I am, as you guessed, a hobbyist, though a halfway experienced one. Don't hesitate to reference formulas or papers :)

OK ! Let us slide into a full relativistic discussion/explanation then ? As you have a compound question I'll answer in parts, satisfying any of your queries before we move to the next bit. :-) :-)

First up then :

Quote:

..... Let a jupiter plunge into a BH from a long distance ... How much mass does the BH gain

Cast your mind back to Einstein's original formulation of his most famous equation [ call it (1) ] :

m = E / c^2

which he derived by considering a massive object that had emitted a single photon. To extract this result he followed the implications of his ( then novel ) approach to space, time and the speed of light. The body loses inertial mass in proportion to the energy of the photon, with the constant of proportionality being the inverse of the speed of light squared. By inertial is meant that prior or subsequent ( to the photon emission ) dynamical interactions would differ in consequence due to that mass change. This is time symmetric. You can run time backwards and have the body absorb the photon and hence gain inertial mass.

You could go plural and calculate some nett gain or loss of photons, each of differing energies if you like, do your flux sums on the emissions and absorptions etc. For example I add ( slightly ) to the mass of buns baking in an oven by a nett irradiation of them, and when they cool on the bench they will ( slightly ) lose mass by nett radiation away. The reason why we hadn't noticed this effect before Einstein's revelation is that with use of human scale units of mass, energy and light speed we would not appreciably notice. Or if you like : everyday life is non-relativistic to the degree that we are sensitive to, else this stuff would be second nature to us.

Now by fairly simple logic one can extend the types of energy that may appear on the right hand side of equation (1). Roughly speaking photons can interact directly or otherwise with anything you may care to name, and thus Einstein's original demonstration may be modified mutatis mutandis. This hence leads to the bold assertion that

any form of energy whatsoever may be considered to have an inertial mass equivalent [ via equation (1) ]

This is an incredibly deep statement about the Universe. Its practical effects have been verified on each and every occasion that experiment has been sufficiently precise to comment on it. No known exceptions to date.

General Relativity ( GR ) retains this principle. GR not so much replaces Special Relativity ( SR ) as extends it, as SR in turn has extended Newtonian or classical dynamics. This encompassing of one theory by its refinement is often misunderstood. In practice it is a matter less of truth in the abstract but sufficient accuracy for some purpose on the day, or the degree to which one chooses to attend to measured outcomes.

So now to GR. The base principle is that inertial mass is identical to gravitational mass. The Equivalence Principle. So the feature of a body which determines its dynamical response [ equation (2) or Newton's Law of Force ] :

F = m_i * a

where a is the acceleration experienced, and the inertial mass is called m_i to distinguish it from m_g or the gravitational mass [ equation (3) or Newton's Law of Universal Gravity ] :

F = G * m_g * M_g / r^2

where M_g is the other mass in the gravitational interaction, G is the constant of proportionality and r is the mutual separation. Einstein insists [ equation (4) ] :

m_i = m_g

Of course Albert was not the first to state this. So by now you sense the inevitable linkage amongst all of these. The incoming Jupiter will encounter the black hole's event horizon with a particular sum total of all it's energies, including kinetic, and that total has a calculable mass equivalent ( call it either/or/both inertial/gravitational ). That increases the BH's mass equivalent, which can now enter any subsequent evolution of interest and be fully valid.

But there is still an implicit issue to deal with, which I now bring forth. In classical mechanics space and time are indifferent to any goings on within. So time is a universal counter that progresses equally in step everywhere, does not depend on location or movements. Any length standard ( rigid rod ) acts likewise. Light travel is instantaneous ie. infinite speed. So in this construct when one says 'the kinetic energy of Jupiter is so & so' then there is no great need to qualify that comment further. Any and all observers may agree about that energy amount, say, by indicating that a common universal rest frame is serving as the reference for all discussion. Of course observers may be darting about within the frame and thus disagree about specific numbers from measurement occasions. But that could still be adjusted/resolved by any particular observer's status with respect to the universal rest frame etc. The null result from the Michelson-Morley and related efforts absolutely exclude such a paradigm.

The upshot is that all relativistic treatments must/should be annotated with the situation and type of observational platform. I won't at present go into the full gore of what defines reference frames and the nuances thereof. One may choose to think in terms of some overall God's Eye View to sort cause from effects etc, but there is no experimental God's Eye View ( that is a residue of classical physics modelling ). As I've indicated with prior answers, and this Jupiter/BH example, the frame I've been using to describe a black hole is the far limit.

In theory that is infinitely distant which raises the obvious point of how could I ever know what is happening down near the event horizon ? For starters light generated there would never reach me, for light is not of infinite speed.

So in practice one can specify some degree of tolerance to which measurements will be made and scenarios described. Then one can calculate how distant one would need to be in order to satisfy said accuracy. What I seek then is a suitably close approximation to an inertial reference frame. SR is deemed to hold exactly in such a frame, spacetime is 'flat' and the metric used is Minkowskian.

But gravity bends/breaks all inertial frames, to some degree, by causing deviation from rectilinear motion in the absence of applied forces. Thus the frame is classified as non-inertial because the law of inertia does not apply. The language is not phrased as gravity being a force encountered by particles in an indifferent background. Gravity is the alteration of the background from the flat case above. Objects in 'free fall' - not subject to the other three forces of nature - traverse within according to Einstein's GR equations. However mathematically horrible those equations are, in those instances where solutions/approximations are obtained to compare with experience the comparison is outstanding eg. Mercury's perihelion advance, Taylor-Hulse system, gravitational lensing, the GPS system etc.

If you've tracked me this far, well done ! Queries please .... ;-)

Cheers, Mike.

( edit ) FWIW : the first book I ever read specifically devoted to SR is 'Special Relativity' by A.P.French ( MIT ) [ ISBN-10: 0412343207 | ISBN-13: 78-0412343209 ], it is a really good bridge for those who already have a decent fist of knowledge of classical physics.

( edit ) .... and if you want to grasp that classical physics you could do a lot worse than 'Newtonian Mechanics (The M.I.T. Introductory Physics Series)' by the same author [ ISBN-10: 0393099709 | ISBN-13: 978-0393099706 ]

( edit ) Whoops. Order of magnitudes. In general for anything :

E^2 = p^2 * c^2 + m_0^2 * c^4

( p is momentum, m_0 is mass measured at rest ie. rest mass ). The rightmost term has c to the fourth power. That factor dominates the right side sum unless p is sufficiently large to compare with it. Or if you like : how fast is your Jupiter coming in ?

BTW, a photon has no rest mass ie. m_0 = 0 and so :

E = p * c

whereas a massive body at rest has p = 0 :

E = m_0 * c^2

I have made this letter longer than usual because I lack the time to make it shorter ...

... and my other CPU is a Ryzen 5950X :-) Blaise Pascal

I tracked you so far, most of what you wrote sounds familiar to me.

One thing that's puzzling me a bit is about kinetic energy. I know that the term "mass" should only be used for "rest mass" and the term "relativistic mass" should be avoided (i.e. impulse divided by velocity).

Though I still wonder about the following thing:

* Have 2 jupiters collide head-on with a velocity high enough so that at the moment of collision the whole energy involved is enough to form a jupiter-sized black hole. I assume they will do exactly that then, right?
* Now, replace one jupiter with a ring of jupiter mass so that the jupiter could just pass through it (and the whole energy still enough to form a black hole including everything the moment they pass). Will GR still cause the formation of a black hole in this case?

I tracked you so far, most of what you wrote sounds familiar to me.

One thing that's puzzling me a bit is about kinetic energy. I know that the term "mass" should only be used for "rest mass" and the term "relativistic mass" should be avoided (i.e. impulse divided by velocity).

Personally I like 'inertial mass' ie. that being relevant to dynamical interactions eg. inertial mass increases with speed, or in accelerators they convert inertial mass to spawn particles. Rest mass is then inertial mass at speed zero ( per frame ). Then in GR Einstein says inertial mass identifies with gravitational mass. But whichever definitions you choose they should be 'tight' so that one neither double counts nor leaves anything out.

Quote:

[ASIDE]Another useful term is 'mass shell', that being the description of the hyperbolic shape - in four dimensions - implied by the equation

E^2 = p^2 * c^2 + m_0^2 * c^4

Here the rest mass ( m_0 ) determines the offset from the ( p = 0 ) origin of the 'shell'. In this approach 'kinetic' energy could be that labelled as arising from motion in a frame, thus separate from that attributed to rest mass alone. Mind you, be careful with rest-mass-less photons, as their energy is independent of speed. All this skirts around the deeper question as to what energy really is ..... I've read viewpoints that put energy as the basic physical quantity ( hence as with any axiom, assumed to need no deeper definition ) from which all other measurables are derived.

Real particles ie. those that interact to produce a directly measurable outcome, have to stay on that surface but those virtual ones ( that live for a short enough time interval so that the energy/time product remains below Planck's constant as per Heisenberg ) don't, and are thus labelled 'off-shell'. The only way that such virtual quantities can have measurable meaning, and one can legitimately challenge such a construct I think, is via contributions to path integrals ( Feynman etc ) and hence quantum expectation values eg. Casimir effect.[/ASIDE]

Quote:

Though I still wonder about the following thing:

* Have 2 jupiters collide head-on with a velocity high enough so that at the moment of collision the whole energy involved is enough to form a jupiter-sized black hole. I assume they will do exactly that then, right?

Above a certain density, yes. You have to get that mass within some given radius, so that the spacetime curvature is extreme enough to trap even light. For black holes I use the approximation of 6km per solar mass, the Sun is about a thousand Jupiters so that would give roughly 12 metres for 2 Jupiters.

Quote:

* Now, replace one jupiter with a ring of jupiter mass so that the jupiter could just pass through it (and the whole energy still enough to form a black hole including everything the moment they pass). Will GR still cause the formation of a black hole in this case?

Same as above really ie. what's the density? I guess there are lots of ways that may lead up to the formation of an event horizon. Once formed then any non-axisymmetric ( quadrupolar and higher terms ) aspects smooth out with gravitational radiation, labelled as 'ring down'. But prior to that horizon formation there will be mechanism dependent energy exchanges going on with the rest of The Universe ( ignoring Hawking radiation for the moment ). Isn't one of the intense gamma ray burst hypotheses the final squawk of some object being swallowed by a black hole ?

Cheers, Mike.

I have made this letter longer than usual because I lack the time to make it shorter ...

... and my other CPU is a Ryzen 5950X :-) Blaise Pascal

For those who may be wondering what is really meant in practical terms by 'inertial' and 'kinetic energy' then check out this nifty demonstration. It will take a while to load into whatever your streaming software is, so I recommend that you go and do something else for awhile and then come back. Lest the pauses while loading annoy you. The whole setup looks quite err ... Electro-Punk ?? :-)

The basic ideas are :

- electrons are the smallest rest mass particles then known ( 1961 ), so are the easiest to accelerate.

- the electron's charge is used by an applied electric field to propel it. Basically the Van de Graaff generator separates charges within a metallic structure, moving them to one end of the gadget by an elongated belt. It actually rubs electrons off one terminal and carries them to the other. The linac section works differently ( unimportant for the result ) but has the same energy boosting effect on the electrons.

- another component 'boils' the electrons off a metal plate, into vacuum, and they are then grabbed and accelerated down the line. This happens in bunches/pulses many times per second.

- the energy given to an electron by the field is it's charge times the voltage.

- this is a direct point-to-point time of flight measurement. The first point is timed by the induction of a small current ( a magnetic effect ) as the electron bunches zip through the space within a short hollow metal cylinder. The second point is timed by the current produced by the arrival of said bunches ie. the movement of the electrons down the line is a current.

- the rigamarole with the cables and the oscilloscope is to ensure that the time of flight in the vacuum tube is also the time difference as sampled at the oscilloscope inputs.

- with knowledge of the thermal properties of the target disc one can check that the energy given to the electrons at one end actually arrives at the other. The electrons collide with the metal disc and thus agitate it ie. heat it up. So one counts the amount of charge transferred to do this tally at both ends, finally seeing that the energies agree ( close enough ).

One could summarise by saying that the inertial mass increases with speed, or 'the faster you go the harder it is to go faster'.

So the speed of light is the limit approached. I think one of the Fermilab accelerator upgrades in the 1970's doubled the beam energy but added a mere 40 miles per hour to the particle speed !!

Cheers, Mike.

I have made this letter longer than usual because I lack the time to make it shorter ...

... and my other CPU is a Ryzen 5950X :-) Blaise Pascal

Correction : a one solar mass black hole has ~ 6km diameter and thus 3km radius .... but you'd still have to stuff 2 Jupiters within the length of my house. :-)

Cheers, Mike.

I have made this letter longer than usual because I lack the time to make it shorter ...

... and my other CPU is a Ryzen 5950X :-) Blaise Pascal

Alternatively if you threw two actual Jupiters at each other to get a black hole ie. each a Jupiter mass ( 1/1000th solar mass ) in a Jupiter width ( 72000km radius ) requires 72000/3 = 24 thousand solar masses worth of inertial mass. To get a Jupiter up to a speed that produces that inertial mass requires a 24 million fold increase above rest mass. That requires a speed just under light speed, to within one part in 10^15 of c ..... solving 2.4 * 10^7 = sqrt[1 - (v/c)^2] here.

Cheers, Mike.

I have made this letter longer than usual because I lack the time to make it shorter ...

... and my other CPU is a Ryzen 5950X :-) Blaise Pascal

## RE: Mike, thank you for the

)

OK ! Let us slide into a full relativistic discussion/explanation then ? As you have a compound question I'll answer in parts, satisfying any of your queries before we move to the next bit. :-) :-)

First up then :

Cast your mind back to Einstein's original formulation of his most famous equation [ call it (1) ] :

m = E / c^2

which he derived by considering a massive object that had emitted a single photon. To extract this result he followed the implications of his ( then novel ) approach to space, time and the speed of light. The body loses inertial mass in proportion to the energy of the photon, with the constant of proportionality being the inverse of the speed of light squared. By inertial is meant that prior or subsequent ( to the photon emission ) dynamical interactions would differ in consequence due to that mass change. This is time symmetric. You can run time backwards and have the body absorb the photon and hence gain inertial mass.

You could go plural and calculate some nett gain or loss of photons, each of differing energies if you like, do your flux sums on the emissions and absorptions etc. For example I add ( slightly ) to the mass of buns baking in an oven by a nett irradiation of them, and when they cool on the bench they will ( slightly ) lose mass by nett radiation away. The reason why we hadn't noticed this effect before Einstein's revelation is that with use of human scale units of mass, energy and light speed we would not appreciably notice. Or if you like : everyday life is non-relativistic to the degree that we are sensitive to, else this stuff would be second nature to us.

Now by fairly simple logic one can extend the types of energy that may appear on the right hand side of equation (1). Roughly speaking photons can interact directly or otherwise with anything you may care to name, and thus Einstein's original demonstration may be modified mutatis mutandis. This hence leads to the bold assertion that

any form of energy whatsoever may be considered to have an inertial mass equivalent [ via equation (1) ]

This is an incredibly deep statement about the Universe. Its practical effects have been verified on each and every occasion that experiment has been sufficiently precise to comment on it. No known exceptions to date.

General Relativity ( GR ) retains this principle. GR not so much replaces Special Relativity ( SR ) as extends it, as SR in turn has extended Newtonian or classical dynamics. This encompassing of one theory by its refinement is often misunderstood. In practice it is a matter less of truth in the abstract but sufficient accuracy for some purpose on the day, or the degree to which one chooses to attend to measured outcomes.

So now to GR. The base principle is that inertial mass is identical to gravitational mass. The Equivalence Principle. So the feature of a body which determines its dynamical response [ equation (2) or Newton's Law of Force ] :

F = m_i * a

where a is the acceleration experienced, and the inertial mass is called m_i to distinguish it from m_g or the gravitational mass [ equation (3) or Newton's Law of Universal Gravity ] :

F = G * m_g * M_g / r^2

where M_g is the other mass in the gravitational interaction, G is the constant of proportionality and r is the mutual separation. Einstein insists [ equation (4) ] :

m_i = m_g

Of course Albert was not the first to state this. So by now you sense the inevitable linkage amongst all of these. The incoming Jupiter will encounter the black hole's event horizon with a particular sum total of all it's energies, including kinetic, and that total has a calculable mass equivalent ( call it either/or/both inertial/gravitational ). That increases the BH's mass equivalent, which can now enter any subsequent evolution of interest and be fully valid.

But there is still an implicit issue to deal with, which I now bring forth. In classical mechanics space and time are indifferent to any goings on within. So time is a universal counter that progresses equally in step everywhere, does not depend on location or movements. Any length standard ( rigid rod ) acts likewise. Light travel is instantaneous ie. infinite speed. So in this construct when one says 'the kinetic energy of Jupiter is so & so' then there is no great need to qualify that comment further. Any and all observers may agree about that energy amount, say, by indicating that a common universal rest frame is serving as the reference for all discussion. Of course observers may be darting about within the frame and thus disagree about specific numbers from measurement occasions. But that could still be adjusted/resolved by any particular observer's status with respect to the universal rest frame etc. The null result from the Michelson-Morley and related efforts absolutely exclude such a paradigm.

The upshot is that all relativistic treatments must/should be annotated with the situation and type of observational platform. I won't at present go into the full gore of what defines reference frames and the nuances thereof. One may choose to think in terms of some overall God's Eye View to sort cause from effects etc, but there is no experimental God's Eye View ( that is a residue of classical physics modelling ). As I've indicated with prior answers, and this Jupiter/BH example, the frame I've been using to describe a black hole is the far limit.

In theory that is infinitely distant which raises the obvious point of how could I ever know what is happening down near the event horizon ? For starters light generated there would never reach me, for light is not of infinite speed.

So in practice one can specify some degree of tolerance to which measurements will be made and scenarios described. Then one can calculate how distant one would need to be in order to satisfy said accuracy. What I seek then is a suitably close approximation to an inertial reference frame. SR is deemed to hold exactly in such a frame, spacetime is 'flat' and the metric used is Minkowskian.

But gravity bends/breaks all inertial frames, to some degree, by causing deviation from rectilinear motion in the absence of applied forces. Thus the frame is classified as non-inertial because the law of inertia does not apply. The language is not phrased as gravity being a force encountered by particles in an indifferent background. Gravity is the alteration of the background from the flat case above. Objects in 'free fall' - not subject to the other three forces of nature - traverse within according to Einstein's GR equations. However mathematically horrible those equations are, in those instances where solutions/approximations are obtained to compare with experience the comparison is outstanding eg. Mercury's perihelion advance, Taylor-Hulse system, gravitational lensing, the GPS system etc.

If you've tracked me this far, well done ! Queries please .... ;-)

Cheers, Mike.

( edit ) FWIW : the first book I ever read specifically devoted to SR is 'Special Relativity' by A.P.French ( MIT ) [ ISBN-10: 0412343207 | ISBN-13: 78-0412343209 ], it is a really good bridge for those who already have a decent fist of knowledge of classical physics.

( edit ) .... and if you want to grasp that classical physics you could do a lot worse than 'Newtonian Mechanics (The M.I.T. Introductory Physics Series)' by the same author [ ISBN-10: 0393099709 | ISBN-13: 978-0393099706 ]

( edit ) Whoops. Order of magnitudes. In general for anything :

E^2 = p^2 * c^2 + m_0^2 * c^4

( p is momentum, m_0 is mass measured at rest ie. rest mass ). The rightmost term has c to the fourth power. That factor dominates the right side sum unless p is sufficiently large to compare with it. Or if you like : how fast is your Jupiter coming in ?

BTW, a photon has no rest mass ie. m_0 = 0 and so :

E = p * c

whereas a massive body at rest has p = 0 :

E = m_0 * c^2

I have made this letter longer than usual because I lack the time to make it shorter ...

... and my other CPU is a Ryzen 5950X :-) Blaise Pascal

## I tracked you so far, most of

)

I tracked you so far, most of what you wrote sounds familiar to me.

One thing that's puzzling me a bit is about kinetic energy. I know that the term "mass" should only be used for "rest mass" and the term "relativistic mass" should be avoided (i.e. impulse divided by velocity).

Though I still wonder about the following thing:

* Have 2 jupiters collide head-on with a velocity high enough so that at the moment of collision the whole energy involved is enough to form a jupiter-sized black hole. I assume they will do exactly that then, right?

* Now, replace one jupiter with a ring of jupiter mass so that the jupiter could just pass through it (and the whole energy still enough to form a black hole including everything the moment they pass). Will GR still cause the formation of a black hole in this case?

## RE: I tracked you so far,

)

Personally I like 'inertial mass' ie. that being relevant to dynamical interactions eg. inertial mass increases with speed, or in accelerators they convert inertial mass to spawn particles. Rest mass is then inertial mass at speed zero ( per frame ). Then in GR Einstein says inertial mass identifies with gravitational mass. But whichever definitions you choose they should be 'tight' so that one neither double counts nor leaves anything out.

Above a certain density, yes. You have to get that mass within some given radius, so that the spacetime curvature is extreme enough to trap even light. For black holes I use the approximation of 6km per solar mass, the Sun is about a thousand Jupiters so that would give roughly 12 metres for 2 Jupiters.

Same as above really ie. what's the density? I guess there are lots of ways that may lead up to the formation of an event horizon. Once formed then any non-axisymmetric ( quadrupolar and higher terms ) aspects smooth out with gravitational radiation, labelled as 'ring down'. But prior to that horizon formation there will be mechanism dependent energy exchanges going on with the rest of The Universe ( ignoring Hawking radiation for the moment ). Isn't one of the intense gamma ray burst hypotheses the final squawk of some object being swallowed by a black hole ?

Cheers, Mike.

I have made this letter longer than usual because I lack the time to make it shorter ...

... and my other CPU is a Ryzen 5950X :-) Blaise Pascal

## For those who may be

)

For those who may be wondering what is really meant in practical terms by 'inertial' and 'kinetic energy' then check out this nifty demonstration. It will take a while to load into whatever your streaming software is, so I recommend that you go and do something else for awhile and then come back. Lest the pauses while loading annoy you. The whole setup looks quite err ... Electro-Punk ?? :-)

The basic ideas are :

- electrons are the smallest rest mass particles then known ( 1961 ), so are the easiest to accelerate.

- the electron's charge is used by an applied electric field to propel it. Basically the Van de Graaff generator separates charges within a metallic structure, moving them to one end of the gadget by an elongated belt. It actually rubs electrons off one terminal and carries them to the other. The linac section works differently ( unimportant for the result ) but has the same energy boosting effect on the electrons.

- another component 'boils' the electrons off a metal plate, into vacuum, and they are then grabbed and accelerated down the line. This happens in bunches/pulses many times per second.

- the energy given to an electron by the field is it's charge times the voltage.

- this is a direct point-to-point time of flight measurement. The first point is timed by the induction of a small current ( a magnetic effect ) as the electron bunches zip through the space within a short hollow metal cylinder. The second point is timed by the current produced by the arrival of said bunches ie. the movement of the electrons down the line is a current.

- the rigamarole with the cables and the oscilloscope is to ensure that the time of flight in the vacuum tube is also the time difference as sampled at the oscilloscope inputs.

- with knowledge of the thermal properties of the target disc one can check that the energy given to the electrons at one end actually arrives at the other. The electrons collide with the metal disc and thus agitate it ie. heat it up. So one counts the amount of charge transferred to do this tally at both ends, finally seeing that the energies agree ( close enough ).

One could summarise by saying that the inertial mass increases with speed, or 'the faster you go the harder it is to go faster'.

So the speed of light is the limit approached. I think one of the Fermilab accelerator upgrades in the 1970's doubled the beam energy but added a mere 40 miles per hour to the particle speed !!

Cheers, Mike.

I have made this letter longer than usual because I lack the time to make it shorter ...

... and my other CPU is a Ryzen 5950X :-) Blaise Pascal

## Correction : a one solar mass

)

Correction : a one solar mass black hole has ~ 6km diameter and thus 3km radius .... but you'd still have to stuff 2 Jupiters within the length of my house. :-)

Cheers, Mike.

I have made this letter longer than usual because I lack the time to make it shorter ...

... and my other CPU is a Ryzen 5950X :-) Blaise Pascal

## Alternatively if you threw

)

Alternatively if you threw two actual Jupiters at each other to get a black hole ie. each a Jupiter mass ( 1/1000th solar mass ) in a Jupiter width ( 72000km radius ) requires 72000/3 = 24 thousand solar masses worth of inertial mass. To get a Jupiter up to a speed that produces that inertial mass requires a 24 million fold increase above rest mass. That requires a speed just under light speed, to within one part in 10^15 of c ..... solving 2.4 * 10^7 = sqrt[1 - (v/c)^2] here.

Cheers, Mike.

I have made this letter longer than usual because I lack the time to make it shorter ...

... and my other CPU is a Ryzen 5950X :-) Blaise Pascal