What is gamma factor of object, which is falling into black hole?

[DimaMazin]

What is gamma factor of object, which is freely falling into black hole, on event horizon?

Edited November 6, 2022 by DimaMazin

[Markus Hanke]

On 11/6/2022 at 12:49 PM, DimaMazin said:

What is gamma factor of object, which is freely falling into black hole, on event horizon?

The gamma factor is used to characterise the relationship between inertial frames in flat spacetime, ie between frames that are related via Lorentz transformations. When you have a test particle freely falling into a black hole, it will trace out a world line in a spacetime that is not flat - you can still choose another far-away frame as reference, and both of these will be locally inertial, but spacetime between them isn’t flat, so these frames are not related by simple Lorentz transformations. Hence, asking about what the gamma factor between these frames will be is meaningless - it is only defined for frames that are related via Lorentz transforms.

[md65536]

On 11/8/2022 at 9:11 AM, Markus Hanke said:

spacetime between them isn’t flat, so these frames are not related by simple Lorentz transformations. Hence, asking about what the gamma factor between these frames will be is meaningless

You could do Lorentz transformations for two local inertial frames at a location the event horizon passes. (However the horizon itself is lightlike so you can't use it as an inertial frame or even describe the speed of the object relative to the horizon except in some other given frame of reference.)

Relativistic effects due to high speeds still occur in curved spacetime. I'm not sure how the effects combine but it seems easy to set up an example with A and B "nearby" and related by a Lorentz transformation, and a distant C related to B by gravitational time dilation say, and if light from A passes through B and reaches C then the total redshift between A and C will be the Doppler shift between A and B multiplied by the gravitational redshift between B and C (you can see this by letting B be a signal repeater and consider what it observes of A and sends to C).

In this way you could separate the relativistic effects due to SR and GR in some specific example, and have "gamma factor" be meaningful, but not only would you have to define frames of reference and/or what the free-falling object's velocity is relative to, but depending on what one means, such as in my example, one might also have to define gamma if it's not standard.

 

On 11/6/2022 at 4:49 AM, DimaMazin said:

What is gamma factor of object, which is freely falling into black hole, on event horizon?

Regardless of whether there's any sense in what I wrote above, you'd at least have to say what frame of reference you're using, and depending on that, different objects could have different velocities as they cross the horizon, so that would also need to be defined.

 

[Markus Hanke]

1 hour ago, md65536 said:

Relativistic effects due to high speeds still occur in curved spacetime.

Yes, of course.

1 hour ago, md65536 said:

I'm not sure how the effects combine

It depends on the effect. In the simplest cases, they just add - for example, the total difference in tick rates between a clock on earth and a clock in an orbiting satellite will just be the sum of gravitational time dilation and kinematic time dilation between these frames.

1 hour ago, md65536 said:

In this way you could separate the relativistic effects due to SR and GR in some specific example

Yes.

1 hour ago, md65536 said:

and have "gamma factor" be meaningful

Personally I associate the gamma factor with inertial frames in Minkowski spacetime, since gamma arises from Lorentz transformations. I think in the interest of clarity and consistency it is best to avoid this terminology when working in curved spacetimes, and just refer to the specific quantity in question instead. For example, in the OP’s scenario it would be best to speak about time dilation, rather than the gamma factor, simply to avoid unnecessary confusion.

There’s also the danger that someone might naively take the gamma factor and apply it to quantities that ‘behave’ differently in the presence of gravity - take for example the OP’s scenario, but use the observed length of the falling object as the quantity in question, rather than time dilation. The result won’t be correct, because in an inhomogeneous gravitational field you have extra tidal effects that don’t exist in Minkowski spacetime. To be fair, you could again separate the various effects, as you suggested - but I think you can see the potential confusion a naive application of gamma to frames in curved spacetimes might cause.

[md65536]

Maybe OP can clarify? But in the meantime, a search shows headlines like, "Matter clocked speeding toward a black hole at 30 percent the speed of light". https://astronomy.com/news/2018/09/matter-clocked-speeding-toward-a-black-hole-at-30-percent-the-speed-of-light

Is the meaning of that speed understood, or does it need more details? It seems ambiguous. I'd assume they mean different things depending on how far from the black hole the matter is.

[swansont]

6 hours ago, md65536 said:

Is the meaning of that speed understood, or does it need more details? It seems ambiguous.

What’s ambiguous about “30% of the speed of light”?

[DimaMazin]

On 11/8/2022 at 7:11 PM, Markus Hanke said:

The gamma factor is used to characterise the relationship between inertial frames in flat spacetime, ie between frames that are related via Lorentz transformations. When you have a test particle freely falling into a black hole, it will trace out a world line in a spacetime that is not flat - you can still choose another far-away frame as reference, and both of these will be locally inertial, but spacetime between them isn’t flat, so these frames are not related by simple Lorentz transformations. Hence, asking about what the gamma factor between these frames will be is meaningless - it is only defined for frames that are related via Lorentz transforms.

I mean    gamma' = 1/(1 - escape velocity²/c²)^1/2

For example distance of the fall is 1 trillion kilometers.

Mass of the black hole is minimal.

There is no initial velocity of the object relative to the black hole.

If the gamma' = about 1 million then let's consider next example when black hole whith minimal mass is falling into black hole with minimal mass.

If their  kinetic energy > 2 millions minimal mass of black hole *c²       then where does the energy go after their collision?

 

[Markus Hanke]

16 hours ago, DimaMazin said:

I mean    gamma' = 1/(1 - escape velocity²/c²)^1/2

For example distance of the fall is 1 trillion kilometers.

Mass of the black hole is minimal.

There is no initial velocity of the object relative to the black hole.

If the gamma' = about 1 million then let's consider next example when black hole whith minimal mass is falling into black hole with minimal mass.

 

I’m sorry, but I don’t understand what you are trying to do here? What do you mean by “minimal mass”? What kind of black hole are talking about (presumably Schwarzschild)?

16 hours ago, DimaMazin said:

If their  kinetic energy > 2 millions minimal mass of black hole *c²       then where does the energy go after their collision?

Kinetic energy is a an observer-dependent concept, so which frame are you working in?

[md65536]

18 hours ago, DimaMazin said:

let's consider next example when black hole whith minimal mass is falling into black hole with minimal mass.

If their  kinetic energy > 2 millions minimal mass of black hole *c²       then where does the energy go after their collision?

If they fall directly toward each other, it goes into the resulting combined black hole. Consider a simpler example of light "falling" into a black hole. Photons have no rest mass, yet adding light to a black hole increases its mass.

Your system of 2 black holes has a rest mass in the frame of the center of momentum of the system, where the total energy (including mass and kinetic energy of the 2 black holes) is the mass of the system, and the system as a whole has net zero kinetic energy. Assuming no energy is radiated away as gravitational waves etc., the mass of the system is the same before and after they collide.

On 11/11/2022 at 6:06 PM, swansont said:

What’s ambiguous about “30% of the speed of light”?

The coordinate system. Is it clear to you what coordinates or frame of reference they're talking about?

[swansont]

4 hours ago, md65536 said:

Your system of 2 black holes has a rest mass in the frame of the center of momentum of the system, where the total energy (including mass and kinetic energy of the 2 black holes) is the mass of the system, and the system as a whole has net zero kinetic energy

Net zero kinetic energy? If they each have KE, the system has KE. 

4 hours ago, md65536 said:

The coordinate system. Is it clear to you what coordinates or frame of reference they're talking about?

Since it was measured by humans, it seems a given that it was from our reference frame.

[md65536]

18 hours ago, swansont said:

Net zero kinetic energy? If they each have KE, the system has KE.

Yes, I mixed up KE and momentum. The kinetic energy of its parts contribute to the system's rest mass. In its center-of-momentum frame, the system has net zero momentum, and the total energy of the system is equivalent to its rest mass.

 

18 hours ago, swansont said:

Since it was measured by humans, it seems a given that it was from our reference frame.

This is galaxy PG1211+143 they're observing, about a billion light years away. Wouldn't they compensate for cosmological redshift? Wouldn't they describe the speed of stuff falling into a black hole relative to the black hole, and not the Earth?

[DimaMazin]

On 11/14/2022 at 10:31 AM, md65536 said:

Yes, I mixed up KE and momentum. The kinetic energy of its parts contribute to the system's rest mass. In its center-of-momentum frame, the system has net zero momentum, and the total energy of the system is equivalent to its rest mass

Then the rest mass after collision should be = m₁ + m₂ - energy of gravitational bubble /c² +KE/c²

But really mass=m₁ +m₂ - energy of gravitational bubble /c² + 0   Why so?

[md65536]

14 hours ago, DimaMazin said:

Then the rest mass after collision should be = m₁ + m₂ - energy of gravitational bubble /c² +KE/c²

But really mass=m₁ +m₂ - energy of gravitational bubble /c² + 0   Why so?

What are you basing this on? You're saying it should be one thing but is "really" another, where do you get the latter from?

If you apply your equations to quarks forming a proton, it really does include the KE. From https://en.wikipedia.org/wiki/Proton "The mass of a proton is about 80–100 times greater than the sum of the rest masses of its three valence quarks, while the gluons have zero rest mass. [...] The rest mass of a proton is, thus, the invariant mass of the system of moving quarks and gluons that make up the particle, and, in such systems, even the energy of massless particles is still measured as part of the rest mass of the system."

In the case of a black hole you can't measure or assert any internal motion, but the externally measurable rest mass is still there and the energy is still conserved.

[DimaMazin]

4 hours ago, md65536 said:

What are you basing this on? You're saying it should be one thing but is "really" another, where do you get the latter from?

https://www.youtube.com/watch?v=6aCc5BzidsM

1.46 minutes

Edited November 22, 2022 by DimaMazin

[DimaMazin]

1 hour ago, DimaMazin said:

 

https://en.wikipedia.org/wiki/GW190521

[md65536]

13 hours ago, DimaMazin said:

https://en.wikipedia.org/wiki/GW190521

These are inspiral orbiting black holes. Yes, I see the numbers in the wiki add up to zero kinetic energy included. However if you look at the references eg. [4], the numbers don't add up exactly, and the uncertainties in the estimates of the black hole masses are so huge that it's not possible to give an exact figure for other energy in the system from the masses. The equation describing this would be more like, remnant BH mass Mf = m1+m2+everything_else_ignored - radiated_energy/c^2. It's likely that whatever is ignored is small on a scale of solar masses.

These aren't two masses falling directly toward each other, they come together because their orbits decay due to energy lost to gravitational waves. The kinetic energy is being radiated away as they approach. When they merge they'd generally have net angular momentum, resulting in a spinning black hole. The rotational energy of a black hole contributes to its mass (https://en.wikipedia.org/wiki/Kerr_metric#Mass_of_rotational_energy). Certainly then, some kinetic energy of the orbiting black holes contributed to the mass of the resulting black hole. In this case, at about 8 solar masses of energy radiated, much much more than any energy not accounted for is lost to gravitational waves.

Edited November 22, 2022 by md65536

[DimaMazin]

23 hours ago, md65536 said:

These are inspiral orbiting black holes.Yes, I see the numbers in the wiki add up to zero kinetic energy included. However if you look at the references eg. [4], the numbers don't add up exactly, and the uncertainties in the estimates of the black hole masses are so huge that it's not possible to give an exact figure for other energy in the system from the masses. The equation describing this would be more like, remnant BH mass Mf = m1+m2+everything_else_ignored - radiated_energy/c^2. It's likely that whatever is ignored is small on a scale of solar masses.

These aren't two masses falling directly toward each other, they come together because their orbits decay due to energy lost to gravitational waves. The kinetic energy is being radiated away as they approach. When they merge they'd generally have net angular momentum, resulting in a spinning black hole. The rotational energy of a black hole contributes to its mass (https://en.wikipedia.org/wiki/Kerr_metric#Mass_of_rotational_energy). Certainly then, some kinetic energy of the orbiting black holes contributed to the mass of the resulting black hole. In this case, at about 8 solar masses of energy radiated, much much more than any energy not accounted for is lost to gravitational waves.

If they fall directly toward each other then what mass would be after such collision?

Why creation of gravitational waves ,before their collision, takes only their KE and no significant mass?

Why creation of gravitational bubble,at the collision, takes only their mass and no significant KE?

[swansont]

15 minutes ago, DimaMazin said:

Why creation of gravitational waves ,before their collision, takes only their KE and no significant mass?

What constitutes “significant” mass? 9 solar masses (GW190521) seems significant to me.

Quote

Why creation of gravitational bubble,at the collision, takes only their mass and no significant KE?

Momentum must be conserved, so you can only have KE in the case where the system’s momentum is nonzero.

Gravitational waves are massless, so p = E/c, so there isn’t much recoil for any asymmetric waves production.

[DimaMazin]

10 hours ago, swansont said:

What constitutes “significant” mass? 9 solar masses (GW190521) seems significant to me.

 

Do you think 9 solar massis were emitted before the collision? Then what their KE was emitted before the collision?

[swansont]

5 hours ago, DimaMazin said:

Do you think 9 solar massis were emitted before the collision? Then what their KE was emitted before the collision?

I missed the “before”

The emitted energy is small before the collision. For rotational KE, the mass would be proportional to this, as E/c^2

[md65536]

23 hours ago, DimaMazin said:

If they fall directly toward each other then what mass would be after such collision?

There are additional replies better than I can give here: https://astronomy.stackexchange.com/questions/32445/head-on-collision-of-two-black-holes

23 hours ago, DimaMazin said:

Why creation of gravitational waves ,before their collision, takes only their KE and no significant mass?

Why creation of gravitational bubble,at the collision, takes only their mass and no significant KE?

If energy is lost from a system (in its CoM frame), the system loses mass. The binary system is losing mass equivalent to the gravitational wave energy radiated away, before the collision.

If an individual component of the system doesn't radiate energy itself, that component don't lose rest mass.

Radiating gravitational waves can reduce angular momentum, so the remnant BH should be decreasing its angular momentum during merger and ringdown.

After they're merged, the system only has the one object in it. Your question is basically asking about the components of the system before the merger, and the system as a whole after, that's the main difference. Also, I think the energy radiated while merging is vastly greater than that radiated during the inspiral phase.

 

 

Edited November 24, 2022 by md65536

[DimaMazin]

12 hours ago, md65536 said:

There are additional replies better than I can give here: https://astronomy.stackexchange.com/questions/32445/head-on-collision-of-two-black-holes

If energy is lost from a system (in its CoM frame), the system loses mass. The binary system is losing mass equivalent to the gravitational wave energy radiated away, before the collision.

If an individual component of the system doesn't radiate energy itself, that component don't lose rest mass.

Radiating gravitational waves can reduce angular momentum, so the remnant BH should be decreasing its angular momentum during merger and ringdown.

After they're merged, the system only has the one object in it. Your question is basically asking about the components of the system before the merger, and the system as a whole after, that's the main difference. Also, I think the energy radiated while merging is vastly greater than that radiated during the inspiral phase.

 

 

Thanks. I thought black hole mass is defined without its KE like partcle mass. It deluded me.