# Is the slowing of time with speed exactly proportional to the increase of mass with speed?

This article on time dilation explains how time slows and mass increases with speed. Are they proportional? And is time dilation with a specific speed-mass equal to the time dilation with a stationary mass (like a planet). If so, can we say that it is mass alone, and not speed, that causes time dilation?

• I see now that relativistic mass is given by m = γm0, where γ = 1/√(1 − v2/c2) which is the same formula as time dilation t' = t/√(1 − v2/c2) So the answer to my first question is yes, they are exactly proportional. So now we are left with the second part of the question, which concerns whether relativistic mass has gravity and gravitational time dilation equal to stationary gravitational mass? And then, is it mass that causes time dilation? Commented Oct 15, 2019 at 22:01

Relativists no longer use the relativistic mass convention: http://physics.stackexchange.com/a/133395/4552 But yes, if you're using that convention, then the factor in both cases is $$\gamma$$.

And is time dilation with a specific speed-mass equal to the time dilation with a stationary mass (like a planet). If so, can we say that it is mass alone, and not speed, that causes time dilation?

Here I assume you're talking about gravitational time dilation. No, that is not related to any kind of mass effect. There is no useful way to define a change in mass with gravitational potential.

The OP asked in a follow-up comment:

I don't understand why "that is not related to any kind of mass effect." If I took a rock 1/10th the mass of the earth, and accelerated it up to 99.5% of c, it would then have the same mass as the earth. Would it have the same gravity (including the gravitational time dilation) as the earth?

No, it wouldn't. The source of the gravitational field is not mass density, it's the stress-energy tensor. Regardless of your frame of reference, the rock does not have the same stress-energy tensor as the earth. You also can't really express this example in terms of gravitational time dilation, because gravitational time dilation is a way of describing how the gravitational potential differs between one observer and another, when both observers are static. (In most spacetimes, there isn't even the notion of a static observer.)

• @bencromwell I don't understand why "that is not related to any kind of mass effect." If I took a rock 1/10th the mass of the earth, and accelerated it up to 99.5% of c, it would then have the same mass as the earth. Would it have the same gravity (including the gravitational time dilation) as the earth? Commented Oct 15, 2019 at 19:52
• @foolishmuse Think about this: Consider two astronauts in a spaceship travelling at $0.99c$. Will the astronauts stick together due to the increased gravitational attraction caused by the increased mass? Commented Oct 15, 2019 at 20:24
• @garyp good question. They wouldn't stick together as seen by themselves, but would they stick together as seen by me on the earth? Exactly the type of thing that makes relativity so darned hard. Commented Oct 15, 2019 at 20:29
• The equivalence principle says that the gravitational and the inertial mass are the same, and the inertial mass is the relativistic mass (therefore a box of heated gas weights more than the same box when it is cold). The only reason why some people don't like the relativistic mass is because you can't plug it into Newton's equations, but that should be clear from the beginning anyway, since the restmass also can't be plugged into Newtonian equations in relativity. Also, if they stick together in one frame, they also have to stick together in every frame, otherwise that would be a contradiction Commented Oct 15, 2019 at 20:41
• mass does not get bigger because you are moving. There are references frames that think you are moving near the speed of light. Do you feel any heavier? No.
– JEB
Commented Oct 15, 2019 at 23:19

Yes.

Both are observed from a frame moving with respect to the frame of the clock and the mass in question. Consider one clock and one mass that stay together, and several observers all moving at different speeds.

The observed slowing of time and the increase in mass both depend on the relative speed with which the observer is moving. Observers moving at different speeds will see different values from one another for the proportion, but none sees any difference between the proportion for time dilation and that for relativistic mass.

• What gravity would the different observers see, let's say if the mass is a rock 1/10th the mass of earth. Would the observer who is traveling at .99c see the clock falling towards the rock at 1G? What would the observer who is traveling at .1c see? Commented Oct 21, 2019 at 19:36
• If the rock was the Earth an observer speeding past would see the falling clock accelerating at 1g but it would be seen in slow motion, and depending on the direction the clock was falling, possibly compressed.
– DrC
Commented Oct 23, 2019 at 17:18

Are they proportional?

Yes.

And is time dilation with a specific speed-mass equal to the time dilation with a stationary mass (like a planet).

It's not clear what you're saying here.

If so, can we say that it is mass alone, and not speed, that causes time dilation?

No, increase in relativistic mass doesn't cause time dilation, they are both caused the same thing. And that thing isn't precisely speed, but rather the Lorentz transformation. Relativistic mass and time dilation are simply caused by the choice of reference frame. Speed changes of what frame of reference is "natural", but it doesn't actually cause mass or time to change, exactly.

• Lorentz transformation shows that $t$ & $t^\prime$ are related to each other as a function of $v$ (velocity), and this exactly means that the relative velocity causes the time dilation and relativistic mass. Your answer is excessive in my view. Commented Oct 15, 2019 at 20:04
• @accumulation If I'm standing on earth I have gravity at 9.8 m/s^2 and a specific time dilation. If a rock (1/10th the mass of earth) speeds by at 99.5% of c, then to me it will have the same mass as earth - to me. Will it have the same gravity including gravitational time dilation - all to me? If I watch a man on the flying rock drop a weight, will it fall with the same acceleration of 9.8 m/s^2 - to me? Will the time dilation on the flying rock be equal to the earth's gravitational time dilation? Commented Oct 15, 2019 at 20:07
• @foolishmuse For further discussion, please refer to this question: physics.stackexchange.com/questions/508397/… Commented Oct 15, 2019 at 22:00

If so, can we say that it is mass alone, and not speed, that causes time dilation?

Speed, esp., the speed of light, is an apriori definition in relativity according to which some relativistic results such as time dilation or mass increase are defined. Therefore, you cannot infer the posterior mass increase prior to the anterior speed.