rest mass/relativistic mass question

Chalky

I was thinking about something and got confused so I thought I will
post it here and hopefully someone more knowledgeable than me can clear
the confusion.

As a thought experiment, suppose, we are accelerating a proton in a
large accelerator (as large as you need it to be). It will continuously
gain mass/energy. Will a time come when it has gained sufficient mass
energy to form a black hole? If not why not?

If yes, then what happens to an observer who is always at rest relative
to the proton? To that observer nothing is happening but then suddenly
the proton turns into a hole? Isn't physics seemingly violated for that
observer?

thanks for any clarification.

Jon Bell

In article <1140127497.964435.327250@f14g2000cwb.googlegroups.com>,
Chalky <utpalchakraborty@gmail.com> wrote:
>
>As a thought experiment, suppose, we are accelerating a proton in a
>large accelerator (as large as you need it to be). It will continuously
>gain mass/energy. Will a time come when it has gained sufficient mass
>energy to form a black hole? If not why not?

This question is discussed in the Usenet Physics FAQ:

http://math.ucr.edu/home/baez/physic...lack_fast.html

--
Jon Bell <jtbell@presby.edu> Presbyterian College
Dept. of Physics and Computer Science Clinton, South Carolina USA

Jonathan Thornburg -- remove -animal to reply

Chalky <utpalchakraborty@gmail.com> wrote:
> As a thought experiment, suppose, we are accelerating a proton in a
> large accelerator (as large as you need it to be). It will continuously
> gain mass/energy. Will a time come when it has gained sufficient mass
> energy to form a black hole? If not why not?
>
> If yes, then what happens to an observer who is always at rest relative
> to the proton? To that observer nothing is happening but then suddenly
> the proton turns into a hole? Isn't physics seemingly violated for that
> observer?

This is a Frequently Asked Question. Fortunately, there are informative
answers to be had in the Physics FAQ (written by one of our esteemed
moderators, no less!). Check out
http://math.ucr.edu/home/baez/physics/
particularly the "Black Holes" section.

ciao,

--
-- "Jonathan Thornburg -- remove -animal to reply" <jthorn@aei.mpg-zebra.de>
Max-Planck-Institut fuer Gravitationsphysik (Albert-Einstein-Institut),
Golm, Germany, "Old Europe" http://www.aei.mpg.de/~jthorn/home.html
"Washing one's hands of the conflict between the powerful and the
powerless means to side with the powerful, not to be neutral."
-- quote by Freire / poster by Oxfam

Pmb

"Chalky" <utpalchakraborty@gmail.com> wrote in message
news:1140127497.964435.327250@f14g2000cwb.googlegroups.com...
>I was thinking about something and got confused so I thought I will
> post it here and hopefully someone more knowledgeable than me can clear
> the confusion.
>
> As a thought experiment, suppose, we are accelerating a proton in a
> large accelerator (as large as you need it to be). It will continuously
> gain mass/energy. Will a time come when it has gained sufficient mass
> energy to form a black hole? If not why not?
>
> If yes, then what happens to an observer who is always at rest relative
> to the proton? To that observer nothing is happening but then suddenly
> the proton turns into a hole? Isn't physics seemingly violated for that
> observer?
>
> thanks for any clarification.

If a object is not a black hole in one frame of reference then it won't be a
black hole in any other frame even though the grazvitational field will
increase with speed. Most people believe an object is a black hole *because*
it has more than a certain amount of mass. But that is not the case. The
mass must be confined within a certain sperical region of space. Take Mount
Everest as an example. We all know that Mt. Everest is not a black hole. But
its theoretically possible to have a black hole with the same mass of Mt.
Everest. Black holes around this size are refered to as mini-black holes.

Pete

John Bell

Chalky wrote:

> As a thought experiment, suppose, we are accelerating a proton in a
> large accelerator (as large as you need it to be). It will continuously
> gain mass/energy. Will a time come when it has gained sufficient mass
> energy to form a black hole? If not why not?

I am in general agreement with the other respondents on this matter.

However, even though I don't know quite where this is taking me, there
is something about physics in the reference frame of the linearly
accelerating proton, which could be very loosely described as vaguely
analogous to black hole physics (despite also being very clearly
different). I am referring to the well-known fact that an observer
linearly accelerating at 1g, will outrun a photon if given a head start
of about a year. Thus, for the linearly accelerating proton, its source
will have an effective 'temporal event horizon' at about t = c/g.
However, this horizon is never observed from the accelerating reference
frame, because, as that horizon is approached, light from the source
continues to become progressively more red shifted. In other words,
light from the source between time 0 and time t = c/g is observed
progressively more stretched out, from time 0 to infinity, in the
reference frame of the constantly accelerating observer. Clearly,
something very similar happens when we look at light from the
continuously accelerating observer, when viewed in the reference frame
of the source. This consideration alone should help to confirm that the
linearly accelerated proton should never become a black hole relative
to the source, and vice versa.

However, if a cyclotron accelerates the proton, the situation could
become significantly different, because the mean velocity of the proton
remains zero, relative to the earth. I thus agree that it then seems
theoretically possible to 'pump up' the total rest mass of the
system indefinitely, provided sufficient energy can be found to
maintain adequate strength in the containment field, as well as to
maintain acceleration of the particle. So, yes, the system might then
eventually turn into a black hole (theoretically).


> If yes, then what happens to an observer who is always at rest relative
> to the proton?

The simple answer is that he will probably have been killed by huge
differential centrifugal forces, long before that system becomes a
black hole. In fact, it is difficult to imagine that those centrifugal
forces will not cause the whole system to explode long before that time
too.

>To that observer nothing is happening

Not so. Massive experienced forces will become progressively vaster.

> but then suddenly
> the proton turns into a hole

I don't think so. If he could survive the forces, that observer should
eventually find himself in the same hole as the proton, and nothing is
particularly strange about that.

Hope this helps.

John Bell

Chalky

ok, i think i can understand why the black hole will not form with a
continuously linearly accelerating mass. John and Greg's reasoning
seemed particularly elucidating.

Blackbird wrote:
--------------------------------------------------------------------------------
Ok. It will be easier to answer that question if you tell us why you
think
acceleration should make a difference.
--------------------------------------------------------------------------------

Correct I should have clarified that at first. That is where the
confusion started. Basically, it seemed to me that as and when I added
mass/energy to any object it contributed to its stress-energy tensor.
You have a static uncharged spehere. You spin it up, it adds to its
stress-energy. You add charges to it, it adds to it stress-energy. In
each case as energy is spent on it some of it is converted into some
form of energy that contributes to the stress-energy tensor and hence
there seems to be no theoritical reason on why that tensor cannot be
made to form a black hole given enough energy is added to it.

Now, with a linearly accelerating mass, energy is definitely being
spent on it. That energy is being transformed into kinetic energy
(which is the other confusing concept because this meaure of energy
seems to be frame dependant) which is as "real" as any other energy
because it can be converted back if necessary to any other form of
energy. However, this kinetic energy for some reason does not
contribute to the stress-energy tensor.

So all kinds of mass-energy do not add to stress-energy tensor? So in
some sort of weird sense energy is not conserved in GR?

Blackbird

Chalky wrote:
> ok, i think i can understand why the black hole will not form with a
> continuously linearly accelerating mass. John and Greg's reasoning
> seemed particularly elucidating.
>
> Blackbird wrote:
> --------------------------------------------------------------------------
------
> Ok. It will be easier to answer that question if you tell us why you
> think
> acceleration should make a difference.
> --------------------------------------------------------------------------
------
>
> Correct I should have clarified that at first. That is where the
> confusion started. Basically, it seemed to me that as and when I added
> mass/energy to any object it contributed to its stress-energy tensor.

The big "curver of space" is the energy-momentum vector. For an isolated
particle, the length of this vector is just its rest mass. For a system of
non-interacting particles, the energy-momentum of the system is simply the
sum of the energy-momentums of its constituents, i.e., the sum of their rest
masses. So the kinetic energy of a system's parts does not per se add to
the system's mass. That is why the statement "energy is mass" is somewhat
misleading, since Einstein only meant E=mc^2 to be applied to a particle in
its rest frame.

Now with the cyclotron someone mentioned in another post, it is a different
matter. Here, the particle interacts with the cyclotron (it has to, since
it's supposed to follow a non-geodesic, circular orbit). This interaction
manifests itself as potential energy (you have to apply a constant inwards
force to keep the particle in orbit). This energy will literally
materialize itself as an increase in the systems mass.

> You have a static uncharged spehere. You spin it up, it adds to its
> stress-energy. You add charges to it, it adds to it stress-energy. In
> each case as energy is spent on it some of it is converted into some
> form of energy that contributes to the stress-energy tensor and hence
> there seems to be no theoritical reason on why that tensor cannot be
> made to form a black hole given enough energy is added to it.

You are right on the target. Sooner or later, this sphere will form a black
hole, unless it explodes first.

> Now, with a linearly accelerating mass, energy is definitely being
> spent on it. That energy is being transformed into kinetic energy
> (which is the other confusing concept because this meaure of energy
> seems to be frame dependant)

Yes it is, that's why energy-momentum, which is frame invariant, and other
frame invariant concepts, are the preferred notions in relativity. I think
Greg Egans introduction is very readable:

http://gregegan.customer.netspace.ne...ound03.html#s3

Chalky

Blackbird Wrote:
----------------------------------------------------------------------------------
http://gregegan.customer.netspace.ne...ound03.html#s3

----------------------------------------------------------------------------------

Great. That helped a lot and cleared my confusion.

markwh04@yahoo.com

Pmb wrote:
> If a object is not a black hole in one frame of reference then it won't be a
> black hole in any other frame even though the grazvitational field will
> increase with speed.

There are subtleties involved here that are not covered in any FAQ; in
large measure because of the vagueness of comments like those above.

As no less than Hawking, himself, has pointed out, the occurrence and
existence of causal horizons is *frame*-dependent [1] and there is, in
fact, no trapped region associated with a black hole -- that is, no
"black hole" in the usual classical sense of the term. Instead, there
is only a causal horizon, and one which is observer-dependent. Even the
event horizon associated with a black hole is frame-dependent.

So the question needs to be closely reexamined, since the FAQ and the
conventional wisdom of the previous century are both a little naive
with respect to the general issue.

This is, of course, not the only place there conventional wisdom will
go wrong and where the FAQ needs to be reexamined. Another good example
concerns the visual appearance of a moving sphere, where there is
actually a double-take end-run on conventional wisdom. The naive
assessment is that such an object appears flattened out. A closer
examination of the optics involved shows that the object would appear
spherical -- which is also naive.

The double-take occurs here in that this, too, is wrong. It actually
appears circular, since the optics is respect to *one* eye.

The actual 3-dimensional *stereographic* view of the moving object
(i.e., that obtained by receiving images in two eyes in real-time) need
not be spherical at all.

Note:
[1] "Frame" when used in the context of quantum theory in curved
spacetime does *not* mean coordinate grid, but global timelike flow.
We're not talking about coordinate dependence above, but dependence on
the selection of a timelike field.