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?
5 Answers
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.
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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.
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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.
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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.
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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
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