The example that bugged me was the post stating that 'MV' increased, therefore MASS increases with velocity increase. Uh, no, it is velocity that got bigger, to cause the parameter 'MV' to increase.
AND injecting relativistic effects into this discussion is just extra confusion, especially when the terms are similar, and often identical. (That being the use of the term 'mass increase'.
It is complicated, because whenever we talk about "bending spacetime", we must talk in relativistic terms; but at the same time, Newtonian physics and relativistic physics do and must agree at the simpler limit (zero velocity, uniform gravitational field, and so on), because system behaviour at simpler limit is measurable and physically verifiable.
No matter how fine the theory, it is only useful if it predicts behaviour physically observed. That's why many think string theory is "just" math and not physics: it cannot really be used to model something physically measurable, so that its predictions could be compared against physical measurements. That makes it "not physics" in the sense that physics describes how the universe behaves; maths folks (including string theorists) can work out the details and the tools, and philosophy folks can deal with the "why" part.
For example:
Mass does not bend space-time. Energy does.In relativistic physics, "mass" is rarely used. Instead a quantity called "invariant mass" or "rest mass" is used; this is the mass of the object at rest, as if observed in a frame where the object is not moving. The invariant mass is invariant, because it is the same quantity no matter what observational frame it is used in.
"Relativistic mass" is the term that best corresponds to what we call "mass" in non-relativistic, or Newtonian, physics. It includes the space-time bending effects.
Because special relativity says that energy ≡ mass, (
mass-energy equivalence,
E=
mc²), the above can sound like quibbling: I should not say mass does not bend space-time and energy does, because the two are equivalent.
But the above is not quibbling: it is worded so because 'mass' as a term is ambiguous. In making that statement,
I was thinking of the wrong kind of mass, you see. See?
Simply put, we should avoid claiming mass bends space-time, because term 'mass' is ambiguous; we don't know whether it is used to refer to relativistic mass (which would be correct and match the classical physics definition of mass, but odd because physicists use 'mass' to refer to invariant or rest mass instead), or to invariant or rest mass (which usually 'mass' alone refers to when used in physics). One interpretation is right but odd, the other is wrong but the more common use for the term.
A case in point: photons.
All electromagnetic radiation from gamma rays to infrared light consists of photons. Photons are massless bosons: their
invariant mass or rest mass is exactly zero (none of that "so tiny we can think of it as zero" stuff; plain and clear zero here), and any number of them can occupy the same state and space (until the energy density is high enough to cause certain interesting stuff to happen). Bosons are the ones who like company, and fermions –– for example electrons being fermions –– are the ones who occupy their own quantum state, and exclude others from it.
In special relativity, the energy \$E\$ of an elementary particle is defined as
$$E = \sqrt{m_0^2 c^4 + p^2 c^2}$$
where \$m_0\$ is the invariant or rest mass, \$c\$ is the speed of light in vacuum, and \$p\$ is the linear momentum of the particle. For particles with nonzero invariant mass, the linear momentum \$p\$ is
$$p = \gamma m_0 v = \frac{m_0 v c}{\sqrt{v^2 - c^2}}$$
where \$v\$ is the velocity of the particle, and \$\gamma\$ is the velocity-dependent Lorenz factor; I expanded that to get the nice simple expression on the right side, but note that usually it is kept contracted to \$\gamma\$ for brevity, so the right side expression may look unfamiliar to many.
Note that for small enough velocities, \$p = m_0 v\$. The correction factor, Lorenz factor \$\gamma = 1 / \sqrt{1 - v^2 / c^2}\$, tells you the error you have for any given velocity if you use Newtonian physics instead of special relativity; see how only the \$p\$ term in total energy has anything to do with particle velocity?
The speed of sound in air is about 330 m/s, so about 0.0000011 of the speed of light. At that speed, \$\gamma = 1.000000000000605\$, or less than one part in 1,000,000,000,000 over one. Double-precision floating point numbers – those used for the vast majority of numerical computation on this planet outside financial stuff which uses decimal formats – have barely enough precision to describe that!
Also see how momentum is related to total energy (for elementary particles). For collections of particles, we'd need to add their internal energy (stored in their structure, their interactions; including angular momentum). In classical physics, we say kinetic energy \$K = m v^2 / 2\$, but it is important to understand how well that matches the relativistic energy. If we use \$E(p) = \sqrt{p^2 c^2 + m_0^2 c^4}\$, then \$K = E(p) - E(0)\$. Expanding the expression for \$K\$ using a Taylor series for small \$v\$ (around \$v = 0\$), we get \$K = m v^2 / 2 + 3 m v^4 / (8 c^2) + \dots\$. Meaning, even the kinetic energy agrees *exactly* at zero velocity, and for larger velocities, just has relatively small additional terms. Classical and relativistic physics agree at small velocities even here, as expected.
Photons have zero invariant or rest mass, and for them, linear momentum is
$$p = \frac{h}{\lambda}$$
where \$\lambda\$ is the wavelength of that photon. Each photon has a single wavelength that only changes when it interacts with other stuff, transferring energy. It does have other properties like polarization, but those do not affect or contribute to the total energy. Thus, their total energy is
$$E = \frac{h c}{\lambda}$$
A mind-bending detail is that because the particle velocity depends on the observational frame used for the measurement, so does the total energy of the particle.
To un-bend ones mind, think of the Doppler effect: the wavelength of the light we observe does depend on our own velocity with respect to the light. We still see it arriving at the speed of light in vacuum, but its energy is shifted. Because photons, having no rest mass, zip everywhere at the speed of light. (The velocity only drops when they interact with other stuff; exactly how that happens is quantum mechanics. That is why we have a constant for it in vacuum, but it drops when zipping through matter.)
In sufficiently small regions, acceleration is indistinguishable from gravity. Because of this, and the fact that Earths gravity well bends space around it, if we were to observe a single photon at a single moment of time (as measured by the position of that photon zipping along at the speed of light) from the surface of the Earth, and from an orbital space station, they would see the photon having different wavelengths. Those two are different frames of reference, so there is nothing wrong in their observations differing (by exactly the relative difference of those two reference frames). That is kinda-sorta one of the basic ideas of relativity.
Also, we have already experimentally proven that even photons themselves do bend space-time. That tells us
invariant mass or rest mass does not bend spacetime, because photons have none (again, not just "close enough to zero for all intents and purposes", but a clear and pure plain real zero), but we have observed them bending space-time. And relativistic mass is even in principle indistinguishable from the energy, because of Einstein's mass-energy equivalence. Thus, to avoid misconceptions, it is – I claim! – a good idea to think of *energy* as bending space-time, and relativistic mass being equivalent to energy per Einstein. As a bonus, we don't have any issues with photons having zero invariant mass, because they are always zipping along at the speed of light (well, duh), and thus always have energy and thus relativistic mass; the zero invariant mass is just not practically relevant other than when working out the details using math. And as a cherry on top, we easily slide into the habit of assuming 'mass' means 'invariant mass' (or rest mass), and instead of talking about 'relativistic mass' can talk about energy; energy being equivalent to mass, but as a term, not at risk of being confused with something else.
Compare to the thought pretzels one has to twist oneself into, if one insists mass is what bends space-time. What 'mass', anyway? No, I think that although you can argue that, that argument is not useful; you need to qualify it with 'invariant' or something else to be precise and not be accidentally misunderstood.
It's the same as when graphics artists talk about 'volume' or 'gravity' as an expression of an emotion evokable by visuals. If everyone agrees on their definition, and nobody accidentally tries to infer anything about related stuff outside that very small domain, there is no problem. So there is no
technical problem in using those terms. But, hilarity ensues when a graphics artist, genuinely puzzled, asks a physicist why they don't just use cut pastel tones to reduce gravity.
It is also okay to now realize that the Higgs boson so much talked about a few years ago, has really nothing to do with the bending of space-time, and everything to do with invariant or rest mass. (Specifically, how "gauge bosons", those that carry the four fundamental forces we know of, have an invariant or rest mass of around 80 GeV/c², while it would be much easier to describe them if they had zero invariant or rest mass.)
See? As in so many other things, if you just understand the terms correctly in their proper context, things just start falling into place.
They matter much more than just whether or not they are technically correct: they are crucial in building correct understanding in the first place, much beyond technical correctness or minute detail.
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