The ether flow

The vector field ${\displaystyle \partial _{t}+cf(\rho )\partial _{\rho }}$ generates radial geodesics parametrized by proper time ${\displaystyle \tau }$, which agrees with coordinate time ${\displaystyle t}$ along the geodesics.

As may be inferred from the graph of ${\displaystyle f^{2}}$, a test particle following one of these geodesics starts from rest at ${\displaystyle \rho =\infty ,}$ falls downward toward the drainhole gaining speed all the way, passes through the drainhole and out into the lower region still gaining speed in the downward direction, and arrives at ${\displaystyle \rho =-\infty }$ with

${\displaystyle \mathrm {speed} =c|f(-\infty )|=c{\sqrt {1-e^{-2m\pi /a}}}<c.}$

The vector field in question is taken to be the velocity field of a more or less substantial 'ether' pervading all of space-time. This ether is in general "more than a mere inert medium for the propagation of electromagnetic waves; it is a restless, flowing continuum whose internal, relative motions manifest themselves to us as gravity. Mass particles appear as sources or sinks of this flowing ether."^[1]

For timelike geodesics in general the radial equation of motion is

${\displaystyle {\begin{aligned}{\frac {d^{2}\rho }{d\tau ^{2}}}&=-\left({\frac {f^{2}}{2}}\right)^{\mathbf {\prime } }(\rho )+(\rho -3m)\left({\frac {d\Omega }{d\tau }}\right)^{2}\\&=-{\frac {m}{r^{2}(\rho )}}+(\rho -3m)\left({\frac {d\Omega }{d\tau }}\right)^{2}\,.\end{aligned}}}$

One sees from this that

• it is the 'stretching' of the ether flow as measured by the term ${\displaystyle -(f^{2}/2)'(\rho )}$ that produces the downward pull of gravity,
• every test particle whose orbit dips as low as ${\displaystyle \rho =3m}$ will fall through the drainhole,
• there are test particles with enough angular speed ${\displaystyle |d\Omega /d\tau |}$ to balance the downward pull that their orbits (circular ones in particular) are confined to the part of the upper region where ${\displaystyle \rho >3m}$,
• the downward pull produces in the upper region acceleration toward the drainhole, thus attractive gravity, but in the lower region acceleration away from the drainhole, thus repulsive gravity,
• the downward pull reaches its maximum where ${\displaystyle r(\rho )}$ is a minimum, namely, at the 'throat' of the drainhole where ${\displaystyle \rho =2m}$, and
• if ${\displaystyle m=0,}$ a test particle can sit at rest (with ${\displaystyle d\rho /d\tau =d\Omega /d\tau =0}$) anywhere in space. (This is the special case of the nongravitating drainhole known as the Ellis wormhole.)

Traversability

It is clear from the radial equation of motion that test particles starting from any point in the upper region with no radial velocity (${\displaystyle d\rho /d\tau =0}$) will, without sufficient angular velocity ${\displaystyle d\Omega /d\tau }$, fall down through the drainhole and into the lower region. Not so clear but nonetheless true is that a test particle starting from a point in the lower region can with sufficient upward velocity pass through the drainhole and into the upper region. Thus the drainhole is 'traversable' by test particles in both directions. The same holds for photons.

A complete catalog of geodesics of the drainhole can be found in the Ellis paper.^[1]

Absence of horizons and singularities; geodesical completeness

For a metric of the general form of the drainhole metric, with ${\displaystyle \partial _{t}+cf(\rho )\partial _{\rho }}$ as the velocity field of a flowing ether, the coordinate velocities ${\displaystyle d\rho /dt}$ of radial null geodesics are found to be ${\displaystyle c(f(\rho )+1)}$ for light waves traveling against the ether flow, and ${\displaystyle c(f(\rho )-1)}$ for light waves traveling with the flow. Wherever ${\displaystyle f(\rho )>-1}$, so that ${\displaystyle c(f(\rho )+1)>0}$, light waves struggling against the ether flow can gain ground. On the other hand, at places where ${\displaystyle f(\rho )\leq -1}$ upstream light waves can at best hold their own (if ${\displaystyle f(\rho )=-1}$), or otherwise be swept downstream to wherever the ether is going (if ${\displaystyle f(\rho )<-1}$). (This situation is described in jest by: "People in light canoes should avoid ethereal rapids."^[1])

The latter situation is seen in the Schwarzschild metric, where ${\displaystyle \textstyle f_{\text{S}}(\rho )=-{\sqrt {2M/\rho }}\,}$, which is ${\displaystyle -1}$ at the Schwarzschild event horizon where ${\displaystyle r_{\text{S}}(\rho )=\rho =2M}$, and less than ${\displaystyle -1}$ inside the horizon where ${\displaystyle \rho <2M}$.

By contrast, in the drainhole ${\displaystyle \textstyle f^{2}(\rho )<1-e^{-2m\pi /a}<1}$ and ${\displaystyle \textstyle f(\rho )=-\left[f^{2}(\rho )\right]^{1/2}>-1}$, for every value of ${\displaystyle \rho }$, so nowhere is there a horizon on one side of which light waves struggling against the ether flow cannot gain ground.

Because

• ${\displaystyle r}$ and ${\displaystyle f}$ are defined on the whole real line, and
• ${\displaystyle r}$ is bounded away from ${\displaystyle 0}$ by ${\displaystyle r(2m)}$), and
• ${\displaystyle f^{2}}$ is bounded away from ${\displaystyle 1}$ (by ${\displaystyle {\sqrt {1-e^{-2m\pi /a}}}}$),

the drainhole metric encompasses neither a 'coordinate singularity' where ${\displaystyle 1-f^{2}(\rho )\to 0}$ nor a 'geometric singularity' where ${\displaystyle r(\rho )\to 0}$, not even asymptotic ones. For the same reasons, every geodesic with an unbound orbit, and with some additional argument every geodesic with a bound orbit, has an affine parametrization whose parameter extends from ${\displaystyle -\infty }$ to ${\displaystyle \infty }$. The drainhole manifold is, therefore, geodesically complete.

Strength of repulsion

As seen earlier, stretching of the ether flow produces in the upper region a downward acceleration ${\displaystyle -m/r^{2}(\rho )}$ of test particles that, along with ${\displaystyle r(\rho )\sim \rho }$ as ${\displaystyle \rho \to \infty }$, identifies ${\displaystyle m}$ as the attractive gravitational mass of the nonlocalized drainhole particle. In the lower region the downward acceleration is formally the same, but because ${\displaystyle r(\rho )}$ is asymptotic to ${\displaystyle -\rho e^{m\pi /a}}$ rather than to ${\displaystyle -\rho }$ as ${\displaystyle \rho \to -\infty }$, one cannot infer that the repulsive gravitational mass of the drainhole particle is ${\displaystyle -m}$.

To learn the repulsive mass of the drainhole requires finding an isometry of the drainhole manifold that exchanges the upper and lower regions. Such an isometry can be described as follows: Let ${\displaystyle M_{m,n}}$ denote the drainhole manifold whose parameters are ${\displaystyle m}$ and ${\displaystyle n}$, and ${\displaystyle M_{{\bar {m}},{\bar {n}}}}$ denote the drainhole manifold whose parameters are ${\displaystyle {\bar {m}}}$ and ${\displaystyle {\bar {n}}}$, where

${\displaystyle {\bar {m}}=-me^{m\pi /a}}$

and

${\displaystyle \;\;{\bar {n}}=ne^{m\pi /a}.}$

The isometry identifies the point of ${\displaystyle M_{m,n}}$ having coordinates ${\displaystyle [T,\rho ,\vartheta ,\varphi ]}$ with the point of ${\displaystyle M_{{\bar {m}},{\bar {n}}}}$ having coordinates ${\displaystyle [{\bar {T}},{\bar {\rho }},{\bar {\vartheta }},{\bar {\varphi }}]=[Te^{-m\pi /a},-\rho e^{m\pi /a},\vartheta ,\varphi ]}$. One infers from it that ${\displaystyle M_{m,n}}$ and ${\displaystyle M_{{\bar {m}},{\bar {n}}}}$ are in fact the same manifold, and that the lower region where ${\displaystyle \rho \to -\infty ,}$ now disguised as the upper region where ${\displaystyle {\bar {\rho }}\to \infty }$, has ${\displaystyle {\bar {m}}}$ as its gravitational mass, thus gravitationally repels test particles more strongly than the true upper region attracts them, in the ratio ${\displaystyle \,|{\bar {m}}|/|m|=-{\bar {m}}/m=e^{m\pi /a}>1}$.

Asymptotic flatness

That the drainhole is asymptotically flat as ${\displaystyle \rho \to \infty }$ is seen from the asymptotic behavior ${\displaystyle r(\rho )\sim \rho }$ and ${\displaystyle f^{2}(\rho )\sim 2m/\rho \sim 0.}$ That it is asymptotically flat as ${\displaystyle \rho \to -\infty }$ is seen from the corresponding behavior as ${\displaystyle {\bar {\rho }}\to \infty }$ after the isometry between ${\displaystyle M_{m,n}}$ and ${\displaystyle M_{{\bar {m}},{\bar {n}}}}$ described above.

The parameter n

Unlike the parameter ${\displaystyle m}$, interpreted as the attractive gravitational mass of the drainhole, the parameter ${\displaystyle n}$ has no obvious physical interpretation. It essentially fixes both the radius ${\displaystyle r(2m)}$ of the throat of the drainhole, which increases from ${\displaystyle n}$ when ${\displaystyle m=0}$ to ${\displaystyle ne}$ as ${\displaystyle m\to n,}$ and the energy of the scalar field ${\displaystyle \phi ,}$ which decreases from ${\displaystyle n\pi /2}$ when ${\displaystyle m=0}$ to ${\displaystyle n/2}$ as ${\displaystyle m\to n}$.

For reasons given in Sec. 6.1 of a 2015 paper,^[5] Ellis suggests that ${\displaystyle n}$ specifies in some way the inertial mass of the particle modeled by the drainhole. He writes further that a "'Higgsian' way of expressing this idea is to say that the drainhole 'acquires' (inertial) mass from the scalar field ${\displaystyle \phi }$".