# Mass increase at relativistic speeds

## Main Question or Discussion Point

Until now, I always that that a relativistic mass increase was literally just the increase in the mass of an object. Today I googled it, and a couple sources say that it is just an increase in energy, not mass. Can anyone confirm this?

At first I thought if the mass increased, then it would take more energy to accelerate that mass, and eventually it would take infinite energy. But if its really energy that increases, then this logic doesn't make sense. Unless it still takes more energy to accelerate a particle with a higher energy?

Finally, why does the energy increase?

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Until now, I always that that a relativistic mass increase was literally just the increase in the mass of an object. Today I googled it, and a couple sources say that it is just an increase in energy, not mass. Can anyone confirm this?

At first I thought if the mass increased, then it would take more energy to accelerate that mass, and eventually it would take infinite energy. But if its really energy that increases, then this logic doesn't make sense. Unless it still takes more energy to accelerate a particle with a higher energy?

Finally, why does the energy increase?
Rest mass stays the same but relativistic mas increases.

Rest mass stays the same but relativistic mas increases.
What is relativistic mass increase. Does it interact with gravitation like rest mass?

What is relativistic mass increase. Does it interact with gravitation like rest mass?
No such thing as "increasing mass" exists in reality and all is just a hypothetical facet of Relativity. The relativistic mass is the one that I measure in my own frame when the object being measured is moving in some other frame along, say, x-axis and of course if my frame was to be inertially moving wrt the object's frame, this axis would show the direction of motion. This kind of mass has nothing to do with any type of interactions and only appears in calculations so outside of the theory we can't feel it! Can you see any difference in the mass of your friend when he is onboard but you're still at rest on earth? Absolutely not and it is why such definition in physics cannot be taken seriously!

AB

George Jones
Staff Emeritus
Gold Member
What is relativistic mass increase. Does it interact with gravitation like rest mass?
In some sense, yes, but it is not just a matter of replacing $m$ by $\gamma m$ in gravitational formulae. See, for example,

http://arxiv.org/abs/gr-qc/9909014

http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=AJPIAS000053000007000661000001&idtype=cvips&gifs=yes [Broken]

http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=AJPIAS000050000006000527000001&idtype=cvips&gifs=yes [Broken].

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What is relativistic mass increase. Does it interact with gravitation like rest mass?
Yes, in the sense that a spinning mass weighs more than when it it is not spinning. However in GR, rest mass and energy and tension and pressure all interact with gravitation and so the increased gravitational effect on the spinning object can be put down to its increased rotational energy.

On the other hand, two particles alongside each other and moving to your right at 0.99c do not gravitate together quicker than two particles that are nearly at rest with respect to you. In fact they come together slower! Certainly, no object can collapse to form a black hole due to its relativistic mass. Like George said, the relationship between "relativistic mass" and gravitational mass is not simple and most modern authors suggest abandoning the relativistic mass concept alltogether.

I don't understand. So does relativistic mass and rest mass refer to the energy of the object or physical mass? And why do people say one's mass increase as one approach the speed of light?

Fredrik
Staff Emeritus
Gold Member
I don't understand. So does relativistic mass and rest mass refer to the energy of the object or physical mass? And why do people say one's mass increase as one approach the speed of light?
I'll quote myself from a post I wrote a minute ago.
See this post. Physicists who use the term "mass" are talking about the "m" in my post, which is independent of speed. The quantity $\gamma m$ on the other hand, depends on the speed, because $\gamma$ does. Some people call $\gamma m$ the "mass". That terminology is considered obsolete and useless by a lot of people, including me. If I have to use a term for $\gamma m$ and I'm not allowed to use units such that c=1, I'll call it "relativistic mass". In units such that c=1, the relativistic mass is equal to the total energy $\gamma mc^2$, so I can just call it "total energy" instead.
The relativistic mass is just the total energy expressed in non-standard units. The total energy, and therefore also the speed of a massive object (relative to say the cosmological microwave background), will have some effect on how this object influences other objects gravitationally, but you can't just plug in the relativistic mass in Newton's law of gravity and expect to get a reasonable result. (Newton's law of gravity would make SR logically inconsistent because of the instantaneous action-at-a-distance, and you can also think about the fact that the Earth and the Moon would both have huge relativistic masses when observed from a spaceship passing us. A naive application of Newton's law would suggest that Earth and the Moon are just going to smash into each other).

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Ok, so rest mass then does not depend on speed. Since relativistic mass is the total energy of the object at a given speed (according to your post), then rest mass is the total energy at rest?

And I don't think you addressed the increase in relativistic mass as velocity increases... why does that happen, conceptually?

Fredrik
Staff Emeritus
Gold Member
Ok, so rest mass then does not depend on speed. Since relativistic mass is the total energy of the object at a given speed (according to your post), then rest mass is the total energy at rest?
Right.

And I don't think you addressed the increase in relativistic mass as velocity increases... why does that happen, conceptually?
I'm not sure what you would consider a conceptual explanation of why

$$\frac{m}{\sqrt{1-v^2}}$$

increases as v increases from 0 and approaches 1 (=c).

Relativistic mass is such a bad concept, the correct way of looking at it is that many of the characteristics we attribute to mass are actually attributed to all forms of energy. Such things are gravitation and inertia, the reason we don't notice them in other forms of energy than mass is that mass is in most cases the dominant term.
And I don't think you addressed the increase in relativistic mass as velocity increases... why does that happen, conceptually?
When you accelererate an object you are adding energy to it, that energy takes the form of a velocity and is called kinetic energy. As I stated above kinetic energy will also come with the properties called inertia and gravitation so these things aren't strange at all.

The best conceptual picture is that mass is momentum in the time direction while kinetic energy is momentum in the space direction. With this picture you move in the direction this vector would point at and you always move at the speed of light, an object in its own rest frame moves fully in time while objects moving a bit more in space will have time dilation etc.

The best conceptual picture is that mass is momentum in the time direction while kinetic energy is momentum in the space direction. With this picture you move in the direction this vector would point at and you always move at the speed of light, an object in its own rest frame moves fully in time while objects moving a bit more in space will have time dilation etc.
How does this relate to the concept of the energy-momentum vector, whose time component is energy (=relativistic mass), whose space components are momentum, and whose magnitude is (rest) mass? Is it just an alternative way of describing the same physics, and are there possible ambiguities to look out for, e.g. if I read something about "momentum", is there potential for confusion if I don't know whether the author is referring to the kind of momentum described in the energy-momentum vector concept, or to the kind of momentum described in this mass/kinetic-energy vector, or are they equivalent?

May I take the opportunity to answer your question in a different approach-
Mass increases with energy. Yes, this is true.
More energy is needed to accelerate more mass. This is also true.
Then what is mass and energy ?
Energy is released when mass is destructed (nuclear bomb). Mass is formed when energy is trapped (nuclear bonding).
So energy and mass are different in forms and can be transformed into one another.
One main point that should be kept at the back of your head is that 'mass and energy exists independently but not simultaneously' . ie., mass and energy cant exist at the same time (same like the wave and particle theory of light).
So whatever it is stated about mass and energy it exists independently.

How does this relate to the concept of the energy-momentum vector, whose time component is energy (=relativistic mass), whose space components are momentum, and whose magnitude is (rest) mass? Is it just an alternative way of describing the same physics, and are there possible ambiguities to look out for, e.g. if I read something about "momentum", is there potential for confusion if I don't know whether the author is referring to the kind of momentum described in the energy-momentum vector concept, or to the kind of momentum described in this mass/kinetic-energy vector, or are they equivalent?
They describe the same things, just that you use a different representation. What you describe is used since it makes the energy momentum vector have Lorentz invariant length which is equal to its rest mass but it might not be the easiest way to conceptualize the physics.

The momentum, energy and rest mass is exactly the same in both. You have rest mass as the time component, momentum as the space component and energy as the total length, this way you got the normal euclidean metric but it isn't Lorentz invariant so it isn't practical from a maths viewpoint.

They describe the same things, just that you use a different representation. What you describe is used since it makes the energy momentum vector have Lorentz invariant length which is equal to its rest mass but it might not be the easiest way to conceptualize the physics.

The momentum, energy and rest mass is exactly the same in both. You have rest mass as the time component, momentum as the space component and energy as the total length, this way you got the normal euclidean metric but it isn't Lorentz invariant so it isn't practical from a maths viewpoint.
In #12, you said that "kinetic energy is momentum in the space direction"; here in #14, you say that "momentum" (3-momentum?) is the space component of this vector which you also call "momentum", but which isn't 4-momentum (energy-momentum), since it isn't Lorentz invariant--so that's at least three distict concepts now going by the name of momentum, including the two conventional ones!--and yet you say that "The momentum, energy and rest mass is exactly the same in both [concepts/viewpoints/representations]." How many components does your vector have in spacetime of 3+1 dimensions? By Euclidean metric, do you mean the fact that p2 + m2 = E2? How does this relate to what you said in #12, where mass and kinetic energy were said to be the time and space components respectively? What does it mean to call something the time component of a Euclidean vector?

DrGreg
Gold Member
Rasalhague, I think Klockan3 is attempting to use the Euclidean "space-propertime" view instead of the conventional Lorentzian "space-time" view. Although that view can give a student an easy-to-understand geometry when you consider a single, isolated vector (whether that is a "space-propertime" vector or a "momentum-restmass" vector), it's not much good for anything else -- it's not a genuine vector space and you can't make much sense of it as soon as you have two "vectors" to compare. Nor does it make much sense if you try to convert from one coordinate system to another. Students really need to take the plunge and use the Lorentzian metric on space-time and momentum-energy vectors.

To answer what you asked earlier, although we have competing definitions for mass (rest mass v. relativistic mass) there's no such competition for energy or momentum. To make any sense at all, these concepts have to have a conservation law and have to approximate to the Newtonian definitions at low velocity, and that effectively fixes their definitions, I think.

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You can have a time dimension in euclidean coordinates as well, all I meant was that the time component wouldn't have a negative metric in this case... That way of putting momentum and mass makes it really obvious that they are components of the same thing, aka different types of energy.

What are you confused about really, this is just the normal relativistic 3 momentum with the mass component in the time direction instead of energy and with the euclidean metric. Two orthogonal directions of momentum have exactly the same relations with each other as momentum and mass has. Energy without momentum is mass, energy without mass is momentum.
Rasalhague, I think Klockan3 is attempting to use the Euclidean "space-propertime" view instead of the conventional Lorentzian "space-time" view. Although that view can give a student an easy-to-understand geometry when you consider a single, isolated vector (whether that is a "space-propertime" vector or a "momentum-restmass" vector), it's not much good for anything else -- it's not a genuine vector space and you can't make much sense of it as soon as you have two "vectors" to compare. Nor does it make much sense if you try to convert from one coordinate system to another. Students really need to take the plunge and use the Lorentzian metric on space-time and momentum-energy vectors.

To answer what you asked earlier, although we have competing definitions for mass (rest mass v. relativistic mass) there's no such competition for energy or momentum. To make any sense at all, these concepts have to have a conservation law and have to approximate to the Newtonian definitions at low velocity, and that effectively fixes their definitions, I think.
In my opinion most classes focuses a bit too much on computations. I don't really see why, computations is the easy and boring part of physics...

Rasalhague, I think Klockan3 is attempting to use the Euclidean "space-propertime" view instead of the conventional Lorentzian "space-time" view.
A cursory google suggests space-propertime might be the same thing as Epstein's method; is that right? This illustration looks like what we've been talking about:

I've read a few basics of how to draw Epstein diagrams, but haven't got very deeply into it yet. I gather one advantage is that it uses Euclidean geometry, and one disadvantage is that a single event in general needs to be represented by more than one point in the diagram.

A cursory google suggests space-propertime might be the same thing as Epstein's method; is that right? This illustration looks like what we've been talking about:

I've read a few basics of how to draw Epstein diagrams, but haven't got very deeply into it yet. I gather one advantage is that it uses Euclidean geometry, and one disadvantage is that a single event in general needs to be represented by more than one point in the diagram.
That looks like it, but I haven't heard about it before, I figured it out myself but I guess that it was to be expected that someone else already had done it as always :tongue2: . I just felt that it was a much more intuitive picture to have rather than the quite abstract picture you get from the tensor formalism.

That looks like it, but I haven't heard about it before, I figured it out myself but I guess that it was to be expected that someone else already had done it as always :tongue2: . I just felt that it was a much more intuitive picture to have rather than the quite abstract picture you get from the tensor formalism.

They say great minds think alike ;-)

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At first I accepted the explanation I saw here that the mass doesn't increase, just the energy. Then I read The Elegant Universe by Brian Greene and I saw this,
The faster something moves the more energy it has and from
Einstein's formula we see that the more energy something has the more massive it becomes. Muons traveling at 99.9 percent of
light speed, for example, weigh a lot more than their stationary cousins. In fact, they are about 22 times as heavy—literally. (The
masses recorded in Table 1.1 are for particles at rest.) But the more massive an object is, the harder it is to increase its speed.
Pushing a child on a bicycle is one thing, pushing a Mack truck is quite another. So, as a muon moves more quickly it gets ever
more difficult to further increase its speed. At 99.999 percent of light speed the mass of a muon has increased by a factor of 224; at
99.99999999 percent of light speed it has increased by a factor of more than 70,000. Since the mass of the muon increases without
limit as its speed approaches that of light, it would require a push with an infinite amount of energy to reach or to cross the light
barrier. This, of course, is impossible and hence absolutely nothing can travel faster than the speed of light.
This is definitely a credible source, so does mass increase or not, and why?

At first I accepted the explanation I saw here that the mass doesn't increase, just the energy. Then I read The Elegant Universe by Brian Greene and I saw this,

[...]

This is definitely a credible source, so does mass increase or not, and why?
Sadly there's no standard terminology. Different authors use the words mass and energy to in different ways, which makes things very hard for students. But the disagreement is only apparent; it's just a matter of words. Useful terms here are "rest mass" and "relativistic mass". These are different things and when you read a statement about mass in special relativity, you need to find out which kind the writer has in mind. For some authors, mass means rest mass (e.g. Taylor & Wheeler). Other authors use the name mass without qualification for relativistic mass (e.g. Rindler, Penrose and Greene). As this thread has shown, there are other quirks, but the distinction between rest mass and relativistic mass is the main one to watch out for. Relativistic mass is the one that increases with speed. Rest mass is the mass of the object in its rest frame (an inertial reference frame in which the object is at rest), and therefore doesn't depend on its speed in some other reference frame.

For an object with mass, relativistic mass is equal to its energy divided by the square of the speed of light in any inertial reference frame. The rest mass of such an object is only equal to its energy divided by the square of the speed of light in the object's rest frame. Some authors use the symbol m0 for rest mass and m for relativistic mass. But this practice isn't universal. Relativistic mass is defined as rest mass times "gamma", where gamma denotes 1/sqrt(1-(v/c)^2).

If the object has no rest mass, it has no rest frame, since it travels as the speed of light. It can still have energy though, hence my continued bemusement at the expression "mass-energy equivalence", which also has ten dozen hundred interpretations ;-)

http://plato.stanford.edu/entries/equivME/

I like the way Fernflores uses the term "inertial mass" here for rest mass, whereas Rindler uses the same term "inertial mass" as a synonym for relativistic mass.

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Relativistic mass is such a dubious concept. Is it not solely for the sake of completeness; to define relativistic mass as the ratio between relativistic momentum and velocity?

Perhaps the confusion is in the definition of 'mass'.

If you define mass as "the energy in an object which is not kinetic energy, divided by c^2", then mass is constant at any speed.

However, if you define mass as "the resistance of an object to change in velocity", then mass changes with speed, because it requires a greater force to accelerate a fast-moving object, compared to an object at rest (relative to the applied force).

In Newtonian physics these two definitions of mass are equivalent, so there is no confusion. However in relativistic physics, we must choose one definition or the other. The first definition (energy which is not kinetic, divided by c^2) is most popular today, I believe.

At first I accepted the explanation I saw here that the mass doesn't increase, just the energy. Then I read The Elegant Universe by Brian Greene and I saw this,

This is definitely a credible source, so does mass increase or not, and why?
Though there is a lot of debate about terminology (with "relativistic" mass falling into disfavor), there is no debate about the experimental verification. See here.