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Here, by inertia, I mean the tendency of an object to retain its state of motion and resist change. Another way of defining inertia is how much force is needed to create a certain amount of acceleration. And so quantitatively, this would be equal to F/a. Then the specific quantity that would determine the amount of inertia I suppose would be $\gamma^3 m$, which can also be called relativistic mass, but I have been advised not to use that term as it is apparently becoming obsolete. However that is a minor issue, and the main point to my question is that this would imply that an object can simultaneously have multiple values for $\gamma^3 m$ and therefore inertia depending on the frame. How can that be? This would imply that there can be multiple possible accelerations for a given force because there will be different values for $\gamma^3 m$ depending on the reference frame. So which $\gamma^3 m$ in which reference frame ultimately determines how much acceleration a certain force will produce? If it depends on the frame of the object that the force is acting on, then it would mean it is the invariant mass of the object that determines the inertia. In that case inertia would not increase with velocity and this would also imply that a finite amount of force and energy can make an object with mass move at speeds equal to or higher than the speed of light. Doesn't this create conflicting realities?

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You seem to be confused about what inertia means and its role in mechanics. Let’s just consider Newtonian mechanics for now, since conceptually there is no substantial difference with regards to your question; Newtonian mechanics still obeys a form of relativity (Galilean relativity). To start, inertia can mean a few different though related things:

  • the concept that bodies maintain their state of motion (in a fixed reference frame), i.e., they resist any change in motion — this is known as the principle of inertia, which is in essence Newton’s first law of motion;
  • the mass of a body — this quantifies how a body accelerates (changes velocity) a given force is applied to it, i.e., Newton’s second law;
  • the momentum of a body — this can be though of as the quantity in which a (net) force directly effects a change, since the net force on a body is precisely the rate of change of the momentum of a body.

So, insofar as it refers to a quantity, inertia can mean either the mass or momentum of a body. In Newtonian mechanics, the first is an absolute quantity (does not change depending on your frame of reference), whereas the second is of course relative to your frame of reference, just like velocities. Even in special relativity, mass is considered invariant, and we only speak of a frame-dependent mass-energy, in modern parlance.

You should not think of a body possessing “many” values of momentum, but rather the property of momentum of a body only making sense when a particular frame of reference is fixed under consideration. Momentum is thus a relation between a body and a frame of reference, just like velocity or kinetic energy.

Noldorin
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  • By inertia I meant the first one i.e. the quantification of the tendency of an object to retain its state of motion and resist changes. But doesn't special relativity say that inertia increases with velocity, and this is what prevents an object from exceeding the speed of light as an infinite amount of energy would be required to do that? Conversely if inertia did not change, it would be possible for an object with mass to exceed the speed of light with a finite amount of energy –  Dec 27 '20 at 19:17
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    @Neelim The modern interpretation of SR is that the mass does not change with velocity. What prevents an object with nonzero mass from reaching the speed of light is that its energy and momentum become infinite as $v\to c$. – G. Smith Dec 27 '20 at 20:02
  • I never said in the question that mass changes with velocity. I am aware that relativistic mass is considered obsolete. I said that the inertia is supposed to change with velocity. As you just mentioned, the energy and momentum approaches infinity as v→c. That would imply that more energy would be needed to create the same change in velocity at a higher v, and therefore inertia, by definition, would increase. This is why I am puzzled because mass should be invariant, and so inertia should not change. But it seems to, according to SR –  Dec 27 '20 at 20:44
  • @Neelim Well, the first definition above is about the principle of inertia; it is not a definition of a quantity. That said, I think I see your problem now. Applying Newton’s 2nd Law in a SR scenario means substituting in relativistic momentum in $F=\tfrac{dp}{dt}$. See this answer for example. https://physics.stackexchange.com/questions/128136/f-ma-calculation-taking-relativity-into-account – Noldorin Dec 27 '20 at 21:20
  • I have read the answer you linked, I was already familiar with this concept. While inertia is not explicitly quantified with dedicated symbols or units, in SR the inertia is determined by the mass, which is invariant, and also the Lorentz factor, which changes with velocity and reference frames. The contradiction then is that an object can have multiple 'degrees' of inertia, depending on the frame of reference –  Dec 27 '20 at 21:37
  • @Neelim I think the problem us with your use of the term “inertia” still. It does not have an unambiguous conventional use as a quantity. Physicists may refer to the principle of inertia, or perhaps bodies possessing the property of inertia, but there is no “measure” of inertia per se. We instead speak of more concrete quantifies like mass or momentum of a body. – Noldorin Dec 27 '20 at 23:13
  • And surely you accept that momentum (in Newtonian mechanics and SR alike) depends on the frame? That’s all there is to it. Your misunderstanding definitely seems to come from some conceptual fuzziness, from what I can tell. Which is why it’s probably best to deal with more concrete quantities and their names rather than resorting to such things as relativistic mass (and calling this “inertia” as maybe you would like to). I think @G.Smith has said it well too, if you prefer to follow his explanation. – Noldorin Dec 27 '20 at 23:16
  • yes I am aware that there is no convention for quantifying inertia as a separate parameter. But basically, the quantity of inertia is determined by the expression γm which can also be called the relativistic mass, which I have been advised not to call it such. But the mass and the Lorentz factor are indeed the factors that determine how much inertia an object has. Hope this clarifies the issue –  Dec 27 '20 at 23:20
  • yes I fully agree that there are some frame dependent elements in Newtonian mechanics. The only such elements I can think of are velocity and momentum, which are relative even in Newtonian mechanics. The difference is that in SR, a lot more things are relative, even the mass and inertia for example. Also Newtonian mechanics uses Galilean transformations for relativistic calculations whereas SR uses Lorentz transformations –  Dec 27 '20 at 23:25
  • @Neelim Right, okay.... I kind of suspected that, per my last comment (which I just edited as you posted yours). I think both “relativistic mass” and “inertia” are problematic terms here, in all honesty, though “relativistic mass” at least had historical currency. I definitely recommend this section on the topic. Indeed, it’s unfortunate that you seem to have been taught using this outdated and problematic concept. https://en.wikipedia.org/wiki/Mass_in_special_relativity#History_of_the_relativistic_mass_concept – Noldorin Dec 27 '20 at 23:34
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But this would mean that an object can simultaneously have multiple values for inertia depending on the frame. How can that be? Doesn't this create conflicting realities?

The quantity $\gamma m$ is commonly called “relativistic mass”, and is not used much in modern physics. It is a frame variant quantity. There are many such quantities: Velocity, Momentum, Energy, Power, E-fields, etc. All of these can “simultaneously have multiple values, depending on the frame”.

There are other quantities that are invariant: invariant mass, charge, proper time, etc. And often several individually frame variant quantities can be combined into one covariant four-vector quantity.

In particular, energy and momentum are combined into a single four-vector quantity called the four-momentum. The Minkowski norm of the four-momentum is the invariant mass, and the relativistic mass is the first component.

The laws of physics are written in terms of the invariant and covariant quantities. Those are seen as the “reality”, and the different values of the individual components no more “create conflicting realities” than two people assigning different coordinates to the same physical vector.

So the fact that one person assigns the relativistic mass a different number than another for the same object is no more controversial or problematic than one person giving a bearing relative to true north and another giving the same bearing a different number relative to magnetic north. It is just two ways to look at the same underlying quantity.

For me the contradiction here seems to be that, if an object has different relativistic masses, the same force could give different possible accelerations. But force and acceleration should be frame invariant and so that can't happen

This contradiction is based on a misconception. Neither force nor acceleration are frame invariant. This should not be surprising since time is dilated and length is contracted and both acceleration and force use length and time. There is a four-vector version of Newton’s 2nd law which is, to our knowledge, the one that is the real law of physics. In this law all of the quantities are invariant or covariant, so different frames are just different coordinates describing the same physical quantities as described above.

Dale
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  • Thanks for the answer, this comes much closer to addressing my main question. I still have some questions though. Is inertia determined by the relativistic mass? If so the relativistic mass would determine how much an object would accelerate when a certain force is applied. For me the contradiction here seems to be that, if an object has different relativistic masses, the same force could give different possible accelerations. But force and acceleration should be frame invariant and so that can't happen –  Dec 28 '20 at 00:08
  • So which mass is ultimately determining how much acceleration is produced by a certain force? Is it the invariant mass of the object? If so doesn't that mean that inertia is unaffected by the relativistic mass? But then this would mean that with a finite amount of energy, it would be possible for an object with mass to exceed the speed of light. Doesn't this seem like a contradiction? –  Dec 28 '20 at 00:11
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    @Neelim “inertia” is a little vague, so you could use any one of a number of quantities to represent it. I would tend to use the invariant mass because Newton’s second law generalizes to a four-vector equation using the invariant mass. But I answered the question using the meaning you intended. – Dale Dec 28 '20 at 00:58
  • @Neelim I added an additional section in the answer to address the concern you mentioned here. – Dale Dec 28 '20 at 01:13
  • By inertia I meant how much force is needed per unit acceleration, which should be the standard definition of inertia –  Dec 28 '20 at 01:16
  • I see. However others, even in the answers here, have mentioned that force and acceleration are invariant of frame. But even if that were not the case, in your modified answer it doesn't address the second issue I have mentioned. If the inertia depends on the invariant mass, then inertia would not increase with velocity and so then a finite amount of force and energy could make an object with mass move at speeds equal to or faster than the speed of light –  Dec 28 '20 at 01:20
  • @Neelim one problem is that different people can mean different things with the same words. I assumed you meant the normal Newtonian force and acceleration, and those are frame variant in relativity. Noldorin explicitly saying that it is not frame variant in Newtonian physics, and he was only talking about Newtonian physics. Valter Moretti is talking about the four-force and four-acceleration, but only says it in the math in the comments, not words in the answer. I don’t think either of them answered the question you intended about the Newtonian concepts of force and acceleration in relativity – Dale Dec 28 '20 at 01:43
  • can you elaborate on the last paragraph in which you mentioned that the four-vector version of Newton's 2nd Law is the correct law? Also which quantities are invariant in that case? –  Jan 06 '21 at 04:18
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Inertia concerns acceleration not velocity. The acceleration of a body does not depend on the choice of the inertial reference frame where you describe the motion of the body. Its velocity instead depends on used inertial the reference frame.

Valter Moretti
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  • Yes both the inertia and acceleration should not depend on the reference frame, I agree with that. But according to SR, it apparently does. That is what is confusing me and why I asked the question –  Dec 27 '20 at 21:05
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    No, in SR it does not depend on the reference frame. The equation is $m dV^a/d\tau = F^a$ and it takes the same form in all reference frames, $m$ does not depend on the reference frame. – Valter Moretti Dec 27 '20 at 21:12
  • I never actually said that mass changes with velocity or the reference frame. I said that according to SR the inertia seems to be changing with velocity and reference frame, which is not the same as mass, by definition, even if they maybe related –  Dec 27 '20 at 21:18
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    You are wrong. There is no statement in SR like the one you are claiming true. – Valter Moretti Dec 27 '20 at 21:20
  • Doesn't SR say that the energy required for an object with mass to travel at the speed of light would become infinite, and therefore it is impossible? Isn't that quantitatively expressed by the Lorentz gamma factor, which combined with mass determines the 'relativistic mass'? I also know that relativistic is considered obsolete and the invariant mass should not actually increase, but the Lorentz factor changes with velocity, and that increases the inertia –  Dec 27 '20 at 21:28
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    No, the formula I wrote down is the correct relativistic formula. The right time to be used is the proper time $\tau$. You are using the wrong notion of time, i.e., the one of the reference frame $t$. That is formally equivalent to replace the invariant mass $m$ for the ill defined object with disputable physical meaning $\gamma m$ called relativistic mass. – Valter Moretti Dec 27 '20 at 21:33
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    The reason why $c$ is maximal speed just concerns kinematics and not dynamics. – Valter Moretti Dec 27 '20 at 21:37
  • https://phys.libretexts.org/Bookshelves/Modern_Physics/Book%3A_Spiral_Modern_Physics_(D'Alessandris)/2%3A_The_Special_Theory_of_Relativity_-_Dynamics/2.1%3A_Relativistic_Momentum%2C_Force_and_Energy#mjx-eqn-Force5

    As shown in the article, the relativistic version of F=ma is: F=(γ^3)ma

    where γ is the Lorentz factor which I mentioned earlier. The Lorentz factor changes with velocity, and therefore force required per unit of acceleration would also change with velocity, according to the equation. The entire premise of this question was that inertia seems to be changing with the frame.

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    That is a disputable explanation in my view. What you should do is asking the author of that explanation. The fact that $c$ is the maximal speed is simply due to the fact that $V_aV^a=-c^2$ and the formula I wrote ( the relativistic version of the 2ndlaw) preserves this purely kinematical constraint since $F_aV^a=0$ is valid just by definition of mechanical four force. – Valter Moretti Dec 27 '20 at 22:07
  • the equation I presented is also confirmed here: https://physics.stackexchange.com/questions/302697/if-velocity-is-relative-how-can-mass-inertia-be-coordinate-independent –  Dec 27 '20 at 22:13
  • Another example: https://physics.stackexchange.com/questions/586867/relativistic-version-of-newtons-2nd-law

    As you see here, the equation for the relativistic momentum is p=γmv. The change in the momentum with respect to v changes with the velocity and reference frame.

     –  Dec 27 '20 at 22:20
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    I am not saying that the equations are wrong. I am saying that that explanation is disputable because the fact that c is the maximal speed is a purely kinematical fact. If you assume that the relativistic mass is a physically meaningful notion and that it is the representation of inertia, then you are right: inertia depends on the inertial reference frame. In my view that is however a misleading approach. At this level the answers is matter of personal taste. – Valter Moretti Dec 27 '20 at 22:21
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    It seems that you already have your preferred answers... – Valter Moretti Dec 27 '20 at 22:25
  • I think the speed of light has neither to do with kinematics nor dynamics. It has to do with space, electromagnetism and it is one of the two foundational postulates for special relativity. It is an observed fact and no one can explain why it is in terms of anything more fundamental. In fact the relativistic equations were derived from the speed of light being constant and ultimate, not the other way round. So if you try to explain it using relativistic kinematics, that would be a circular explanation which assumes it to be so in the first place. –  Dec 27 '20 at 22:51
  • I don't really have a preferred answer. I just haven't gotten an answer to my main question yet. And I feel like people are focusing more on irrelevant things like whether to call the expression γm by the name of 'relativistic mass' or not etc. without trying to answer the actual question –  Dec 27 '20 at 22:56
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The inertia, meaning the resistance of an object to a change of momentum, is frame invariant in Newtonian mechanics. $F = \frac{dp}{dt}$. The magnitude and direction of the momentum changes, but not the expression for the second law.

What changes is the energy required to put a movable body to a rest, for each selected frame.

Claudio Saspinski
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  • I have a pretty clear understanding of Newtonian mechanics and the fact that inertia is frame invariant in it. The amount of energy required to bring an object to rest in Newtonian mechanics is exactly (momentum)*v/2, which can be derived using P = mv and K.E. = (mv^2)/2. But what about in special relativity, which is considered the more complete theory? In SR it seems like the amount of energy to bring an object to rest does not change linearly with velocity –  Dec 27 '20 at 22:42
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Lets start off by understanding what intertial mass is in newtonian physics. Newton noticed that the net force acting on an object is always directly proportional to its acceleration (special case of 2nd law). This constant of proportionality is called intertial mass. You could call this an “intrinsic property” of an object, but at it’s core, mass is just a name given to a numerical ratio.

Now, it was later noticed that this ratio, at any instant, actually depends on the measured velocity. Note: the intertial mass is NOT equal to the ratio of net force and acceleration; you cannot take out gamma out of the derivative operator in F=dp/dt. Infact, for an object accelerated by a constant force, the ratio is actually (gamma^3 x rest mass). The point is, that it the rate of change of velocity seems to depend on the measured velocity. This may seem extremely counterintuitive at firs, but the question is, why do you think this ratio should be an intrinsic property of an object and invariant in the first place? Other Quantities like Kinetic energy are frame dependent even in newtonian physics, aren’t they? The reason for this counter-intuitiveness is that you are so used to “mass” of an object being constant in newtonian physics and forget that it is just a mathematical model that used to fit observations and is now succeeded by special relativity.

The thing is, special relativity has many unintuitive things that you just have to accept. Change in relative velocity is intuitive (from Newtonian mechanics), but how can 2 photons moving in opposite direction have a relative velocity of C instead of 2C ( see relativistic relative velocity formula)? How can the time slow down or speed up depending on the frame of reference? At its core, Einstein derived his special relativity from 2 simple axioms that have no intuitive justification whatsoever but perfectly fit observational data: 1) principal of relativity & 2) invariance of speed of light. These axioms result in a number of kinematic and dynamic consequences that, although extremely non-intuitive, have to be accepted as this theory seems to accurately model reality.

So multiple values would create conflicting realities even within the same frame

I think this point is the crux of your confusion. You don’t even need intertia for this. Consider a runner competing in a 100m race at 0.86c. The runner will observe half time of completion as observed by us and measure a distance of only 50m. Although kinematic quantities have changed, there is no change in “reality”; the outcome i.e runner reaching the finish line is same in all frames of reference. Another interesting situation is two like charges moving parallel to each other: the magnetic force snd thus the net force between them actually depends on the frame of reference. This doesnt change the “reality” that these charges will repel, not attract, as the (relative) velocity required for the magnetic force to cancel the electric force is C, which is impossible. See Magnetic force between 2 moving charges.

Perhaps the best example to clarify the invariance if “reality” is the barn-pole paradox. https://m.youtube.com/watch?v=YVhI45_WzJ4

Vulgar Mechanick
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  • Your answer is most in line with the how I was thinking about this subject. However I still have a few things to say:

    1. "Now, it was later noticed that this ratio, at any instant, actually depends on the measured velocity." - the measured velocity in which frame determines this?

    2. I get that the consequences of relativity seem counterintuitive, but seem to be accurate regardless and therefore should be accepted. However I feel like there can be more intuitive explanations and models that would produce the same result, but only if the postulates were actually modified.

     –  Dec 28 '20 at 01:55
  • I suppose the second point is probably less relevant to the topic of this particular question.
    1. Wouldn't that mean that nothing is invariant and is therefore frame-dependent, even the force and change in velocity? It is possible, but it seems rather strange. But in that case there would be no such thing as invariant or rest mass, and relativistic mass would be the only mass that can be measured, and it should be considered to be real and significant. Why do people have problems with the concept of relativistic mass then?
     –  Dec 28 '20 at 02:01
  • Yes. I get your confusion, but remember that although many quantities vary depending on the frame of reference, quantities in one frame are related to the other in a fixed way through lorentz transformations. Invariant mass is simply defined as relativistic mass measured from a stationary frame. One of the obvious problems with relativistic mass is that you can’t even use it in F=ma, since this special case is no longer valid. Moreover, the mass you actually measure using a balance is always rest mass. – Vulgar Mechanick Dec 28 '20 at 02:17