Surface tension and wave propagation

3.10.1 Discussion

Assume that we have a water bubble of radius R in the form of a hemisphere with the flat surface being located on the water surface. The temperature of the air within the bubble is T. The volume of the hemispherical bubble is

V = 2tt7?³/3

Let denote the surface tension along the circular boundary of the flat surface part of the bubble. If p is the pressure within the bubble, then we have

pV = nKT

where K is Boltzmann’s constant and n is the number of air molecules within the bubble. The total pressure force that pulls the spherical surface part of the bubble away along the z axis is given by

/ p2'KR?sin(O)cos(ff)d0

Jo

= 7rR²p

This pressure force must be balanced by the surface tension force, ie,

npR² = t.2ttR

or equivalently,

t = pR/2

Note that the surface tension acts on the circular boundary along the negative z direction. Now we consider small fluctuations in the radius of the bubble. If R(t) denotes the bubble radius at time t, then the pressure force along the z direction is

p(t)vR² ,p(t) = nkT/V(t) = nkT/(4nR³(t)/3) = 3nkT/^R³(t)

and if r(t) is the surface tension, we have the following relations: The total internal energy of the air molecules within the bubble is given by Ui(t) = mCT(t) where C is the specific heat per unit mass of air. The total gravitational potential energy of the air within the bubble is

_rR(t)

= / p(j}g^z'K(R²(t) — z²)dz, p(t) = 3m/47r/?³(t) Jo

Since no heat is added from outside to the bubble, the thermodynamic relation

0 = dQ = dU + pdV, U = Ui + U_g

gives us one relation between 7?(t) and T(t). The other relation is r = p(t)R(t)/2 assuming that the surface tension t does not vary with time.

Exercise: Derive using the above formulas, a differential equation satisfied by 7?(t) and using small oscillation theory, solve approximately for 8R(t) = R(t) — R(0) by the method of linearization.

3.10.2 Points to remember

[1] The surface tension on the circumference of the circular base of a water bubble balances the air pressure within the bubble so as to prevent it from breaking. At a given air temperature, using the ideal gas law, we can compute the pressure as a function of the bubble radius and then if we assume the bubble radius to be oscillating, we can compute its equation of motion using energy conservation taking into account the gravitational potential energy of the air within the bubble. The total kinetic energy of the air molecules within the bubble can be computed using Maxwell’s equipartition formula giving the relation between the kinetic energy of a molecule and the temperature. The rate of change of the kinetic plus potential energy of the bubble must be equal to the net rate at which pressure forces from outside and the surface tension on the bottom circumference do work on the bubble and this leads to bubble oscillations.

Klein-Gordon equation in the Schwarzchild space-time with a radial time independent electromagnetic field and its application to computing the Hawking temperature at which massless/massive particles are emitted from a blackhole

3.11.1 Discussion

The KG equation for a complex scalar field r, 0. ) = ip(t, r) = V>(a:) in the presence of an external em field described by the static four potential Ag(r) is given by

(dp. + ieA^g¹¹¹'y/^g{d„ + ieA^fx) + p²-i^(a-’) = 0

This can be derived from the variational principlewhere

S[V’,V’*] = I L[i^{x), ip*(x), d/W>*(x^d⁴x

and

L = (l/2)p^/"^y((^ ~ ieX^)V>)*((^ + ⁱ⁽:A,AA)A=-9 - ii²ip*ip/=^/2

We now look at the Schwarzchild metric which is diagonal in the time and spherical polar coordinates:

poo = a(r),pn = -a(r)^-1,p₂2 = -r²,p33 = -r²sm²(0),

so that

P°° = «(r)-¹^¹¹ = —a(r),p²² = -l/r²,p³³ = -l/(rW(0))

The KG equation becomes for the Schwarzchild metric in the frequency domain, ie dt —> ia>,

-9°°V-g(^ + eA₀)²^ + (d_r + ieA_r)p¹¹y⁼p(V’,r + ieA^ib)

+(<% + ieX₂)p²²/^rp(V’,0 +

+ ieX₃)p³³y^p(V’,i> + ieA₃ip) =

This becomes

a(r)^-1(cv’ + e^o)²^ + ~^r + ieAi)a(r)r²(V>,r + ieApib)

+ , . Ade + ¿eA₂)sm(0)(V’,e + ieA₂-tp) r^zsint)

+ 2 ■ 2ia^ +

r^zsin^z{9)

If we set a(r) = 1, ie m = 0, then the usual Klein-Gordon equation of special relativity is obtained. We substitute into this equation

V»(w, r, 9d>) = exp(iS(cj, r, 0, ))

and obtain using

= iS,,

V’.rr = (iS._r - S²_r)^,A,&e = (iS^e - S²_e)^,

then we get

(d_r + zeTi)a(r)r²(V^,,r + ieAiip) =

{d_r + ieAAA² ~ + ieAi4>) =

2(r — + ieAW>) + (r² — 2mr)(V’,rr + ieAi^ + 2ieAiV»._r — e² A^pA)

= (r²—2mr)V’,rr+(2(i—m)+(r²—2/rw’)2ieyli)V’.r+(2(i—m)ieA

+(r ²—2mr ) ie A i, _r—e² (r²—2mr ) A² )il>

(d_e + ieA₂)sm(0)(^ +

= sm(0)'0_j00 + (2ieA2sin(9) + cos(0'))ip, o

+(ie(sin(#)yl₂).0 — e² A^sin^O))^

and finally,

(d3)²ip =

+ ZieAsV’,^ + (ieA-i_>(f, — e² A²)4>

The resulting problem is one in which there are two perturbation parameters m, e. We can formulate this problem in the following form:

(Lo + rnL + eL₂ + emL-i + e²mL + err²L-, + e²m²L$pl> = 0

where some of the linear partial differential operators Lk,k = 0,1, ....6 depend on the frequency and also on the particle mass p. To solve this pde perturbatively, we need to make some assumption regarding the relative order of magnitudes of m and e. Let us say that m = O(e^s), ie, we put m = ke^s where

is an integer or a fraction. Then, the above pde acquires the form

• (L_o + fee’Lj + eL-₂ + Le^s+1L₃ + fce^s+²L₄ + k²e^2s+1L$ + fc²e^2s+²L₆)V- = 0
• 3.11.2 Points to remember

[1] The Klein-Gordon equation for a particle in the gravitational field defined by a metric and also in the presence of an electromagnetic field can be expressed using the Laplace-Beltrami operator, ie ordinary divergence of the scalar field taking into account the em potential as a connection followed by a covariant divergence with a connection term coming from the electromagnetic potential. This equation described the dynamics of a scalar field in the presence of gravity and electromagnetic forces. When solved using perturbation theory, it can be used to compute the probability density of scalar particles outside the event horizon of a Schwarzchild blackhole after time t and if we work in the frequency domain, assuming a static em field, it can be used to compute the probability density of particles just outside the event horizon at a given frequency which on equating to exp(—hai/kT) can be used to compute the Hawking temperature at a given frequency in terms of the background em field.