I have an embarrassing query... Firstly:
An elementary particle falling effectively "from infinity" toward a
neutron star (NS), say, will hit its its surface with a relativistic speed equal to the star's escape velocity. From its relativistic mass increase (maybe by a factor of ~1.5 or so), the particle will then add that extra
amount of relativistic energy/mass-equivalent onto the NS. Similarly
with trillions of other particles raining down onto the NS. They confer
not only their rest or invariant mass/energy, but something extra due
to spec. relativity. Okay so far, I think. Great.

...But, then:
I have a conundrum when the massive body is a black hole, say
Cyg-X1. If I understand correctly, elementary particles falling toward
its event horizon, will reach ultra-relativistic speeds. Perhaps even
reaching light speed at the EH? Which would seem to add extremely ultra-relativistic mass to each such particle. Since, roughly, ~10^58
mass particles would have experienced this in the history of Cyg-X1,
from its time zero, beginning as a micro black hole, to the present,
this would seem also to confer a mass M for this black hole vastly
greater than its actual mass by orders of magnitude. As though
the BH should be bending the scales to the tune of many thousands
of solar masses. Yet Cyg-X1 has a "mere" 14.8 solar masses,
latest measurement.

I know that there is something wrong with my thinking, but it eludes
me.
Thanks,
Gene

[[Mod. note --

1. Phrases like "ultra-relativistic speeds" have to be used with
some care when near a BH. That is, we need to be careful to define
the (a) reference frame for such a speed measurement. Close to the
BH, it's not sufficient to just say "speed relative to the BH" --
there's more than one way (in fact, there are infinitely many ways)
to define a coordinate system near the BH, and any speed measurement
needs to implicitly or explicitly specify which coordinate system
it's referred to.

2. For a reasonable choice of coordinates for radial motion near the
horizon (say, areal radius and Eddington-Finkelstein time), I don't
think an infalling particle (with nonzero rest mass) will actually
reach "ultra-relativistic speeds" (special-relativistic gamma >> 1)
before crossing the event horizon.

3. In analyzing this system, we need to do the "energy accounting"
consistently. What we can actually measure (e.g., by observing
stars or gas orbiting the BH) is mass-energy-at-infinity; any energy
"down near the BH" is effectively redshifted in its contribution
to mass-energy-at-infinity. So, if an infalling particle has a
non-trivial special-relativistic gamma factor when close to the BH,
that kinetic energy needs to be redshifted (divided) by that same gamma
factor before being included in the total mass-energy-at-infinity.

4. Conservation of mass-energy still applies to the system consisting
of the BH and the infalling matter. Thus 1 kg of infalling matter
can't add more than 1kg to the BH mass; if it adds less then the
difference must be radiated away (e.g., as photons).

5. If the infalling matter has *zero* angular momentum relative to
the BH, and we treat it as "dust" (a set of non-interacting particles),
then we have an "advection-dominated accretion flow", where all
(or almost all) of the infalling matter's mass-energy is captured
by the BH.

6. If the infalling matter has *nonzero* angular momentum relative
to the BH, then it won't fall straight in to the BH, but will rather
orbit the BH, forming accretion disk. Any individual matter particle
will then spiral in over a relatively long time-scale, and its
interaction ("friction") with other infalling matter will heat the
accretion disk to high temperature, so the accretion disk will radiate
photons. This is the underlying process behind (e.g.) Cygnus X-1's
observed X-ray luminosity. This process can in theory radiate
a substantial fraction of the mass-energy in the infalling matter.

-- jt]]

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