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So according to the principle of relativity there is no preferred reference frame, and thus from the reference frame of a fast-moving particle all objects would be moving close to the speed of light. The mass dilation transformation says that objects gain relativistic mass when they approach the speed of light, so do all objects have a high relativistic mass? I know I'm misunderstanding Special Relativity, but how can mass differ between reference frames?

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    $\begingroup$ The concept of relativistic mass, in spite of being scientifically correct and frequently used in many textbooks and works on special relativity by luminaries like Pauli, Born, Tolman, etc., is anathema to some one-sided zelots here on this site. They try to enforce that mass signifies only rest mass and insist that special relativity effects have to be expressed in terms of 4-vectors. This has advantages and disadvantages, but is not universally accepted in the physics community. So be careful and better don't mention relativistic mass, you will be mercilessly ostracized. $\endgroup$ – freecharly Dec 10 '16 at 5:59
  • $\begingroup$ In spite of this, I will give you an answer to your question addressing "relativistic mass" below. $\endgroup$ – freecharly Dec 10 '16 at 6:40
  • $\begingroup$ Vote to close the question. It's more opinion and interpretation that physics. $\endgroup$ – Bob Bee Dec 10 '16 at 6:45
  • $\begingroup$ See some of the answers and comments under physics.stackexchange.com/questions/133376/… to get an idea of how the kind of physicist who do relativity all the time feel about relativistic mass. You can certainly do physics that way, but until you know the subject well the concept is a trap that will lead you into error over and over again: it simply makes unreasonable and even silly questions sound profound. $\endgroup$ – dmckee Dec 10 '16 at 7:13
  • $\begingroup$ As long as you know what is meant by "relativistic mass" as opposed to the Lorentz invariant "rest mass" there should be no problem in its use, similarly to the relative concepts of length or time. It has not only disadvantages but also definite advantages. See e.g., T. R. Sandin,"In defense of relativistic mass", American Journal of Physics 59, 1032 (1991). $\endgroup$ – freecharly Dec 10 '16 at 7:28
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Energy, in particular kinetic energy, and momentum depends on the reference frame. In special relativity, when a body moves close to the speed of light $c$ with speed v, its momentum is given by $$p=\frac {m_0}{\sqrt{1-(v/c)^2}}v$$ where $m_0$ is the rest mass of the body and its total energy is given by $$E=\frac {m_0 c^2} {\sqrt{1-(v/c)^2}}$$ Thus both the momentum and the total energy of the body approach infinity when $v \to c$. When you interpret in these formulae the term $$ \frac {m_0} {\sqrt{1-(v/c)^2}}$$ as the "relativistic mass" this mass goes to infinity when $v \to c$ explaining the impossibility to accelerate the body to $c$. This is equivalent to stating that the total energy goes to infinity. This "relativistic mass" is obviously reference frame dependent, just like length and time intervals, in contrast to the rest mass, which is frame independent. Thus if you move close to the speed of light to all other bodies, all these bodies gain energy and thus "relativistic mass" relative to you.

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That is why in special relativity the concept of invariant mass is the relevant one, not the relative mass between moving systems , but the mass of the object at its rest frame.

Relativistic mass is mathematical concept that demonstrates the mass/energy equivalence but is a confusing concept. It is called "mass" because when studied from another frame of reference it reacts to changes in momentum ( force=dp/dt) as the Newtonian kinematics mass :

In physics, mass is a property of a physical body. It is the measure of an object's resistance to acceleration (a change in its state of motion) when a force is applied.

you ask:

when they approach the speed of light, so do all objects have a high relativistic mass?

All objects have their fixed invariant mass in their own system. The mathematics of special relativity is such that this mass measure is invariant even when measured on moving systems . We measure the invariant mass of elementary particles with great accuracy using relativistic kinematics on moving systems. These elementary particles move with respect to us usually close to the velocity of light. We do not calculate their relativistic mass because it is not an invariant, it changes according to

emc

where m_0 is the invariant mass we measure and identify the particle.

Why don't we constantly experience a mass dilation?

The particle experiences no changes in its invariant mass due to the existence of fast moving other particles. It will simply observe other particles as having high relativistic mass with respect to it. At the center of mass system of the particle, if it could measure, it would observe everything as having an enormous relativistic mass.

Objects observed from a moving system may have a very large relativistic mass. The relativistic mass is not an invariant measure of special relativity but depends on the relative velocities between the observing system and the observed system.

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so do all objects have a high relativistic mass?

No, because there are very few objects moving at speeds close to the speed of light due to Aristotle's principle.

according to the principle of relativity there is no preferred reference frame

In special relativity there is not. In general relativity, there is.

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    $\begingroup$ @Annix Special relativity has no preferred reference frame among inertial reference frames. In GR there are no preferred frames, the Einstein equations hold in all coordinate systems. And what did Aristotle say that in any way means that very few objects travel at close to the speed of light? $\endgroup$ – Bob Bee Dec 10 '16 at 6:41
  • $\begingroup$ @Bob Bee in GR there is the preferred reference frame, that is the cosmological rest frame aka comoving frame. core.ac.uk/download/pdf/35472181.pdf $\endgroup$ – Anixx Dec 10 '16 at 7:12
  • $\begingroup$ I know GR. That's only for the cosmological solution, or work in cosmology. For instance for GR on the earth it is totally irrelevant, and you can use the Schwarzschild coordinates to get for instance the time difference due to GR for the GPS satellite. GR has no preferred coordinates, but a physical situation with some symmetries can lead to one. For cosmology it's the universe's homogeneity and isotropy, for static spherical symmetry it is a different coordinate system, etc. $\endgroup$ – Bob Bee Dec 10 '16 at 7:33
  • $\begingroup$ @Bob Bee well, yes, I meant the cosmological solution of the GR has a preferred coordinate frame (honestly, any GR solution has a preferred coordinate frame, does not it?) $\endgroup$ – Anixx Dec 10 '16 at 7:47
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    $\begingroup$ @Bob Bee any GR solution has a preferred frame. So, if our universe is described by GR, it consequently has a preferred frame. $\endgroup$ – Anixx Dec 10 '16 at 7:56

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