The speed limit is with respect to what?

As I understand, an object with mass cannot reach the speed of light because its (relativistic) mass increases "exponentially" as it approaches light speed.

So there is a relation between mass and speed. But if speed is a relative measurement (it depends on a point of reference) how does that affect the mass?
Is the mass also relative to a point of reference?
Or is there some absolute speed (wrt some fixed point) which affects the mass of an object?

Follow-up:
What I have trouble to understand is... given these two facts (I hope I got the facts right):

1. Light always moves at the same speed (independently of the inertial frame of reference)
2. The speed of a mass object is relative to a frame of reference

then if an object is moving at 99.999% of light-speed for a given observer, what will a different observer see?
Will that other observer see the object moving at a different percentage of light speed?
(assuming the observers are moving at different speeds, of course)

• Einstein was faced with exactly the same question. The answer wasn't obvious, and it took Einstein to see it. The answer he gave caused a revolution in science, and as a result can be found in any introductory text book on the subject. May 31, 2014 at 3:30
• @Nathaniel, what was that answer he gave? can you post it as an answer? May 31, 2014 at 3:46
• The thing is, you can find it explained in the first chapter or two of virtually any introductory text book on relativity. The explanation found in such a book will be much more comprehensive and better written than anything I could write, with much better diagrams. May 31, 2014 at 3:56
• @Nathaniel, unfortunately I don't have any books on relativity. I'm not a physics student, I ask this question just because I'm curious about these things. May 31, 2014 at 4:10

The answer by Alfred Centauri is correct, it is how physicists now view the use of the term "mass", but it needs some clarification on the usage.

$$\vec p = \frac{m\vec v}{\sqrt{1 - \frac{v^2}{c^2}}} \tag{1}$$

When special relativity was first studied the equation

$$E = \frac{m{c^2}}{\sqrt{1 - \frac{v^2}{c^2}}} \tag{2}$$

(where m is the rest mass the measure of the four dimensional vector describing the particle/system in motion), gave a new definition for the energy of moving objects.

From Newtonian physics we know that $\vec{p} = m \vec{v}$, so it was "natural" to think of the value multiplying the velocity vector in momentum as a mass: they called it "relativistic mass" because as far as classical inertial systems go the particle will behave to changes in motion as if its mass is increasing according to formula 2): more and more energy has to be supplied to reach the value of v. It is the relativistic mass that increases. In the rest mass of the particle, its mass is always m.

The use of the term "relativistic mass" is being dropped, because it gives rise to confusions as in your question. Each particle has a mass in its center of mass system that does not change .

A better term might be "apparent mass", how an observer would see a particle approaching the speed of light : as if it had more and more difficulty reaching there.

By the way, in the lab we observe relativistic elementary particles which are point particles, like the electrons.

The particle accelerator known as the Large Electron Positron (LEP) collider at the Centre Européenne pour la Recherche Nucléaire (CERN) laboratory near Geneva could propel electrons to 99.999999999 percent of the speed of light.

Their relativistic mass increases, but they are still measured as point particles with rest mass m.

• Can we think of this "apparent mass" as the inertia of an object? May 31, 2014 at 2:12
• @GetFree yes, for the observers seeing the object it acts/is the inertial mass. It is the rest mass that is the new concept, i.e. the energy momentum vector and its measure the rest mass. But since the rest mass is the fundamental attribute/scalar describing any object physicists who need relativistic physics to describe their data and form their theories are giving up on paying special attention to the relativistic mass. After all at the non relativistic limit they are equal to each other, and the concept is not useful in situations where special relativity has to be considered. May 31, 2014 at 4:17

According to SR, an object's momentum increases without bound as $v \rightarrow c$

$$\vec p = \frac{m\vec v}{\sqrt{1 - \frac{v^2}{c^2}}}$$

and indeed, an object's momentum is frame dependent.

However, an object's invariant mass $m$, (a Lorentz scalar) , is not frame dependent.

And that's all that really needs to said about that.

The speed limit is with respect to what?

Speed is relative - this means that you have to specify which frame an object is moving with respect to. So the speed limit is with respect to whichever frame you are talking about.

E.g., Two spacecraft are moving in opposite directions at $\frac{2}{3}c$ with respect to the Earth. With respect to one of the spacecraft, however, the Earth is moving at $\frac{2}{3}c$, but the other spacecraft is not moving at $\frac{4}{3}c$. It is moving at a speed given by a formula called the velocity addition Lorentz transformation. The need for this Lorentz transformation arises at speeds comparable to c, due to the fact that objects travel through spacetime at a constant rate.

If a baseball was passing right in front of you but suddenly all time surrounding you came almost to a standstill, you would see the ball hovering in the air in front of you. The ball is now in a very slow time frame.

Due to it being in a slow time frame, if you try to push it or grab it and throw it, an enormous amount of energy would be required to do so. Even if you did manage to throw the ball at a typical speed from your point of view, from its point of view it is moving at an enormous speed since it is now crossing large distances in very short time periods, thus an enormous amount of energy would be required to throw the ball while at the same time be satisfying the laws of physics as seen from the balls point of view.

Thus the closer an object approaches the speed of light, the slower a time frame it is existing within. This therefore simulates there having been an increase in its mass.

How much of an increase of mass there seems to be, all depends upon how close your frame of reference is to that of the object, meaning how fast is your clock ticking relative to the clock of the object.

As mentioned by physicist Brian Greene in his book The Elegant Universe ( See The_Elegant_Universe-B.Greene.pdf Motion Through Space-Time pages 26 and 27 ) all objects are constantly on the move within Space-Time at the speed of light.

Thus it is to be noted that all objects are moving at the speed of light to begin with. What can be done however, is change the direction of that constant motion within the 4 dimensional Space-Time environment.

I don't know many equations so I think I'll give an example. Suppose you wanted to travel to the future. The best way to do so is to travel close to the speed of light. So imagine that you gear up your rocket to such a speed that for you one year in the moving rocket is equal to a million years on earth. Now from the perspective of those on earth, you need to pack in more fuel to compensate for the million minus one extra years you will be travelling. However from both perspectives (those on earth and yours) the rocket is moving at the same high speed with respect to the earth. So from your perspective you are apparently carrying enough fuel to last for a million years although your journey lasts but one year. So perhaps that's because the mass of your rocket increases due to its speed, in your perspective. In that case what fuel was used to compensate in the increase in time from the stationary (not really stationary as we mentioned the earth) point of view is used in your moving point of view to compensate for increase in mass. So I guess the mass of moving objects do increase but only from the perspective of the one moving. However this entire answer is only conjecture on my part!!