Relativistic Mass: Does Spacetime Distort at High Velocities?

[K.J.Healey]

After reading a lot of posts lately, I believe I recall someone mentioning that physicists no longer look at the relativistic effect of mass-"dialation" but rather velocity. I am not 100% sure I understand this so firstly, please explan a little. The way I understood it was that the more relativist speeds you reached, others viewed you as gaining more mass. Feel free to use heavy math, it always seems to help me understand when I can see the math behind it.

Now my question about this is also:
Do mass-full observers witness a change in gravitational spacetime distortion when a relativistic mass-full object passes by (more than they would if the object moved slowly?).
A way to impossibly measure this:

have two masses arranged in space such that they are connected in a line by a rope/mechanism that will expand but not contract. That way the masses don't fall into each other, but can stil lbe pulled around by different means. A relativistic mass of the same amount (rest) passes by within close enough range where it can feel the effects noticeably.

>>>>>>-----O------>>>>>

O
|
|
|
O

Would, since it is moving fast, and relativistically its "heavier", the other two masses notice these effects physically. Would there be a greater pull than one would expect non-relativistically. Of course it would be in the system for only a short amount of time, and the time it takes for spacetime distortions to have effect is c. (i assume)

[As an afterthough, the whole reason why I made two mases is I couldn't think of another way to measure a spatial field gradient than to measure the differences in changes of momentum (differences in potential). When there's no other objects you can't measure agains anything.

Is this question making sense? It's nothing too crazy but I'm tryign to be clear. I know length contraction or time dilation is all how you choose to think about the problem, and from where you're watching. What about mass and the effects of gravity?

 

[lightarrow]

After reading a lot of posts lately, I believe I recall someone mentioning that physicists no longer look at the relativistic effect of mass-"dialation" but rather velocity. I am not 100% sure I understand this so firstly, please explan a little. The way I understood it was that the more relativist speeds you reached, others viewed you as gaining more mass. Feel free to use heavy math, it always seems to help me understand when I can see the math behind it.

Now my question about this is also:
Do mass-full observers witness a change in gravitational spacetime distortion when a relativistic mass-full object passes by (more than they would if the object moved slowly?).
A way to impossibly measure this:

have two masses arranged in space such that they are connected in a line by a rope/mechanism that will expand but not contract. That way the masses don't fall into each other, but can stil lbe pulled around by different means. A relativistic mass of the same amount (rest) passes by within close enough range where it can feel the effects noticeably.

>>>>>>-----O------>>>>>

O
|
|
|
O

Would, since it is moving fast, and relativistically its "heavier", the other two masses notice these effects physically. Would there be a greater pull than one would expect non-relativistically. Of course it would be in the system for only a short amount of time, and the time it takes for spacetime distortions to have effect is c. (i assume)

[As an afterthough, the whole reason why I made two mases is I couldn't think of another way to measure a spatial field gradient than to measure the differences in changes of momentum (differences in potential). When there's no other objects you can't measure agains anything.

Is this question making sense? It's nothing too crazy but I'm tryign to be clear. I know length contraction or time dilation is all how you choose to think about the problem, and from where you're watching. What about mass and the effects of gravity?

A related question in the thread "Relative Weight":

https://www.physicsforums.com/showthread.php?t=160221

 

[pervect]

I don't really understand how the proposed mechanism using "ropes" works.

However, the easy (and as far as I know, the only) way to measure the effect of gravity on a small region of space-time (a point and its neighborhood) is to look at tidal gravity.

For instance, we could study the effect of the Sun's gravity on the Earth by looking at the tides.

More precisely, you take a uniform rod, and put accelerometers on both ends

(a)----------(a)

The rod is assumed to be in free-fall, and have no angular momentum. In this case, when there are no masses nearby, both accelerometers will read zero.

In the limit of a short rod, the two ends of the rod will read equal and opposite accelerations when the rod points twoards the large mass M, which will be equal to +GM/r^3 and -GM/r^3, for a difference of 2GM/r^3

(If this isn't obvious, consider the Newtonian case, and/or ask more questions).

So, how does motion affect tidal gravity? If we have the following situationM------------(a)(b)

where two of our rods (a) and (b) are at essentially the same location in space-time compared to some mass M, at some Schwarzschild coordinate 'r'

then if (b) moves "up the page", it will experience a larger tidal force. This increase, though, will not be the factor of gamma that's associated with the relativistic mass.

see also
https://www.physicsforums.com/showthread.php?t=87991

for an expression that shows how the tidal force actually varies, and

https://www.physicsforums.com/showpost.php?p=1347429&postcount=10 for
for a post by Chris Hillman which addresses the specific case of the tidal force when (b) is in a circular orbit.

 

[pervect]

One other point I should make.

Working out the tidal gravity of a moving mass is an interesting exercise in GR. But there are some simpler results that are also interesting. Suppose you have a pressure vessel containing a gas. Then you heat up the gas. The only difference between the hot gas and the cold gas is due to the fact that the atoms or molecules in the hot gas have more kinetic energy while the atoms in the cold gas have less.

Does the hot gas weigh more? In this example the gas molecules are all moving, but they move in different directions. The system on a whole is stationary and symmetrical which makes analyzing it in familiar terms of force, mass, and weight much easier.

The answer to this quiestion is yes - the hot gas does weigh more than the cold gas. See for instance http://arxiv.org/abs/gr-qc/9909014

 

[lightarrow]

One other point I should make.

Working out the tidal gravity of a moving mass is an interesting exercise in GR. But there are some simpler results that are also interesting. Suppose you have a pressure vessel containing a gas. Then you heat up the gas. The only difference between the hot gas and the cold gas is due to the fact that the atoms or molecules in the hot gas have more kinetic energy while the atoms in the cold gas have less.

Sorry pervect, but is that the only difference? Shouldn't we also consider, at least, the different tension on the vessel's walls?

 

[pmb_phy]

After reading a lot of posts lately, I believe I recall someone mentioning that physicists no longer look at the relativistic effect of mass-"dialation" but rather velocity.

Not quite right. Most particle physicists don't use the concept and most physicists work in particle physics. But there are plenty of physicists who do use it.

Now my question about this is also:
Do mass-full observers witness a change in gravitational spacetime distortion when a relativistic mass-full object passes by (more than they would if the object moved slowly?).

The gravitational field (acceleration due to gravity rather than tidal forces aka spacetime curvature) will increase the faster the body moves. The details of what that means is to be found in the journal article

Measuring the active gravitational mass of a moving object, D.W. Olson and R.C. Guarino, Am. J. Phys. 53(7), July 1985. The abstract reads

If a heavy object with rest mass M moves past you with a velocity comparable to the speed of light, you will be attracted gravitationally towards its path as though it had an increased mass. If the relativistic in active gravitational mass is measured by the transverse (and longitudinal) velocities which such a moving mass induces in test particles initially at rest near its path, then we find, with this definition, that M_rel = g(1 + [itex]\beta[/itex])M. Therefore, in the ultrarelativistic limit, the active gravitational mass of a moving body, measured in this way, is not [itex]\gamma[/itex]M but is 2[itex]\gamma[/itex]M .

If you'd like I can e-mail this article to you. The article is about 3.2 Mb in size so your e-mail ISP must be able to accept/retrieve that size of one file. If you can/wish to do this then send me your e-mail address in PM and I'll e-mail the article to you.

Best wishes

Pete

 

[lightarrow]

The gravitational field (acceleration due to gravity rather than tidal forces aka spacetime curvature) will increase the faster the body moves. The details of what that means is to be found in the journal article

Measuring the active gravitational mass of a moving object, D.W. Olson and R.C. Guarino, Am. J. Phys. 53(7), July 1985. The abstract reads

If a heavy object with rest mass M moves past you with a velocity comparable to the speed of light, you will be attracted gravitationally towards its path as though it had an increased mass. If the relativistic in active gravitational mass is measured by the transverse (and longitudinal) velocities which such a moving mass induces in test particles initially at rest near its path, then we find, with this definition, that Mrel = g(1 + LaTeX graphic is being generated. Reload this page in a moment.)M. Therefore, in the ultrarelativistic limit, the active gravitational mass of a moving body, measured in this way, is not LaTeX graphic is being generated. Reload this page in a moment.M but is 2LaTeX graphic is being generated. Reload this page in a moment.M

Hello Pete.
So, a question arises: if a particle is fast enough, it becomes a black hole? If a body is near its trajectory, it get captured inside its horizon and can't escape any longer?

 

[pervect]

Sorry pervect, but is that the only difference? Shouldn't we also consider, at least, the different tension on the vessel's walls?

[expanded & revised]

The reference I quoted goes into that to some extent, actually. It gets a bit complicated. Here's the condensed version, which is still rather complex.

In GR, pressure (and tension) do affect "mass", but the mass being talked about is actually subtly different than the "mass" defined by SR. There are actually several sorts of mass in GR, the most relevant one is the Komar mass. You'll probably be relieved to know that for isolated systems, the various notions of mass (the SR mass and the Komar mass for instance) agree. But, for non-isolated systems, they don't agree.

Philosophically, it's probably best in general to regard the mass of a non-isolated system as not being well-defined. (We'll see that there are some exceptions to this general rule, however).

When talking about the gravitational field, it makes more sense to use the GR notion of mass (in this case, the Komar mass) rather than the SR notion. SR does not make any predictions about gravity, that's the realm of GR. The reference I quoted above analyzes the mass only from this GR point of view, using the Komar mass, but doesn't make a direct point of pointing out that it is using a different defintion of mass.

Using the GR defintions, and considering the entire system, the tension terms inside the pressure vessel are exactly canceled by the pressure terms, so there is no net effect of the pressure terms on mass. This leaves only the kinetic energy terms.

Now, for the SR point of view. The SR definition of mass does not include any pressure terms. Since they don't exist, they don't need to be accounted for at all in the SR analysis. The main point is that the SR mass does approach the GR mass for complete small systems as one might expect. "Small" here actually means that the Newtonian gravitational binding energy can be neglected, so that we can ignore self-gravitational effects in determining the mass of the system.

Now, let's talk about what might be described as "gravitational field", by which I mean for the purposes of this post the reading on an accelerometer of a test body held stationary with respect to the mass, and not anything else.

External to the walls of the pressure vessel, SR and GR give the same answers - if you have a "small" pressure vessel (i.e one with negligible gravitational binding energy). the gravitational acceleration of a test body distance away from it will be GM/r^2, where M is the mass of the pressure vessel, and M has the same value for both SR and GR.

If we add heat energy of an amount E to the above pressure vessel, the gravitational acceleration rises to G(M+E/c^2) / r^2, where E is the amount of added energy. So the SR and GR masses both increase by E/c^2.

To get correct predictions for the gravitational field (measured in the same way) INSIDE the walls of the pressure vessel, one has to use the more complete GR definitions, the SR notions fail.

One finds in the limit of a relativistic gas that the gravitational field at the surface of the sphere for a relativistic gas (including photon gasses) is 2G(E/c^2) / r^2, twice the value that one would expect from using the SR definitions.

So, to summarize:

SR and GR have different definitions for mass (GR has several sorts of mass). These differing definitions agree for complete systems in the limit for "small" systems with negligible gravitational binding energies.

The external gravitational field depends on the complete system, so the SR and GR masses and the field computed from them also agree.

The internal gravitational field is dependent only on parts of the system, and the SR definitions can fail fairly badly.

I've written more on the topic in the Wikipedia article below

http://en.wikipedia.org/w/index.php?title=Mass_in_general_relativity&oldid=134753971

note that this article should not be considered as an "independent source", though I've provided references.

 

[pmb_phy]

Hello Pete.
So, a question arises: if a particle is fast enough, it becomes a black hole? If a body is near its trajectory, it get captured inside its horizon and can't escape any longer?

An object is not an object because it has a certain amount of mass. The proper mass has to be confined within the Schwarzschild radius as observerd in Schwarzschild coordinates (i.e. rest frame of black hole). From a moving frame of reference the mass increases, but as I said, it is the proper mass within the Schwarzschild radius, not the relativistic mass, which determines when an object is a black hole.

The article I mentioned covers your question. See

http://home.comcast.net/~peter.m.brown/Olson_Guarino_1985.pdf

Pete

 

[lightarrow]

Thank you Pervect.
Thank you Pete.

 

[pmb_phy]

Thank you Pervect.
Thank you Pete.

You're most welcome Sir.

I assume that the article from the American Journal of Physics was enlightening. Was it useful to you?

Best regards

Pete

 

[Ransom]

Alright, in order to not start a new thread I am going to post my question here.

(Note)[ I am having trouble understanding the difference between SR mass and GR mass.]

My main question is if mass increases the faster an object is traveling, than what increases? For example if a "Ball" of 5 H atoms is thrown, as it reaches c, the mass is supposed to increase. So is the ball now more Hydrogen than when thrown? Or is the atomic mass increasing? If this is the case do elements change chemically as they reach c?

I hope i am being clear, thanks for all the advice in advance.

 

[pmb_phy]

My main question is if mass increases the faster an object is traveling, than what increases? For example if a "Ball" of 5 H atoms is thrown, as it reaches c, the mass is supposed to increase. So is the ball now more Hydrogen than when thrown?

I believe that you're confusing the concept of "quantity of matter" with "amount of inertial mass." The term "quantity of matter" is an ill-defined term and used only as a non-technical term (i.e. we use it in terms of a vauge idea of what we think it means - but that runs into problems). The actual inertial mass (aka relativistic mass) actually does increase in magnitude. This means that the greater the inertial mass of a body the more difficult it is to change its momentum. As for gravitational mass increase - this refers to that quantity of gravitational mass responsible for the intensity of a gravitational field. The faster a point particle moves the greater is the magnitude of a gravitational field. You can read all about this in the American Journal of Physics Article which I made available online at

http://home.comcast.net/~peter.m.brown/Olson_Guarino_1985.pdf

Or is the atomic mass increasing? If this is the case do elements change chemically as they reach c?

The resistance of the atom to resist changes in momentum increases with increasing speed so yes, that part is true. This is a direct result of the combination of time dilation and Lorentz contraction (and Lorentz transformation).

I hope i am being clear, thanks for all the advice in advance.

Very clear. I hope my response was helpful.

Best wishes

Pete

 

[jostpuur]

For example if a "Ball" of 5 H atoms is thrown, as it reaches c, the mass is supposed to increase. So is the ball now more Hydrogen than when thrown?

I never believed somebody would actually consider the possibility of number of atoms changing as result of increased velocity. Ransom, did you seriously have this difficulty when trying to understand mass, or did you just somewhere read that "relativistic mass is confusing: for example, it could lead somebody to believe that atom number is changing"?

 

[Ransom]

@Jostpuur, What happened was that i got confused. I confused the Lorentz Contraction with how the objects relativistic mass increases. I was under the impression that if an objects appears smaller it would actually be smaller. Then I found out that mass increases as your approach c. I thought well how does the mass increase thinking in terms of non relative mass. My conclusion was that either there is more atoms that make up the ball, more protons that make up the atom, thus transforming the element, or more quarks (if that makes any sense since I know little about them) that make up the proton. I now understand that relative mass, is more like inertia, right? Since we do not have a full grasp on what matter is we define it in terms of c, that is, relative mass?

 

[pervect]

I'd suggest taking a look at "Does mass change with speed" for more info. The term "relativistic mass" is favored by some, and disliked by others, who regard it as outdated. It is possible to define mass in a way such that it does not change with speed, this is the so-called "invariant mass".

 

[pmb_phy]

My conclusion was that either there is more atoms that make up the ball, more protons that make up the atom, thus transforming the element, or more quarks (if that makes any sense since I know little about them) that make up the proton.

If you believed that then you'd have to believe that if it was you who changed to a frame of reference which was moving relative to the atom etc then as you picked up speed then you'd observe more atoms popping into existence whereas the observer at rest in the atoms frame would never make such an observation. How would you reconcile this contradiction??

I now understand that relative mass, is more like inertia, right?

If by "like" you mean "is a measure of" then yes, I agree that's true.

Since we do not have a full grasp on what matter is we define it in terms of c, that is, relative mass?

I disagree. If you don't have a definition of "quantity of mass" then you can't say how that quantity is related to the inertial mass. That's why I said its ill-defined. Otherwise one could define "quantity of matter" to be defined as a synonym for "mass."

There is another point I'd like to make and that is that "inertial mass" is also that quantity which defines momentum. This should be kept in mind when considering single photons since a photon does have momentum and yet it can't change its speed. But it can transfer momentum to other objects.

Best regards

Pete

 

[Ransom]

"My conclusion was that either there is more atoms that make up the ball, more protons that make up the atom, thus transforming the element, or more quarks (if that makes any sense since I know little about them) that make up the proton.
If you believed that then you'd have to believe that if it was you who changed to a frame of reference which was moving relative to the atom etc then as you picked up speed then you'd observe more atoms popping into existence whereas the observer at rest in the atoms frame would never make such an observation. How would you reconcile this contradiction??"

Exactly I why I came here to ask, I didn't think it made sense. Now I know it does not.

 

[pmb_phy]

Exactly I why I came here to ask, I didn't think it made sense. Now I know it does not.

Its a wonderful thing to figure out an answer to one's query. Especially when you don't have to do anything more than think about it, isn't it? Kinda gives you a warm fuzzy feeling of success!

Best regards

Pete