Questions and Answers of Modern Classical Physics

(a) Show that the coordinate transformation (24.60a) brings the metric ds2 = ηαβdxαdxβ into the form of Eq. (24.60b), accurate to linear order in separation xĵ from the origin of
Inside a laboratory on Earth’s surface the effects of spacetime curvature are so small that current technology cannot measure them. Therefore, experiments performed in the laboratory can be
By introducing a specific but arbitrary coordinate system, show that among all timelike world lines that a particle could take to get from event P0 to event P1, the one or ones whose proper time
Show that in an arbitrary coordinate system xα(P) the geodesic equation (25.11c) takes the form of Eq. (25.14). V₂ P=0 (25.11c)
By inserting the coordinate transformation (25.8b) into the Robertson-Walker metric (25.8a), derive the metric (25.8c), (25.8d) for a local Lorentz frame.Equations 25.8. ds²=a² (n) [-dn²+dx² +
Consider a thin disk with radius R at z = 0 in a Lorentz reference frame. The disk rotates rigidly with angular velocity Ω. In the early years of special relativity there was much confusion over the
With the help of the Newtonian limit (25.35) of the Riemann curvature tensor, show that near Earth’s surface the radius of curvature of spacetime has a magnitude R ∼ (1 astronomical
(a) Place a local Lorentz frame at the center of Earth, and let εjk be the tidal field there, produced by the Newtonian gravitational fields of the Sun and the Moon. For simplicity, treat Earth as
(a) Let a and b be scalar fields with arbitrary but smooth dependence on location in curved spacetime, and let A(vector) and B(vector) be vector fields. Show that(b) Use Eq. (25.32) to show that (i)
Since Maxwell’s equations, written in terms of the classically measurable electromagnetic field tensor F [Eqs. (2.48)] involve only single gradients, it is reasonable to expect them to be lifted
On the surface of a sphere, such as Earth, introduce spherical polar coordinates in which the metric, written as a line element, takes the formwhere a is the sphere’s radius.(a) Show (first by hand
Consider two neighboring geodesics (great circles) on a sphere of radius a, one the equator and the other a geodesic slightly displaced from the equator (by △θ = b) and parallel to it at ∅ = 0.
By evaluating expression (25.30) in an arbitrary basis (which might not even be a coordinate basis), derive Eq. (25.50) for the components of the Riemann tensor. In your derivation keep in mind that
(a) In the Newtonian theory of gravity, consider an axisymmetric, spinning body (e.g., Earth) with spin angular momentum Sj and time-independent mass distribution ρ(x), interacting with an
Derive Eqs. (25.98a) for the trace-reversed metric perturbation outside a stationary (time-independent), linearized source of gravity. More specifically, do the following.(a) First derive h̅00. In
In this exercise we illustrate linearized theory by computing the gravitational field of a moving particle with finite rest mass and then that of a zero-rest-mass particle that moves with the speed
Consider a system that can be covered by nearly globally Lorentz coordinates in which the Newtonian-limit constraints (25.75) are satisfied. For such a system, flesh out the details of the text’s
(a) Derive the Bianchi identity (25.70) in 4-dimensional spacetime.(b) By contracting the Bianchi identity (25.70) on ∈αβμν∈νγδ∈, derive the contracted Bianchi identity
(a) Show that the “infinitesimal” coordinate transformation (25.87) produces the change (25.88) of the linearized metric perturbation and that it leaves the Riemann tensor (25.80) unchanged.(b)
(a) Derive the equation △Ωi = Bijξj for the precession angular velocity of a gyroscope at the tip of ξ as measured in an inertial frame at its tail. Here Bij is the frame-drag field introduced
Show that in linearized theory, for a spinning particle at the origin with mass M and with its spin J along the polar axis, the orthonormal-frame components of ε and B areWhat are the eigenvectors
(a) It turns out that the following line element is a solution of the vacuum Einstein field equation G = 0:Since this solution is spherically symmetric, Birkhoff’s theorem guarantees it must
Bruno Bertotti (1959) and Ivor Robinson (1959) independently solved the Einstein field equation to obtain the following metric for a universe endowed with a uniform magnetic field:HereIf one computes
Consider a photon emitted by an atom at rest on the surface of a static star with mass M and radius R. Analyze the photon’s motion in the Schwarzschild coordinate system of the star’s exterior, r
Show that in the coordinate system {x0̂, x1̂, x2̂, x3̂} of Eqs. (26.25b), the coordinate basis vectors at xĵ = 0 are Eqs. (26.24), and, accurate through first order in distance from xĵ = 0,
(a) Use index manipulations to show that in general (not just inside a static star), for a perfect fluid with Tαβ = (ρ + P)uαuβ + Pgαβ, the law of energy conservation uα Tαβ;β = 0 reduces
The equation of state of a neutron star is very hard to calculate at the supra-nuclear densities required, because the calculation is a complex, many-body problem and the particle interactions are
(a) Show that the embedding surface of Eq. (26.48) is a paraboloid of revolution everywhere outside the star.(b) Show that in the interior of a uniform-density star, the embedding surface is a
The fluid treatment of the neutrino component would only be adequate if the neutrinos were self-collisional, which they are not. The phenomenon of Landau damping alerts us to the need for a kinetic
(a) Using an order-of-magnitude analysis based on Eq. (27.60), show that the strongest gravitational waves that are likely to occur each year in LIGO’s HF band have h+ ∼ h× ∼ 10−21—which
Burke’s radiation-reaction potential (27.64) produces a force per unit volume −ρ∇Φreact on its nearly Newtonian source. If we multiply this force per unit volume by the velocity v = dx/dt of
(a) Compute the net rate at which the quadrupolar waves (27.57) carry energy away from their source, by carrying out the surface integral (25.102) with T0j being Isaacson’s gravitational-wave
Show that the Euler-Lagrange equation for the action principle (27.8) is equivalent to the geodesic equation for a photon in the static spacetime metric g00(xk), gij(xk). Specifically, do the
The isotropic-coordinate line element (26.16) describing the spacetime geometry of a Schwarzschild wormhole is independent of the time coordinate t . However, because gtt = 0 at the wormhole’s
(a) Near the event {r = 2M, θ = θo, ∅ = ∅o, t finite}, on the horizon of a black hole, introduce locally Cartesian spatial coordinatesaccurate to first order in distance from that event. Show
Derive Eq. (11.56) for the free energy of a beam that is compressed with a force F and has a critical compression Fcrit = π2D/ℓ2, where D is its flexural rigidity. V
(a) Verify that −∇ · TM = j × B, where TM is the magnetic stress tensor (19.11).(b) Take the scalar product of the fluid velocity v with the equation of motion (19.10) and combine with mass
Consider a normal shock wave (v perpendicular to the shock front), in which the magnetic field is parallel to the shock front, analyzed in the shock front’s rest frame.(a) Show that the junction
Consider an infinitely long cylinder of plasma with constant electric conductivity, surrounded by vacuum. Assume that the cylinder initially is magnetized uniformly parallel to its length, and assume
Consider a stably stratified fluid at rest with a small (negative) vertical density gradient dρ/dz.(a) By modifying the analysis in this section, ignoring the effects of viscosity, heat conduction,
Make an order-of-magnitude estimate of the size of the fingers and the time it takes for them to grow in a small transparent jar. You might like to try an experiment.
Consider a small bubble of air rising slowly in a large expanse of water. If the bubble is large enough for surface tension to be ignored, then it will form an irregular cap of radius r. Show that
The density and temperature in the deep interior of the Sun are roughly 0.1 kgm−3 and 1.5 × 107 K.(a) Estimate the central gas pressure and radiation pressure and their ratio.(b) The mean free
Consider a knife on its back, so its sharp edge points in the upward, z direction. The edge (idealized as extending infinitely far in the y direction) is hot, and by heating adjacent fluid, it
Use the Rayleigh criterion to estimate the temperature difference that would have to be maintained for 2mm of corn/canola oil, or water, or mercury in a skillet to start convecting. Look up the
In Sec. 14.4, we introduced the notion of a laminar boundary layer by analyzing flow past a thin plate. Now suppose that this same plate is maintained at a different temperature from the free flow. A
In astrophysics (e.g., in supernova explosions and in jets emerging from the vicinities of black holes), one sometimes encounters shock fronts for which the flow speeds relative to the shock approach
Consider the 1-dimensional flow of shallow water in a straight, narrow channel, neglecting dispersion and boundary layers. The equations governing the flow, a derived and discussed in Box 16.3 and
Consider a black hole or neutron star with mass M at rest in interstellar gas that has constant ratio of specific heats γ. In this exercise you will derive some features of the adiabatic, spherical
Use the development of relativistic gas dynamics in Sec. 13.8.2 to show that the cross sectional area of a relativistic 1-dimensional flow tube is also minimized when the flow is transonic. Assume
(a) Show that viscosity damps a monochromatic deep-water wave with an amplitude e-folding time τ∗ = (2νk2)−1, where k is the wave number, and ν is the kinematic viscosity.(b) As an example,
Consider small-amplitude (linear) shallow-water waves in which the height of the bottom boundary varies, so the unperturbed water’s depth is variable: ho = ho(x, y).(a) Using the theory of
A toy boat moves with uniform velocity u across a deep pond (Fig. 16.2). Consider the wave pattern (time-independent in the boat’s frame) produced on the water’s surface at distances large
Verify that expression (16.33) does indeed satisfy the dimensionless KdV equation (16.32). az az +5 + ат an a3s an3 = 0. (16.32)
(a) Verify, using symbolic-manipulation computer software (e.g., Maple, Matlab, or Mathematica) that the two-soliton expression (16.36) satisfies the dimensionless KdV equation.(b) Verify
For the radially oscillating ball as analyzed in Sec. 16.5.3, verify that the radiation reaction acceleration removes energy from the ball, plus the fluid loaded onto it, at the same rate as the
In Sec. 7.5, we explored five elementary (generic) caustics that can occur in geometric optics. Each is described by its phase φ(ã , b̃; x̃, ỹ, z̃) for light arriving at an observation point
Telescopes can also look down through the same atmospheric irregularities as those mentioned in Sec. 8.3.2. In what important respects will the optics differ from those for ground-based telescopes
In Ex. 3.17, we studied the diffusion of particles through an infinite 3-dimensional medium. By solving the diffusion equation, we found that, if the particles’ number density at time t = 0 was
(a) Enthalpy H is a macroscopic thermodynamic variable defined byShow that this definition can be regarded as a Legendre transformation that converts from the energy representation of thermodynamics
Polarization observations of the CMB provide an extremely important probe of fluctuations in the early universe.(a) By invoking the electromagnetic features of Thomson scattering by free electrons,
Use a similarity analysis to derive the solution (17.27) for the shock-tube flow depicted in Fig. 17.9.Figure 17.9.Equation 17.27. Po Co rarefaction t=0 (a) 1>0 (b) vacuum 2Cot (x-1) X
Many stars possess powerful stellar winds that drive strong spherical shockwaves into the surrounding interstellar medium. If the strength of the wind remains constant, the kinetic and internal
A simple analytical solution to the Sedov-Taylor similarity equations can be found for the particular case γ = 7. This is a fair approximation to the behavior of water under explosive conditions, as
Use the quoted scaling of N-wave amplitude with cylindrical radius ω̅ to make an order-of-magnitude estimate of the flux of acoustic energy produced by the Space Shuttle flying at Mach 2 at an
Consider a shock tube as discussed in Sec. 17.4.2 and Fig. 17.9. High density “driver” gas with sound speed C0 and specific heat ratio γ is held in place by a membrane that separates it from
(a) Almost all equations of state satisfy the condition (∂2V/∂P2)s > 0. Show that, when this condition is satisfied, the Rankine-Hugoniot relations and the law of entropy increase imply that
We have computed the velocity field for a freely expanding gas in 1 dimension, Eqs. (17.27).Use this result to show that the path of an individual fluid element, which begins at x = x0Equation 17.27.
Use the rough figures in Box 17.4 to estimate the energy released per unit mass in burning the fuel. Does your answer seem reasonable?Box 17.4. BOX 17.4. SPACE SHUTTLE NASA's (now retired) Space
For γ = 3 and for a channel with A = A∗(1+ x2), solve the flow equations (1) of Box 17.3 analytically and explicitly for v(x), and verify that the solutions have the qualitative forms depicted in
Consider an ideal gas consisting of several different particle species (e.g., diatomic oxygen molecules and nitrogen molecules in the case of Earth’s atmosphere).Consider a sample of this gas with
Redo the computation of radiation reaction for a radially oscillating ball immersed in a fluid without imposing the slow-motion assumption and approximation. Thereby obtain the following coupled
Consider the emission of quadrupolar soundwaves by a Kolmogorov spectrum of free turbulence. Show that the power radiated per unit frequency interval has a spectrumAlso show that the total power
Idealize the trumpet as a bent pipe of length 1.2 m from the mouthpiece (a node of the air’s displacement) to the bell (an antinode). The lowest note is a first overtone and should correspond to B
Consider the G string (196Hz) of a violin. It is ∼30 cm from bridge to nut (the fixed endpoints), and the tension in the string is ∼40 N.(a) Infer the mass per unit length in the string and
Viscosity and thermal conduction will attenuate sound waves. For the moment just consider a monatomic gas where the bulk viscosity can be neglected.(a) Consider the entropy equation (13.75), and
Consider a sound wave propagating through a static, inhomogeneous fluid with no gravity. Explain why the unperturbed fluid has velocity v = 0 and pressure Po = constant, but can have variable density
In the film Fultz (1969), about 20 min 40 s into the film, an experiment is described in which Rossby waves are excited in a rotating cylindrical tank with inner and outer vertical walls and a
Consider deep-water gravity waves of short enough wavelength that surface tension must be included, so the dispersion relation is Eq. (16.14). Show that there is a minimum value of the group
For a soap film that is attached to a bent wire (e.g., to the circular wire that a child uses to blow a bubble), the air pressure on the film’s two sides is the same. Therefore, Eq. (16.16) (with
Consider a point P in the curved interface between two fluids. Introduce Cartesian coordinates at P with x and y parallel to the interface and z orthogonal [as in diagram (b) in Box 16.4], and orient
What is the maximum size of water droplets that can form by water very slowly dripping out of a syringe? Out of a water faucet (whose opening is far larger than that of a syringe)?
(a) Show that in a gravity wave in water of arbitrary depth (deep, shallow, or in between), each fluid element undergoes forward-rolling elliptical motion as shown in Fig. 16.1. (Assume that the
By using the matrix techniques discussed in the next-to-the-last paragraph of Box 12.2, deduce that the general solution to the algebraic wave equation (12.6) is the sum of a longitudinal mode with
Optical fibers in which the refractive index varies with radius are commonly used to transport optical signals. When the diameter of the fiber is many wavelengths, we can use geometric optics. Let
Consider a harmonic oscillator (e.g., a pendulum), driven by bombardment with air molecules. Explain why the oscillator’s position x(t) and velocity ν(t) = dx/dt are random processes. Is
Re-express Box 4.3’s pedagogical example of quantum decoherence and entropy increase in the language of the quantum mechanical density operator p̂ (Box 4.2). Use this example to explain the
Earth has normal modes of oscillation, many of which are in the milli-Hertz frequency range. Large earthquakes occasionally excite these modes strongly, but the quakes are usually widely spaced in
(a) Evaluate the scaling of the rate of mass flow (discharge) Ṁ(x) along the 3- dimensional turbulent jet of the previous exercise. Show that Ṁ increases with distance from the nozzle, so that
Consider a 2-dimensional turbulent jet emerging into an ambient fluid at rest, and contrast it to the laminar jet analyzed in Ex. 15.3.(a) Find how the mean jet velocity and the jet width scale with
(a) Derive the time-averaged Navier-Stokes equation (15.13b) from the time dependent form [Eq. (15.11b)], and thereby infer the definition (15.13c) for the Reynolds stress. Equation (15.13b) shows
One does not have to be a biologist to appreciate the strong evolutionary advantage that natural selection confers on animals that can reduce their drag coefficients. It should be no surprise that
Consider a narrow, 2-dimensional, incompressible (i.e., subsonic) jet emerging from a 2-dimensional nozzle into ambient fluid at rest with the same composition and pressure. (By 2-dimensionalwemean
Repeat Ex. 15.1 for the 3-dimensional laminar wake behind a sphere.Data from Exercises 15.1.We explore the structure of the wake behind the cylinder when the Reynolds number is high enough that the
(a) In a shower or bathtub with the drain somewhere near the center, not the wall, set water rotating so a whirlpool forms over the drain. Perform an experiment to see where the water going down the
Consider low-Reynolds-number flow around a sphere. Derive the velocity field (14.30) using the stream function of Box 14.4. This method is more straightforward but less intuitive than that used in
Fluid flows down a long cylindrical pipe of length b much larger than radius a, from a reservoir maintained at pressure P0 (which connects to the pipe at x = 0) to a free end at large x, where the
Many microorganisms propel themselves, at low Reynolds number, using undulatory motion. Examples include the helical motion of E. coli’s corkscrew tail (Box 14.3), and undulatory waves in a forest
When an appropriately curved airfoil (e.g., an airplane wing) is introduced into a steady flow of air, the air has to flow faster along the upper surface than along the lower surface, which can
Consider an ideal fluid interacting with a (possibly dynamical) gravitational field that the fluid itself generates via ∇2Φ = 4πGρ. For this fluid, take the law of energy conservation, ∂U/∂t

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