Physics

Physics · modern-physics

Problem

2. Now consider the opposite point of view. If an observer at the initial point (x=0)(x=0) is observing the particle's clock when the observer's time is tt, what is the reading of the particle's clock τ0\tau_{0} ? After a long period of time, will this reading approach a certain value? If so, what is the value? Given constants: Global Positioning System (GPS) is a navigation technology which uses signal from satellites to determine the position of an object (for example an airplane). However, due to the satellites high speed movement in orbit, there should be a special relativistic correction, and due to their high altitude, there should be a general relativistic correction. Both corrections seem to be small but are very important for precise measurement of position. We will explore both corrections in this problem. First we will investigate the special relativistic effect on an accelerated particle. We consider two types of frame, the first one is the rest frame (called SS or Earth's frame), where the particle is at rest initially. The other is the proper frame (called S′S^{\prime} ), a frame that instantaneously moves together with the accelerated particle. Note that this is not an accelerated frame, it is a constant velocity frame that at a particular moment has the same velocity with the accelerated particle. At that short moment, the time rate experienced by the particle is the same as the proper frame's time rate. Of course this proper frame is only good for an infinitesimally short time, and then we need to define a new proper frame afterward. At the beginning we synchronize the particle's clock with the clock in the rest frame by setting them to zero, t=τ=0t=\tau=0 ( tt is the time in the rest frame, and τ\tau is the time shown by particle's clock). By applying equivalence principle, we can obtain general relativistic effects from special relavistic results which does not involve complicated metric tensor calculations. By combining the special and general relativistic effects, we can calculate the corrections needed for a GPS (global positioning system) satellite to provide accurate positioning. Some mathematics formulas that might be useful - sinh⁡x=ex−e−x2\sinh x=\frac{e^{x}-e^{-x}}{2} - cosh⁡x=ex+e−x2\cosh x=\frac{e^{x}+e^{-x}}{2} - tanh⁡x=sinh⁡xcosh⁡x\tanh x=\frac{\sinh x}{\cosh x} - 1+sinh⁡2x=cosh⁡2x1+\sinh ^{2} x=\cosh ^{2} x - sinh⁡(x−y)=sinh⁡xcosh⁡y−cosh⁡xsinh⁡y\sinh (x-y)=\sinh x \cosh y-\cosh x \sinh y - ∫dx(1−x2)32=x1−x2+C\int \frac{d x}{\left(1-x^{2}\right)^{\frac{3}{2}}}=\frac{x}{\sqrt{1-x^{2}}}+C - ∫dx1−x2=ln⁡1+x1−x+C\int \frac{d x}{1-x^{2}}=\ln \sqrt{\frac{1+x}{1-x}}+C Part A. Single Accelerated Particle Consider a particle with a rest mass mm under a constant and uniform force field FF (defined in the rest frame) pointing in the positive xx direction. Initially (t=τ=0)(t=\tau=0) the particle is at rest at the origin (x=0)(x=0). Context question: 1. When the velocity of the particle is vv, calculate the acceleration of the particle, aa (with respect to the rest frame). Context answer: a=Fγ3ma=\frac{F}{\gamma^{3} m} Context question: 2. Calculate the velocity of the particle β(t)=v(t)c\beta(t)=\frac{v(t)}{c} at time tt (in rest frame), in terms of F,m,tF, m, t and cc. Context answer: β=Ftmc1+(Ftmc)2\beta=\frac{\frac{F t}{m c}}{\sqrt{1+\left(\frac{F t}{m c}\right)^{2}}} Context question: 3. Calculate the position of the particle x(t)x(t) at time tt, in term of F,m,tF, m, t and cc. Context answer: x=mc2F(1+(Ftmc)2−1)x=\frac{m c^{2}}{F}\left(\sqrt{1+\left(\frac{F t}{m c}\right)^{2}}-1\right) Context question: 4. Show that the proper acceleration of the particle, a′≡g=F/ma^{\prime} \equiv g=F / m, is a constant. The proper acceleration is the acceleration of the particle measured in the instantaneous proper frame. Context answer: 证明题 Context question: 5. Calculate the velocity of the particle β(τ)\beta(\tau), when the time as experienced by the particle is τ\tau. Express the answer in g,τg, \tau, and cc. Context answer: β=tanh⁡gτc\beta=\tanh \frac{g \tau}{c} Context question: 6. ( 0.4 pts)\mathbf{0 . 4} \mathbf{~ p t s )} Also calculate the time tt in the rest frame in terms of g,τg, \tau, and cc. Context answer: t=cgsinh⁡gτct=\frac{c}{g} \sinh \frac{g \tau}{c} Extra Supplementary Reading Materials: Part B. Flight Time The first part has not taken into account the flight time of the information to arrive to the observer. This part is the only part in the whole problem where the flight time is considered. The particle moves as in part A. Context question: 1. At a certain moment, the time experienced by the particle is τ\tau. What reading t0t_{0} on a stationary clock located at x=0x=0 will be observed by the particle? After a long period of time, does the observed reading t0t_{0} approach a certain value? If so, what is the value? Context answer: 证明题

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