# Does mass increase proportionally to speed?

## Main Question or Discussion Point

If I understand correctly, an object with mass can't reach the speed of light because at the speed of light it would have infinite mass (and presumably fill the infinite universe). But what happens as an object approaches the speed of light? Eg., if an object traveled at 1% of the speed of light, how much would its mass increase? Certainly it wouldn't fill 1% of the entire universe at that speed. If an object were to reach 99% of the speed of light, would its mass increase proportionately so it theoretically filled 99% of the universe? That seems wrong for a lot of reasons, but what does theory say? Is there an exponential ratio that makes it conceivable for an object to approach the speed of light without noticeably increasing mass? At what point does it become an impossibility? Only at the speed of the light, but at no point before?

Which raises the dependent paradox that if the speed of light has a definite value, wouldn't that imply that infinity has a definite value? If an object at the speed of light fills an infinite universe, wouldn't an object at 99.9999% the speed of light fill up 99.9999% of the universe? What would 99.9999% of infinity be?

I'm not sure if there's an explanation I'll understand, but it's worth a try.

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Nugatory
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The relationship between mass and velocity is:

$$m=\frac{m_{0}}{\sqrt{1-\frac{v^{2}}{c^{2}}}}$$

Here, m is the mass that will be measured by an observer moving with speed v relative to the object and m0 is the mass that will be measured by an observer at rest relative to the object. Some noteworthy things about this formula:

1) If v is small compared to the speed of light, the difference between m and m0 will be even smaller. This is why we never notice the mass increase in everyday life.

2) You can get a mass as large as you want even when v never reaches, let alone exceeds, c. That's probably the answer to your question... Try plugging in v=.9900c and v=.9999c, notice how much the mass changes as we go from "close to the speed of light" to "even closer".

3) This formula does not apply to light (or anything else that moves at the speed of light), nor would it apply if you could make something move faster than light.

4) Most important - The moving object doesn't see its own mass increasing. You may see me moving at .99c and measure my mass as increased, but as far as I'm concerned, I'm at rest and you're moving away from me at .99c.

Nugatory
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If an object at the speed of light ....
There's no such thing as an object traveling at the speed of light (as opposed to some speed that's very close to but still less than the speed of light), so any paradox of the form "an object moving at the speed of light will [something absurd and paradoxical]" is like saying "if 1=0 then [something absurd and paradoxical]". That is the problem isn't in the conclusion, it's in the starting assumption.

The relationship between mass and velocity is:

$$m=\frac{m_{0}}{\sqrt{1-\frac{v^{2}}{c^{2}}}}$$
This is only true for a so called "transversal mass", which is the proportionality constant between the component of the force perpendicular to the velocity, and the perpendicular acceleration.

There is a different "longitudinal mass" given by:
$$m_{\mathrm{long.}} = m_{0} \, \left( 1 - \frac{v^2}{c^2} \right)^{-\frac{3}{2}}$$
which is the proportionality factor between the component of the force parallel to the velocity and the parallel acceleration.

These two become equal only when $v = 0$, and they are both equal to $m_{0}$ - the rest mass of the particle. This is the only physically important characteristic of the particle and it is a Lorentz invariant.

Sorry, if I phrased that poorly: I may have misunderstood other explanations, but I've been told, and read, that an object with mass can't reach the speed of light because it would then have infinite mass, which is impossible. Isn't that the same statement as "if an object with mass reached the speed of light it would have infinite mass"? It's just a different way of phrasing the same question. I don't think that changes the nature of the question. I'm probably misunderstanding the meaning of mass. Am I incorrect in picturing an object with greater mass taking up more space? Even if an object could reach a tiny fraction of the speed of light, say one millionth the speed of light, it wouldn't increase mass in any way that we could say it filled some measurable percentage of the space of the universe, would it?

(The paradox I'm confused by isn't that at the speed of light mass would be infinite, it's that at something less than the speed of light mass would be some definable percentage of infinite. Infinity as I understand it can't be measured in percentages. I understand that mass increases as its speed increases. I don't understand the nature of the steps between a measurable mass and an infinite mass.)

I admit the question may be meaningless. I'm just hoping someone can explain how this works in a way I can grasp. My math skills aren't on your level, so I don't get a clear picture of what the formula means.

Sorry, if I phrased that poorly: I may have misunderstood other explanations, but I've been told, and read, that an object with mass can't reach the speed of light because it would then have infinite mass, which is impossible. Isn't that the same statement as "if an object with mass reached the speed of light it would have infinite mass"?
Well, you've been told wrong.

The relativistic energy of a particle with (rest) mass m, travelling at a speed v which is a significant fraction of the speed of light c, is given by:
$$E = \frac{m c^{2}}{\sqrt{1 - \frac{v^2}{c^2}}}$$

As you can see, it takes an infinite amount of energy to accelerate an object to a speed of light.

PeterDonis
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(The paradox I'm confused by isn't that at the speed of light mass would be infinite, it's that at something less than the speed of light mass would be some definable percentage of infinite. Infinity as I understand it can't be measured in percentages. I understand that mass increases as its speed increases. I don't understand the nature of the steps between a measurable mass and an infinite mass.)
First of all, I'm going to say "energy" instead of "mass" because it's less confusing; "mass" could also mean "rest mass", whereas "energy" is unambiguous.

That said, do you understand the concept of a limit? The more correct way to say "at the speed of light energy would be infinite" is this: as an object's speed approaches the speed of light, its energy increases without bound; in the limit as the object's speed goes to the speed of light, the object's energy goes to infinity. But at any speed less than that of light, the energy has some finite value. It's not a "percentage of infinity"; "infinity" here is just a shorthand for "increases without bound".

What's the distinction you're (Dickfore) trying to make? Are you just saying that the correct statement is "an object can't reach the speed of light because it would require infinite energy to accelerate it to the speed of light"? But the the explanation "an object can't reach the speed of light because its mass would be infinite" was an incorrect description? Is it untrue then?

What happens to mass as it approaches the speed of light? That's all I'm trying to get at. Does mass increase with velocity or not? I think (but I'm not entirely sure) you're trying to point out to me that since mass couldn't be infinite it can't "approach infinity" as velocity increases? So it's illogical for me to phrase the problem in terms that imply a fraction of the speed of light is related to a fraction of infinity, since no such correlation actually exists?

(PeterDonis) Thanks, that makes a bit more sense to me. I'm going to try to make sense of this before I try again, since I think I'm having trouble making the question clear.

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PeterDonis
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Are you just saying that the correct statement is "an object can't reach the speed of light because it would require infinite energy to accelerate it to the speed of light"?
That's correct, yes.

But the the explanation "an object can't reach the speed of light because its mass would be infinite" was an incorrect description?
It depends on what you mean by "mass". If by "mass" you just mean "energy" (but measured in different units), then this description isn't incorrect (though it would be more correct to say mass, or energy, increases without bound as speed approaches the speed of light).

What happens to mass as it approaches the speed of light? That's all I'm trying to get at. Does mass increase with velocity or not?
Energy increases with velocity. "Mass" is an ambiguous term; it has several possible meanings, so whether it increases with velocity or not depends on which meaning you are using. One possible meaning of "mass" is "energy"; with that meaning, yes, mass increases with velocity. But another possible meaning is "rest mass"; with that meaning, no, mass does not increase with velocity.

I think (but I'm not entirely sure) you're trying to point out to me that since mass couldn't be infinite it can't approach infinity as velocity increases?
No, I'm trying to say that "mass" is an ambiguous term. See above.

So it's illogical for me to phrase the problem in terms that imply a fraction of the speed of light is related to a fraction of infinity, since no such correlation actually exists?
It's illogical because "a fraction of infinity" is meaningless. (More precisely, "a fraction of infinity" is still infinity; dividing infinity by any finite number does not change it. So it's meaningless to try to compare "a fraction of infinity" with a fraction of some finite number such as the speed of light.)

[Edit: I see that what I was responding to above was actually directed at Dickfore. But hopefully my responses are still helpful.]

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@jfraze: What we are trying to say is that when you talk about mass changing with speed, you are talking about what is called "relativistic mass" whereas most physicists today prefer talking about "invariant mass" or equivalently "rest mass". The relativistic mass changes depending on the frame, so that it has different values for different speeds. Relativistic mass is the formula in post #2. This concept suggests that something changes in the internal structure of the accelerated object (something which is not really a correct description) and easily leads to misunderstandings, which is why most of us have abandoned this term in favour of invariant mass (this is the $m_0$ in the equation in #2, or equivalently the $m$ in Dickfore's post). I suggest you read the following wiki page which explains well the ambiguity about the term mass in Special Relativity which Peter Donis talks about: http://en.wikipedia.org/wiki/Mass_in_special_relativity

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It's illogical because "a fraction of infinity" is meaningless. (More precisely, "a fraction of infinity" is still infinity; dividing infinity by any finite number does not change it. So it's meaningless to try to compare "a fraction of infinity" with a fraction of some finite number such as the speed of light.)
Exactly, which is why I'm confused about the idea that mass (or anything else) could increase towards infinity, when infinity is not a set value as light speed is. I'm having trouble with the idea that light speed and mass are directly correlated when one value seems to be finite and the other is not. You can have something measurably attaining a speed of 99% of light speed, but 99% of infinity is obviously a meaningless term.

Is it important that there are infinite steps between 99% of the speed of light and 100% of the speed of light? Velocity could increase towards c infinitely without ever attaining c. At least speaking mathematically. Practically, though, I still have a hard time wrapping my mind around what the difference would be as you reach ever decreasing fractions. It's easier to grasp with the knowledge that relativistic mass appears to be more a measure of the energy in an "object." But still confusing. This has helped a lot though, and cleared up a few of my misconceptions. Thanks.

Exactly, which is why I'm confused about the idea that mass (or anything else) could increase towards infinity, when infinity is not a set value as light speed is. I'm having trouble with the idea that light speed and mass are directly correlated when one value seems to be finite and the other is not. You can have something measurably attaining a speed of 99% of light speed, but 99% of infinity is obviously a meaningless term.

Is it important that there are infinite steps between 99% of the speed of light and 100% of the speed of light? Velocity could increase towards c infinitely without ever attaining c. At least speaking mathematically. Practically, though, I still have a hard time wrapping my mind around what the difference would be as you reach ever decreasing fractions. It's easier to grasp with the knowledge that relativistic mass appears to be more a measure of the energy in an "object." But still confusing. This has helped a lot though, and cleared up a few of my misconceptions. Thanks.
You need to take Calculus 1 and learn about infinite limits and asymptotes. Your confusion is not with Physics, as it is with a mathematical concept.

HallsofIvy
Homework Helper
"Increase toward infinity" just means "increases without bound". It does NOT imply that "infinity" is a specific thing.

You also say, "But what happens as an object approaches the speed of light? Eg., if an object traveled at 1% of the speed of light, how much would its mass increase? Certainly it wouldn't fill 1% of the entire universe at that speed." "Mass" has nothing to do with volume. in fact, as an object's volume increases, relative to an observer, that observer would see its mass increasing and its volume decreasing.

Your difficulties seem to be more with English grammer than with physics.

PeterDonis
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Is it important that there are infinite steps between 99% of the speed of light and 100% of the speed of light? Velocity could increase towards c infinitely without ever attaining c. At least speaking mathematically. Practically, though, I still have a hard time wrapping my mind around what the difference would be as you reach ever decreasing fractions.
This is why I asked if you were familiar with the concept of a limit. It doesn't seem like you are, so I would recommend, as others have, that you learn about that concept, which is a basic concept of calculus. If you want to get some idea of how it works in this specific case, you could try graphing the following function:

$$\gamma (v) = \frac{1}{\sqrt{1 - v^{2}}}$$

the relevance of which should be obvious. I have written this in units where the speed of light is 1, so try plotting this function, with $\gamma$ on the y axis vs. v on the x axis, as v gets closer and closer to 1. What happens to $\gamma$?