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This question says that, at relativistic speeds, an object's increased mass will result in increased weight or gravitational force. But if we have increased mass and corresponding gravitational effects, does that mean that the object actually contains more matter? Does the increase in velocity result in an increase in the number of atoms in the object?

  1. If so, where does the additional matter come from? How is it distributed? E.g. if I drive my car at 0.9c, is there some law that decides whether the added matter goes to my engine block or to my headlights?
  2. If not:
    1. How does mass increase if there is no increase in matter? The SI system defines mass in terms of a physical artifact (or a number of atoms, according to one proposal), but Wikipedia has seven different definitions of mass, so I'm not sure which one(s) applies here.
    2. Does it make sense to ask how the added mass is distributed throughout the object?
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    $\begingroup$ There is not such a thing like relativistic mass (common misconception). Please read, for example, this post (and/or this one, this one). $\endgroup$ – AccidentalFourierTransform Jan 3 '16 at 23:25
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    $\begingroup$ @WillO the fact that we can talk about relativistic mass (and write down a formula) doesn't mean that its an accepted notion in physics. We used to talk about caloric, aether, plum atoms, etc. We used to write down formulas fot these concepts. Yet, as for today, we consider them to be wrong concepts. The same can be said about $m/\sqrt{1-v^2}$. Nobody today thinks relativistic mass is a thing. Its just a useless concept. $\endgroup$ – AccidentalFourierTransform Jan 3 '16 at 23:39
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    $\begingroup$ Confusions like this are a big part of the reason why the concept of "relativistic mass" has been pretty much thrown on the scrap heap. The term "mass" as generally defined by physicists does not change as a function of velocity. A great deal has been written on this subject. But the short answer is: Its mass doesn't actually change. $\endgroup$ – elifino Jan 4 '16 at 0:38
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    $\begingroup$ Here's an example clarifying why the modern terminology is what it is: phys.ncku.edu.tw/mirrors/physicsfaq_old/Relativity/SR/mass.html To directly answer your question: The number of atoms doesn't change. The quantity of matter doesn't change. The mass doesn't change. It's the energy that's changing. $\endgroup$ – elifino Jan 4 '16 at 0:47
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    $\begingroup$ @WillO I disagree that relativistic mass is perfectly well defined, unless you just mean total energy divided by $c^2$ in which case it doesn't exist as a separate concept. All it does is allow people to think that things that are equivalent in Newtonian mechanics such as $m\vec a$ and $d\vec p/dt$ can continue to be equivalent when they can't. A theory that assumed mass didn't change and one where $E/c^2$ does change are different theories and people need to learn them as different theories. It is horrific terminology, literally as it causes actual horrors. $\endgroup$ – Timaeus Jan 4 '16 at 4:39
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When an object reaches speeds of the order of the speed of light one no longer uses Newtonian mechanics, but has to use special relativity. The the E=mc^2 is unfortunately a hybrid between Newtonian and special relativity , in order to describe the effect of force on an object moving with speeds close to the velocity of light.

The basic relativistic formula to keep in mind is the one that contains the invariant mass, i.e. a mass that is invariant to Lorenz transformations and can define a matter object :

$$m_0^2 = E^2 - ||\mathbf p||^2\;.$$

A particle, a proton let us say, has a fixed rest mass, called "rest" because it is the value the energy of the proton is in the rest frame ( momentum zero). When accelerated close to the speed of light it is the energy that changes, not the rest mass. The equivalence of mass and energy is what , when working in a Newtonian frame, responds as a classical mass to forces .

Does the increase in velocity result in an increase in the number of atoms in the object?

No. Matter (ensembles of protons and neutrons and electrons) have a fixed rest mass and the number of constituents cannot change.

The answer leads to 2)

How does mass increase if there is no increase in matter? The SI system defines mass in terms of a physical artefact (or a number of atoms, according to one proposal), but Wikipedia has seven different definitions of mass, so I'm not sure which one(s) applies here.

It only "appears to increase" if one tries to fit Newtonian physics to the interaction, i.e. work with an $F=m\cdot a$ formula.That is why the m is qualified as "relativistic". It is the total energy that is increasing with an increase of the momentum:

$$E= mc^2=\sqrt{p^2c^2 + {m_0}^2 c^4} $$

Does it make sense to ask how the added mass is distributed throughout the object?

As it is really the kinetic energy that is increasing with the increase in speed, this question is answered already.

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Don't use relativistic mass. Not ever.

When you increase your kinetic energy, the magnitude of your momentum increases.

Your momentum is $\vec p=\vec v E/c^2$ and its magnitude increased because your total energy $E$ increased (and if you weren't already moving at the speed of light then it also increased because the magnitude of your velocity increased too but your total energy actually increased less due to your rest energy).

A force can change your momentum. It will have to do this by changing your energy and maybe your velocity (if massive by definitely changing your velocity).

In many cases what you mean by relativistic mass is just to take energy $E$ divided by $c^2$ such as $\vec p=\vec v E/c^2.$ However a big issue is that things you thought were equivalent (because they approximately were at slow speeds) need to actually be kept distinct at high speed. And there is no consistent way around this.

So how about weight? Well firstly you can just say that in the absence of any forces besides gravity it accelerates with the gravitational acceleration. Or even better, you could just say that Newton's laws hold in a frame that is freely falling with the gravitational acceleration.

That is actually how General Relativity does it. And you can do it too. For instance air floats because the pressure below it is bigger than the pressure above it by just enough to make it accelerates upwards with $1g$ in the freely falling frame. That's why it floats.

There aren't more atoms. That wouldn't make sense because someone moving will see different relativistic masses. And they can't see different numbers of atoms. Plus, just one atom could move and if it moved a tiny bit, the energy increases just a little bit, way less than the smallest atom.

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  • $\begingroup$ Relativistic mass is $E/c^2$. Surely this is a meaningful expression. No harm can come from giving a name to a meaningful expression. Of course harm can come from treating things as equivalent that need to be kept distinct, but that has absolutely nothing to do with the question of which expressions we choose to give names to. $\endgroup$ – WillO Jan 4 '16 at 6:28
  • $\begingroup$ Moreover, the OP's (deep) confusion has nothing to do with relativistic mass. If you tell him it's not mass but energy that's increasing, he'll want to know how the extra energy is distributed. $\endgroup$ – WillO Jan 4 '16 at 6:29
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    $\begingroup$ @WillO It fundamentally is bad. Firstly it avoids learning that it was energy all along that was responsible for some effects. It also makes it seem like just one tweak gives you relativity instead of learning that whereas before many equations were equivalent only some of them actually work. For instance you need to learn the correct expressions in terms of mass, velocity, energy, and momentum not just think that any of many Newtonian expressions can be "fixed" by using $E/c^2.$ That's the true error. The horrific idea that SR is "about" replacing m with $E/c^2$ as if that's even consistent. $\endgroup$ – Timaeus Jan 4 '16 at 6:34
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    $\begingroup$ @WillO And yes the confusion is 100% and totally about the mass. Because Newton and Chemists both defined mass in terms of how much stuff you have, so it becomes natural to ask where the extra stuff is. Asking where the extra energy is is fine. Because then you can learn that energy is frame dependent and that each part can have a different kinetic energy in a different frame. And so on. That would be an excellent question. And it would be a question, not a confusion. Whereas relativistic mass makes people, like the OP, think there is more stuff when it is moving. $\endgroup$ – Timaeus Jan 4 '16 at 6:38

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