# Schwarzschild Radii and Relativistic Mass

• Lelan Thara
In summary, the conversation discusses the concept of relativistic mass and its relationship to the Schwarzschild radius and black holes. It is mentioned that special relativity states that as a massive body accelerates, its mass increases, and general relativity equates the force of acceleration with gravity. The question is raised whether any accelerating body would eventually reach a speed where its relativistic mass would surpass the Schwarzschild radius and cause it to become a black hole. The conversation also delves into the idea of relativistic mass being replaced by invariant mass, and whether time dilation and length contraction are still valid concepts. The experts in the conversation discuss the experimental evidence for time dilation and the use of momentum 4-vectors in interactions, rather than relativ
Lelan Thara
Hello, folks - here is a question that I have been pondering for about 20 years.

As I understand it, the Schwarzschild radius can be thought of as a measure of how much mass can fit within a given space before that space warps from gravity to such an extent that it acquires an event horizon and collapses into a black hole.

- Special relativity tells us that as a massive body accelerates, it's mass, as seen from a slower frame of reference, will increase. If the body could accelerate to the speed of light, it's relative mass would become infinite.

- General relativity tells us that the force of acceleration is equivalent to the force of gravity.

So, my question is - shouldn't any accelerating body have a finite speed below light speed at which its relativistic mass would surpass the SChwarzchild radius - causing the accelerating body to become a black hole?

This seems such an inescapable conclusion to me - is there something fundamental I am misunderstanding?

Science fiction often dwells on the problem of spacecraft traveling faster than the speed of light - if I am right, isn't this a moot point? If a spacecraft would become a black hole at a finite speed below light speed - what would be the theoretical implications for space travel? Do we have any idea what a rapidly moving black hole would be like? Do we have any sound theories on how spaces beyond an event horizon might be related to normal spacetime?

If we imagine travellers on this spacecraft - wouldn't they survive this gravitational (accelerational) collapse because their mass relative to the craft remains unchanged?

I apologize if the space travel parts of this question are too speculative. But I'd be grateful to hear what you folks have to say about my major question -

- if acceleration and gravitation are equivalent, shouldn't every accelerating body have a Schwarzschild radius?

(As usual, I must remind you I am a layman and ask you to indulge me with answers in plain English. Thanks.)

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Lelan Thara said:
shouldn't any accelerating body have a finite speed below light speed at which its relativistic mass would surpass the SChwarzchild radius - causing the accelerating body to become a black hole?

See http://www.physics.adelaide.edu.au/~dkoks/Faq/Relativity/BlackHoles/black_fast.html from the Usenet Physics FAQ.

jtbell said:
See http://www.physics.adelaide.edu.au/~dkoks/Faq/Relativity/BlackHoles/black_fast.html from the Usenet Physics FAQ.

Oh well. Thanks for the link, jtbell. I may have additional questions when I've had a chance to digest this all a bit.

I read the Usenet FAq directly related to my question, but then I followed a link to an additional FAQ, "does mass change with velocity"?

The answer implies that the whole concept of relativistic mass has fallen into disfavor of late. The impression this article gives is not that some observation has refuted the idea of relative mass - but rather that relative mass is mathematically inconvenient, and has been replaced with a different mathematical formalism of invariant masses described as four-vectors in Minkowski space.

This leads me to two important questions:

I have always thought of relative mass, time dilation and length contraction as one indivisible package. Now I am unsure whether I should believe any of these concepts.

My questions:

1. If you accelerate a massive particle in a particle accelerator until it reaches a significant fraction of light speed - does that accelerated particle interact with other particles as if it were more massive than its rest mass - or not? Is relative mass experimentally observed?

2. I have been led to believe that time dilation has been experimentally observed and proven in a variety of different experiments. Is this correct? Is time dilation experimentally observed?

Lelan Thara said:
1. If you accelerate a massive particle in a particle accelerator until it reaches a significant fraction of light speed - does that accelerated particle interact with other particles as if it were more massive than its rest mass - or not? Is relative mass experimentally observed?

No. In interactions we only have the invariant mass. You have to think about the meaning of the momentum 4-vector in relativity.
Naturally, if one wishes it is possible in some cases to try explaining some things (illogically) with the relativistic mass.

Lelan Thara said:
2. I have been led to believe that time dilation has been experimentally observed and proven in a variety of different experiments. Is this correct? Is time dilation experimentally observed?
Yes it is.

You could also look at https://www.physicsforums.com/showthread.php?t=144545

Los Bobos said:
No. In interactions we only have the invariant mass. You have to think about the meaning of the momentum 4-vector in relativity.

I don't see the logic of this. Why should interactions only take into account invariant mass ? I think the relativistic mass should be used to obtain the gravitational interaction. Why would gravitation make a distinction between the invariant mass and the mass corresponding to the kinetic energy of the particle. It doesn't make sense to me. Can you prove your assertion ?

Los Bobos said:
No. In interactions we only have the invariant mass. You have to think about the meaning of the momentum 4-vector in relativity.
Naturally, if one wishes it is possible in some cases to try explaining some things (illogically) with the relativistic mass.

Let me make sure I understand you clearly -

If I accelerate a massive body, there is absolutely no measureable effect on its mass? It's gravitational attraction to other masses will not increase? The momentum it imparts to other bodies it strikes will not increase? It's ability to resist acceleration will not increase?

What, then, would be these "things" that I could try to explain "illogically" with relativistic mass?

If there is no observable validity at all to relativistic mass, why do all the links say things like "relativistic mass is not wrong - it's just another perspective"?

Why do the links say that relativistic mass is still taught as a starting point before teaching invariant mass with four-vectors?

Why do luminaries like Hawking and Feynman write about relativistic mass in their books? Are they trying to deceive us poor dumb laymen? Are they just a lot stupider than you?

Why do I have to "think about the meaning of momentum 4 vector in relativity" if there is no objective, observed, measureable phenomenon related to accelerated masses to even contemplate?

Do you understand what I am asking you? I am not asking you what equations you prefer. I am asking you if there is anything observable in the physical world that corresponds to the idea that "accelerated bodies are more massive than the same bodies at rest."

notknowing and Lelan Thara,
if read the faqs I think you noticed the main point that relativistic mass is used when trying to find correspondences with Newtonian mechanics.

When we are using special relativity we can only use invariant mass, as it is the mass of the particle. For example in QED we have the Lagrangian density
$$\mathcal{L} =\overline{\psi}(i \gamma^\mu D_\mu -m)\psi -\frac{1}{4} F_{\mu\nu} F^{\mu\nu},$$
where m is the mass of the particle. And this is the one and only mass the particle has. It would make no sense to handle things like this with relativistic mass.
The effects which people try explain using relativistic mass as an analogy with the classical case are encoded in the 4-momentum. Naturally these unphysical things as the increase of the mass do not happen, but something similar happens due the time dilatation(or actually all this is kind of encoded in the structure of spacetime). So the effect is kind of the same and so it is used in teaching.

In general relativity gravitation affects everything and the source of the gravitation is the so called energy-momentum(-tension)-tensor. Here we have the invariant masses, but also the 4-momentum again. The mass used in the equation is again the invariant mass.

And again, if mass would rise with velocity, photons would see only black holes.

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Lelan Thara said:
Let me make sure I understand you clearly -

If I accelerate a massive body, there is absolutely no measureable effect on its mass? It's gravitational attraction to other masses will not increase? The momentum it imparts to other bodies it strikes will not increase? It's ability to resist acceleration will not increase?

What, then, would be these "things" that I could try to explain "illogically" with relativistic mass?

You want to be very careful about thinking about the gravitational field of a moving body. Thinking of it as being the same as the gravitational field of a stationary body, but larger, is in fact incorrect, and leads to confusion on issues like "why doesn't a rapidly moving body turn into a black hole" that you brought up earlier. This sort of confusion comes from assuming that one can replace mass in Newtonian equations with "realtivistic mass" and get the correct relativistic answer. This is also an incorrect idea - the ratio of acceleration to force gives rise to "longitudinal" and "transverse" masses, both of which are different numerically (it turns out the transverse mass equals the relativistic mass equals gamma times the invariant mass, while the longitudinal mass is gamma^3 times the invariant mass).

This sort of confusion is unfortunatly quite common for people who have been taught relativistic mass, BTW, it's one reason that many feel that the concept should not be taught.

As far as the gravity of a rapidly moving body goes -- the most conveinent thing to measure for a rapidly moving body is not the gravity, but the tidal gravity. This is convenient because tidal gravity can be measured essentially at a point (one actually needs a point and some small neighborhood around it), for instance with a Forward mass detector. Measruing "gravity" is not something that can be done at a point in this way, and due to reasons I won't get into becomes quite a messy problem that can be nicely sidestepped by measuring tidal gravity instead.

If one measures the tidal gravity of a rapidly moving object, one finds that the field is not spherically symmetrical. Much like the electric field of a moving charge, it's transverse field is strengthened. (If you're not familiar with the electrostatic field of a moving charge, I can provide some links on that subject, if it's of interst).

As far as the mass issue goes, relativistic mass is a coordinate dependent concept. If you have a body just sitting in empty space, one observer co-moving with the body will measure one value for relativistic mass, while another observer, moving at some velocity v with respect to the body, will measure a difrerent value for the relativistic mass. But both observers will measure the SAME invariant mass for that body, as long as the body in question is an isolated system.

This means that relativistic mass is not just a property of the body alone - one needs to specify an observer, also. Invariant mass, in contrast, is a property of the body, and only of the body (at least for an isolated system) - it does not depend on the observer. I find it reasonable, but imprecise, to call such coordinate independent quantites "more physical" than qunatities which depend on the observer. YMMV.

Note that a certain amount of the debate on this topic has reached the literature. Some links to the abstracts (many of the articles may not be readable without going to a university library, the first arxiv link is readable though) are:

http://arxiv.org/abs/physics/0504110
http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=AJPIAS000059000011001032000001&idtype=cvips&gifs=yes
http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=AJPIAS000055000008000739000001&idtype=cvips&gifs=yes
and also of some interest is
http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=PHTEAH000044000001000040000001&idtype=cvips&gifs=yes

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Mass, kinetic energy, ultraboosts

Hi, Lelan,

Lelan Thara said:
I read the Usenet FAq directly related to my question, but then I followed a link to an additional FAQ, "does mass change with velocity"?

The answer implies that the whole concept of relativistic mass has fallen into disfavor of late.

Actually, it fell into disfavor long ago.

The book Spacetime Physics by Taylor & Wheeler is an excellent introduction to str. As the authors explain, the old term "relativistic mass" really should refer to "mass plus relativistic kinetic energy". That is, as Einstein himself knew in 1905, if you expand the factors in a "boost" type Lorentz transformation wrt velocity, to lowest order you recover mass plus the Newtonian expression for kinetic energy.

Lelan Thara said:
1. If you accelerate a massive particle in a particle accelerator until it reaches a significant fraction of light speed - does that accelerated particle interact with other particles as if it were more massive than its rest mass - or not? Is relative mass experimentally observed?

If you admit that mass and energy (in particular, kinetic energy) are "equivalent" in the sense introduced by Einstein, then there is no problem with saying that mass is invariant under boosting--- but not kinetic energy! That is, when we boost a particle in a particle accelerator, the added kinetic energy makes it seem to behave as if, speaking in qualitative Newtonian terms, the "inertial mass" of the particle has increased "as measured in the lab frame".

In the early 20th century, most physicists naturally preferred to try to stay close to their Newtonian intuition. After the geometric innovations of Minkowski in 1907, it began to become clear to leading physicists that it is better to embrace Minkowski's geometric imagery, in which "inertial mass" is just the "length" of the four-momentum vector, an invariant quantitly, while the timelike component (-not- invariant!) gives the sum of the inertial mass plus the relativistic kinetic energy, and the spacelike components give the relativistic momentum.

notknowing said:
I don't see the logic of this. Why should interactions only take into account invariant mass ? I think the relativistic mass should be used to obtain the gravitational interaction. Why would gravitation make a distinction between the invariant mass and the mass corresponding to the kinetic energy of the particle. It doesn't make sense to me. Can you prove your assertion ?

Well, we don't "take into account only invariant mass"; we also take into account the kinetic energy, which is -not- invariant under boosts, whereas we sensibly treat the mass of the particle as being invariant under boosts.

Lelan Thara said:
General relativity tells us that the force of acceleration is equivalent to the force of gravity.

Well... sort of. It depends on how you think of it.

One observation which should be helpful here is to recall that there are no "absolute" velocities in relavistic physics. Another is that (as I have pointed out several times in recent threads), you are both tacitly referring here to "speed in the large", and thus to "distance in the large", which can be tricky.

notknowing said:
Why would gravitation make a distinction between the invariant mass and the mass corresponding to the kinetic energy of the particle.

-We- choose to make that distinction, if you like. This issue comes down to the question of which terminology is most consistent and convenient. Modern mainstream opinion is that it is best to consider mass to be an invariant property. (In gtr, there are additional wrinkles in defining the mass-energy of an isolated system which I hope we can avoid here, although this point has often been discussed in other threads.)

One way to compute what gtr predicts about what gravitational field corresponds to an ultrarelavistic isolated massive object would be to study the gravitational field measured by an ultrarelativistic observer (riding a test particle) in the Schwarzschild vacuum.

Another is to try to concoct a coordinate transformation of the Schwarzschild vacuum and take a suitable limit to obtain a model of an ultrarelavistically boosted massive object. This gets rather tricky, but the Aichelburg-Sexl ultraboost accomplishes this task; see the version of the Wikipedia article by that title listed at http://en.wikipedia.org/wiki/User:Hillman/Archive
(crucial caveat: I have no idea what the current version says so I can't vouch for that!).

Both of these approaches yield the interesting conclusion that the gravitational field of an isolated object which whizzes by an observer at ultrarelativistic "speed in the large" will closely resemble a particular linearly polarized axisymmetric gravitational pp-wave which has the character of a "pulse wave" with its energy concentrated in a single wavefront. The effect on the world lines of observers is to bend them as they cross this wavefront.

For more details, see the version of the Wikipedia article by that title listed at the page above, with the same caveat as noted above), or see the book Black Hole Physics by Frolov & Novikov for more about ultraboosts.

Because of the multiplicity of competing definitions of "distance in the large" and thus "speed in the large", ultraboosts can be tricky and different limiting procedures can give different looking results. There are various distinct "ultraboosted Kerr vacuums" discussed in the literature, for example, which again have the character of an axisymmetric pp-wave. Indeed, any reasonably ultraboosted Ernst vacuum (axisymmetric stationary vacuum solution) should have this character.

Lelan Thara said:
Why do luminaries like Hawking and Feynman write about relativistic mass in their books? Are they trying to deceive us poor dumb laymen?

Quite the contrary. Rather, none of us can do anything about the fact that any attempt to describe physics in non-mathematical language almost invariably is highly misleading, no matter how hard on tries to avoid this.

I like to define mathematics as the art of "thinking about simple situations without getting confused". That is, discussions of even simple situations (as we have seen in this thread) tend to quickly become terribly confusing if you do not have recourse to mathematical reasoning. Complicated situations are generally too hard even if you DO have recourse to mathematical reasoning; existing mathematics and its applications in mathematical physics (and other subjects) is mostly limited to simple situations, for this reason.

pervect said:
This sort of confusion is unfortunately quite common for people who have been taught relativistic mass, BTW, it's one reason that many feel that the concept should not be taught.

Exactly, it causes uneccessary confusion and has no justification whatever that I can see.

pervect said:
If one measures the tidal gravity of a rapidly moving object, one finds that the field is not spherically symmetrical. Much like the electric field of a moving charge, it's transverse field is strengthened.

Exactly. In the AS ultraboost, the gravitational field (Riemann tensor, projected into the frame field of our observer) is essentially compressed into a single "wavefront", and within that wavefront (think of this as the coordinate plane $$z-t=0$$ in a cylindrical coordinate chart) it is concentrated near the axis of symmetry, decaying like $$m/r^2$$, i.e. less rapidly than the Coulomb tidal field which decays like $$m/\rho^3$$ where $$\rho^2=z^2+r^2$$. This is analogous to what happens in electromagnetism, and it is consistent with the picture in which we simply consider an ultrarelavistic observer in the usual Schwarzschild vacuum solution.

pervect said:
This means that relativistic mass is not just a property of the body alone - one needs to specify an observer, also. Invariant mass, in contrast, is a property of the body, and only of the body (at least for an isolated system) - it does not depend on the observer.

Exactly--- this is the essential point here.

Chris Hillman

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Thank you to all who have tried to explain this to me. I do believe I understand now.

Chris, you answered the question that bothered me most when you said:

"If you admit that mass and energy (in particular, kinetic energy) are "equivalent" in the sense introduced by Einstein, then there is no problem with saying that mass is invariant under boosting--- but not kinetic energy! That is, when we boost a particle in a particle accelerator, the added kinetic energy makes it seem to behave as if, speaking in qualitative Newtonian terms, the "inertial mass" of the particle has increased "as measured in the lab frame".

I will confess that when I first read "oh, we don't teach relativistic mass any more", I actually became angry. It made me feel that the books for laymen I had read were making fantastical claims that had no experimental validity. What I quoted above makes it clear that I haven't been completely deceived.

If I might add a small point I came across that made a lot of sense to me - to people who have a beginner's understanding of physics, "mass" and "matter" can seem like synonyms - or at least, inextricably intertwined. So, when such folks are told that mass increases with acceleration - it creates a false impression that somehow traveling fast can create more matter - that objects moving close to c magically acqure "more stuff".

So treating mass as invariant, but discussing a characteristic of mass - namely momentum - that changes with speed is a better description of the truth.

It also is very helpful to me to learn that the transverse and longitudinal momenta can differ - I must admit, it wouldn't make much sense to talk about an object having more mass in one direction than another.

Thanks again to all who helped me understand this!

## 1. What is a Schwarzschild radius and how is it related to black holes?

A Schwarzschild radius is a measure of the size of a black hole based on its mass. It is the distance from the center of a black hole at which the escape velocity is equal to the speed of light. This means that anything within this radius, including light, cannot escape the gravitational pull of the black hole.

## 2. How is the Schwarzschild radius calculated?

The Schwarzschild radius is calculated using the formula Rs = 2GM/c^2, where G is the gravitational constant, M is the mass of the black hole, and c is the speed of light. This formula is derived from Einstein's theory of general relativity.

## 3. What is relativistic mass and how is it different from rest mass?

Relativistic mass is a concept in Einstein's theory of relativity that describes how an object's mass changes as it approaches the speed of light. It takes into account the effects of time dilation and length contraction on the object's mass. Rest mass, on the other hand, is the mass of an object when it is at rest and is constant.

## 4. How does an object's relativistic mass change as it approaches the speed of light?

As an object approaches the speed of light, its relativistic mass increases exponentially. This means that the object becomes more and more difficult to accelerate, and it would require an infinite amount of energy to reach the speed of light. This is why it is impossible for anything with mass to travel at the speed of light.

## 5. Can an object's relativistic mass exceed its rest mass?

Yes, an object's relativistic mass can exceed its rest mass as it approaches the speed of light. However, this does not violate the principle of mass conservation, as the increase in mass is due to the object's kinetic energy. Once the object slows down, its relativistic mass will return to its rest mass.

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