changing clock reference systems has recently been developed 35 , 70 , 71 , We present a Hamiltonian constraint formulation of N relativistic particles with internal degrees of freedom.

A complementary approach has been taken in ref. Note that Eq. In terms of the parameters t n the action takes the form. Expressed in terms of the single parameter t 73 , the action in Eq. This Lagrangian treats the temporal, spatial, and internal degrees of freedom as dynamical variables on equal footing described by an extended phase space interpreted as the description of the particles with respect to an inertial observer.

The Hamiltonian associated with L t is constructed by a Legendre transform of Eq. Upon substituting Eq. Furthermore, using Eq. In Eq. The constraint functions in Eqs. The quantum analog of the constraints is to demand that physical states of the theory are annihilated by these constraint operators.

In this subsection, we recover the standard formulation of relativistic quantum mechanics with respect to a center-of-mass coordinate time using the Page—Wootters formalism. Equation 36 defines the conditional state. This implies that the physical states are normalized with respect to the inner product Given this observation and the definition of the conditional state in Eq.

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Formulate conclusions of the theory of special relativity, noting the assumptions that were made in deriving it. According to the theory of special relativity, it is impossible to say in an absolute sense whether two distinct events occur at the same time if those events are separated in space, such as a car crash in London and another in New York. The question of whether the events are simultaneous is relative: in some reference frames the two accidents may happen at the same time, in other frames in a different state of motion relative to the events the crash in London may occur first, and still in other frames, the New York crash may occur first. If we imagine one reference frame assigns precisely the same time to two events that are at different points in space, a reference frame that is moving relative to the first will generally assign different times to the two events. This is illustrated in the ladder paradox, a thought experiment which uses the example of a ladder moving at high speed through a garage.

Time and the Metaphysics of Relativity pp Cite as. The relativity of simultaneity and the relativity of length lead naturally to the strangest consequences of relativity theory: time dilation and length contraction. Time dilation means that relative to a clock taken to be at rest, a moving clock runs slow, so that relative to the moving clock the amount of time recorded by the clock at rest expands or dilates. Let us suppose that we have two clocks A and B in motion relative to each other Figure 3. Unable to display preview.

In special relativity, an observer in inertial i. A second inertial observer, who is in relative motion with respect to the first, however, will disagree with the first observer regarding which events are simultaneous with that given event. Neither observer is wrong in this determination; rather, their disagreement merely reflects the fact that simultaneity is an observer-dependent notion in special relativity. A notion of simultaneity is required in order to make a comparison of the rates of clocks carried by the two observers. A closely related phenomenon predicted by special relativity is the so-called twin paradox. Suppose one of two twins carrying a clock departs on a rocket ship from the other twin, an inertial observer, at a certain time, and they rejoin at a later time. In accordance with the time-dilation effect, the elapsed time on the clock of the twin on the rocket ship will be smaller than that of the inertial observer twin—i.

Time Dilation and Length Contraction

The length of any object in a moving frame will appear foreshortened in the direction of motion, or contracted. The amount of contraction can be calculated from the Lorentz transformation. The length is maximum in the frame in which the object is at rest. A clock in a moving frame will be seen to be running slow, or "dilated" according to the Lorentz transformation. The time will always be shortest as measured in its rest frame.

How does the elapsed time that the astronaut measures in the spacecraft compare with the elapsed time that an earthbound observer measures by observing what is happening in the spacecraft? Examining this question leads to a profound result.

Time dilation

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Time dilation

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  1. Nafihole

    Length contraction is the phenomenon that a moving object's length is measured to be shorter than its proper length , which is the length as measured in the object's own rest frame.

    30.04.2021 at 21:23 Reply

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