qvis.kgwave

kgwave = “Klein-Gordon Wave”

Package for plotting and visualizing solutions of massive KG wave equation.

Intro

Formalism follows conventions of Birrel and Davies (1982) Chapter 2 with metric signature \((+---)\). Units \(\hbar=c=1\).

......................................................
 Solve KG on interval:

           0             L/4            L/2
           |--------------|--------------|
          y=0                           y=0

 m=2pi
 L>>(2pi/m)
......................................................

Consider the flat-space (1+1)d massive KG wave equation

\[(\partial_t^2 - \partial_x^2 + m^2) \, \varphi = 0\]

for a real scalar field \(\varphi\), on the interval (0, L/2) with closed (\(\varphi\) =0) boundary conditions.

Choose mass \(m=2\pi\) and measure space and time in units of the Compton wavelength \(\lambda_m = 2\pi/m =1\). The Compton wavenumber is \(k_m = 2 \pi / \lambda_m = m\).

You can immediately read off the dispersion relation

\[\omega^2 = k^2 + m^2\]

which is made precise later.

The group velocity is

\[v_g = \tfrac{d\omega}{dk} = \tfrac{k}{\omega} = \tfrac{k/m}{\sqrt{1+(k/m)^2}} < 1,\]

and the phase velocity is \(v_p = \tfrac{\omega}{k} = \tfrac{1}{v_g}\).

In true units, \(\omega/k\) has units of \(c\), and \(v_g = c^2 k/\omega\).

Energy

The equation of motion is obtained from the Lagrangian density

\[\mathcal{L} = \tfrac{1}{2} \, \left( (\partial_t \varphi)^2 - (\partial_x \varphi)^2 - m^2 \varphi^2 \right).\]

The corresponding Hamiltonian density is

\[\mathcal{H} = \tfrac{1}{2} \, \left( (\partial_t \varphi)^2 + (\partial_x \varphi)^2 + m^2 \varphi^2 \right)\]

with canonical momentum density \(\pi = \partial_t \varphi\).

The Hamiltonian, which depends on the (1+1) splitting provided by the coordinates, is

\[H = \int_{0}^{L/2} \mathcal{H} \; dx.\]

The gravitational stress-energy-momentum (stress) tensor is

\[T_{ab} = \partial_a \varphi \, \partial_b \varphi - \mathcal{L} \, \eta_{ab} ,\]

so that

\[\begin{split}T_{tt} &= \mathcal{H} \\ T_{tx} = T_{xt} &= \partial_t \varphi \; \partial_x \varphi \\ T_{xx} &= \tfrac{1}{2} \, \left( (\partial_t \varphi)^2 + (\partial_x \varphi)^2 - m^2 \varphi^2 \right).\end{split}\]

The energy flux vector \(F^\mu = {T^\mu}_t\) relative to a stationary observer (sitting at x=const) is

\[\begin{split}F^t &= \mathcal{H} \\ F^x &= - \, \partial_t \varphi \; \partial_x \varphi .\end{split}\]

More generally, \({T^t}_\nu = T_{t\nu}\) and \({T^x}_\nu = -T_{x\nu}\), so the up-down stress tensor in the Minkowski coordinates is

\[\begin{split}{T^\mu}_\nu = \begin{pmatrix} T_{tt} & T_{tx} \\ -T_{xt} & -T_{xx} \end{pmatrix}.\end{split}\]

The proper density \(\rho\) and (minus the) proper pressure \(-p\) are defined as the timelike and spacelike eigenvalues of \({T^a}_b\), respectively. The energy conditions in this context are: NEC (\(\rho+p\geq 0\)), WEC (\(\rho \geq 0\) +NEC), FEC (\(\rho^2 \geq p^2\)).

Define \(\alpha=- \, \partial_t \varphi/\partial_x \varphi\). One generally expects \(|\alpha|>1\) (since \(|\omega| > |k|\)). Let us suppose, for the time being, this holds true. Then by diagonalizing the stress tensor one obtains

\[\begin{split}\rho &= \mathcal{H} - (\partial_x \varphi)^2 \qquad & &\textrm{with eigenvector} \qquad & &(\alpha,1) \\ -p &= \mathcal{H} - (\partial_t \varphi)^2 \qquad & &\textrm{with eigenvector} \qquad & &(1,\alpha).\end{split}\]

It’s easy to check that the eigenvectors are orthonormal, and also that both \(\rho>0\) and \(\rho+p>0\) explicitly, so both NEC and WEC are satisfied. Also, \(\rho^2 - p^2 = m^2 \varphi^2 \, ((\partial_t \varphi)^2 - (\partial_x \varphi)^2) \geq 0\), so FEC is also satisfied.

Energy conditions violations are only possible, therefore, if \(|\alpha| \leq 1\).