Increased Mass

[a1call]

* How fast would a 100 kg (at rest) object have to travel to achieve 101 kg of mass?

* What would be the mass of a 100 kg (at rest) object when it is traveling at a speed of 30 km/s?

* What is the general formula?


Thanks for the lesson.

[Ken G]

You need to calculate what is called the "Lorentz factor". Take v/c, square it, and subtract it from 1. Now take the square root-- you have a number less than 1. Now take the inverse of that number (one over it), and you'll have a number larger than 1. That's the factor to multiply the mass by.

[a1call]

Much obliged Ken G.

I was hoping there would be a more noticeable mass change at lower speeds.

Well, now I know.

[speedfreek]

You have to be travelling at around 86.6% of the speed of light before your mass apparently doubles. The Lorentz transformation introduces an exponential curve to the equation. Between 86.6% of the speed of light and light-speed itself, the mass increases from double, to infinite.

[Ken G]

I was hoping there would be a more noticeable mass change at lower speeds.

Yeah, as a check on your result, I get that at 30 km/s the mass change is 1 part in 200 million. Now you know why we usually don't have to worry about relativity!

[a1call]

Basically what is bothering me is that increased mass will violate the law of preservation of momentum. This is because we will have to multiply the Lorentz factor by the product of mass and velocity. Now this law breaking is fine with me except that the law of the preservation of momentum is generally referred to as an absolute and is the basis of many ATM debunking arguments.

I guess my follow up questions are:

* Isn't the law of preservation of momentum an approximation which only holds pretty accurate at low velocities?
* If so how can it be argued that there can be no device which alters the momentum of a system?


Added:
I do not understand these stuff well at all. I did some more reading and it appears that Conservation of four-momentum is more of an absolute law. So:

* Why not spend energy to gain momentum? In other words why don't we have a mechanical space propulsion system?

[Senor Molinero]

According to my calcs a 1% increase in mass is achieved at 14%c.

[a1call]

A gyroscopic thruster consists of a system of gyroscopes connected to a central body. When the central body is torqued, the gyros move in a way that appears to defy gravity. Actually the motion is due to gyroscopic precession and the forces are torques around the axes of the gyros' mounts. There is no net thrust created by the system.

To keep an open, yet rigorous, mind to the possibility that there has been some overlooked physical phenomena with such devices, it would be necessary to explicitly address all the conventional objections and pass at least a pendulum test. Any test results would have to be impartial and rigorously address all possible false-positive conclusions. There has not yet been any viable theory or experiment that reliably demonstrates that a genuine, external, net thrust can be obtained with one of these devices. If such tests are ever produced, and if a genuine new effect is found, then science will have to be revised, because it would then appear that such devices are violating conservation of momentum.

I don't understand the "science will have to be revised" part.

A system that spends energy to increase it's net momentum won't violate the "Conservation of four-momentum" (law?). Same way that nuclear energy does not violate the preservation of energy and matter law (it does violate the preservation of energy law). It spends matter to increase a system's net energy.

[G O R T]

Basically what is bothering me is that increased mass will violate the law of preservation of momentum. This is because we will have to multiply the Lorentz factor by the product of mass and velocity. Now this law breaking is fine with me except that the law of the preservation of momentum is generally referred to as an absolute and is the basis of many ATM debunking arguments.

I guess my follow up questions are:

* Isn't the law of preservation of momentum an approximation which only holds pretty accurate at low velocities?
* If so how can it be argued that there can be no device which alters the momentum of a system?


Added:
I do not understand these stuff well at all. I did some more reading and it appears that Conservation of four-momentum is more of an absolute law. So:

* Why not spend energy to gain momentum? In other words why don't we have a mechanical space propulsion system?

Conservation of momentum defines total energy interactions in a closed system. It does not define the magnitude of momentum. Momentum is a product of mass, velocity, and energy. It takes huge amounts of energy to accelerate mass to relativistic speeds, the increased mass (momentum) reflects this energy input.


All space propulsion systems are mechanical. You must push at something to change your own vector acceleration.

[a1call]

All space propulsion systems are mechanical. You must push at something to change your own vector acceleration.

Hi GORT,

True. Bad word to describe the concept. By mechanical I mean a system which changes its net momentum by converting mechanical energy to momentum. An isolated system similar to a gyro propulsion system which has two wheels can potentially produce an alteration in momentum balance between the two wheels which would result in creation of a net thrust. It would do so by increasing the momentum of one wheel by increasing its spin to relativistic speeds (actually any increase should have a minute effect in momentum increase). Alternatively it can reduce the momentum of one wheel by reducing its spin speed. It can alternate the process between the two wheels to create a continuous thrust.
The increased mass of one wheel can be used to push against or perhaps could be used to attract the other wheel with a greater force. Many scenarios are possible (it seems).
My point:
Where did we defy the conservation of four-momentum?

Added:

I can see many problems with the gibberish that I wrote above. Please ignore it.
What I am trying to say is IMO if a system can be shown to create net thrust somehow, it does not mean we have to revise science. This is because the law of conservation of momentum is an approximation which only holds approximately true at low velocities. Potentially a system can exist which defies the law of conservation of momentum but not the conservation of four-momentum.

[Ken G]

What I am trying to say is IMO if a system can be shown to create net thrust somehow, it does not mean we have to revise science. This is because the law of conservation of momentum is an approximation which only holds approximately true at low velocities. Potentially a system can exist which defies the law of conservation of momentum but not the conservation of four-momentum.

No, that's not correct. The existence of a Lorentz factor in the momentum does not in any way invalidate the conservation of momentum-- indeed, the whole reason you need it is to conserve momentum. If a closed system starts out as a very high-rest-mass object and a low-rest-mass object both at rest, and ends up having the high mass object moving at low speed one way and the low mass object zooming off at a relativistic speed in the opposite direction, you will need to include the Lorentz factor with the resulting rest masses to get the momentum to add to zero. You will also find that the total relativistic mass is the same as before, so if you find that the low-mass object's relativistic mass increased due to the Lorentz factor, then the high mass object's mass must have decreased (that's where the energy "came from" if there's nothing else going on in the system). So we still have conservation of both energy and momentum, but we need relativistic corrections in both.

When you are talking about four-vectors, you are talking about something rather different-- you are talking about what stays invariant when you look at the whole process from a different reference frame. The relativistic masses will obviously be different if you watch this entire process from another frame, and so will the momenta, but the square of the rest masses minus the square of the momentums (times c^2) will stay the same when you transform to the new frame. That does not change the fact that both energy and momentum will be conserved in the new frame as well, but the individual numbers do change when you change frames. Only the invariant does not.

[Grey]

* Isn't the law of preservation of momentum an approximation which only holds pretty accurate at low velocities?

No. However, your confusion here might be from the fact that at relativistic velocities, there's a more complex formula for momentum than p = mv. Just as with things like addition of velocities and the value for kinetic energy, when dealing with things moving relativistically, you have to use the full relativistic formula for momentum. Nevertheless, all experiments to date have shown that momentum is conserved.

* If so how can it be argued that there can be no device which alters the momentum of a system?

We can't be absolutely certain that there's no such device, but it's fantastically unlikely. We have no evidence of a process anywhere in nature that does not conserve momentum. It's also interesting to note that if you assume a few of the basic principles of quantum mechanics (which are themselves extremely well-supported by experiment) and add the assumption that the laws of physics are the same here as they are where you are as they are on the moon and everywhere else, you can show that conservation of momentum is a necessary consequence. In a similar manner, conservation of energy can be derived from a few of the basic principles of quantum theory and the postulate that the laws of physics do not change over time.

* Why not spend energy to gain momentum? In other words why don't we have a mechanical space propulsion system?

You can't. At least, all the evidence points that way. Energy and momentum are both conserved separately. It is not possible to simply transform energy into momentum or momentum into energy. That hasn't stopped quite a few people from trying, but there's simply no evidence to suggest that it's possible at all.

[mugaliens]

You have to be travelling at around 86.6% of the speed of light before your mass apparently doubles. The Lorentz transformation introduces an exponential curve to the equation. Between 86.6% of the speed of light and light-speed itself, the mass increases from double, to infinite.

So - would a hypothetically possible 1G rocket become less than a 1G rocket? Or would the mass of it's propellant, since it's also accelerated, and flying out the back only slight less velocity compared to C, continue to accelerate the rocket and almost nearly 1 G (the difference being the loss of mass of the slightly lesser velocity in the propellant compared to the rocket)?

[mugaliens]

All space propulsion systems are mechanical. You must push at something to change your own vector acceleration.

Wasn't this the justification against Goddard?

How does one push against a vacuum?

Perhaps just poorly worded, so I'll definately give you the benefit of the doubt, as I'm sure you're well aware of "for every action there must be an equal and opposite reaction" which works as well in space where there's nothing to push against.

[a1call]

Gentlemen,

First, it is quite obvious that you are at your elements and I am like a fish in the desert. I fully acknowledge that and appreciate your replies.

In the light of the points made I will revise my question:

The formula

p = mv

is accurate for v=0


As you mentioned the momentum at relativistic velocities will be calculated with inclusion of Lorentz factor.


Now how is the momentum preserved in a system with initial velocity of greater than 0 when chemical energy say from a battery is used to spin 1/2 of the system relative to rest at a velocity of 86.6% of the speed of light some time later. The mass will increase while the net velocity will remain at what it was initially.

In other words it appears to me that the momentum will increase because the introduction of a spin at relativistic speeds will increase the mass of the system. Think of such a system where energy is converted to mass resulting in momentum gain.

* Now what did I overlook?

[speedfreek]

So - would a hypothetically possible 1G rocket become less than a 1G rocket? Or would the mass of it's propellant, since it's also accelerated, and flying out the back only slight less velocity compared to C, continue to accelerate the rocket and almost nearly 1 G (the difference being the loss of mass of the slightly lesser velocity in the propellant compared to the rocket)?

No, if you want to accelerate at a constant 1g you will get closer to the speed of light and you will require more fuel to accelerate your mass and more fuel to accelerate the extra fuel!

This article has a good explanation of the concepts involved.

[Grey]

Now how is the momentum preserved in a system with initial velocity of greater than 0 when chemical energy say from a battery is used to spin 1/2 of the system relative to rest at a velocity of 86.6% of the speed of light some time later. The mass will increase while the net velocity will remain at what it was initially.

In other words it appears to me that the momentum will increase because the introduction of a spin at relativistic speeds will increase the mass of the system. Think of such a system where energy is converted to mass resulting in momentum gain.

Let me see if I'm imagining the same sort of object you're suggesting. You have a sphere split in half, with the two halves connected by an axle. You first accelerate the sphere to a substantial fraction of the speed of light, and then you start up a motor which rotates the axle so that one half of the sphere starts turning. Is that what you have in mind? If so, no trouble at all. First, linear momentum and angular momentum are conserved separately. The sphere's rotation, whether we're talking about the whole sphere are various parts of it, has no bearing on the linear momentum of the sphere. Second, angular momentum is indeed conserved, just like linear momentum is. If the sphere is free in space and you start spinning one half in one direction, the other half will inevitably start spinning in the other direction at exactly the same speed, leaving the total angular momentum at exactly zero, just as it was when it started. You'll find that, no matter how hard you try, you won't be able to get part of the sphere spinning without some other part spinning in a way that exactly balances the total angular momentum. Well, that's not quite true: you could have attitude jets that eject propellant out into space, but then your system is no longer closed. If you were to carefully account for the angular momentum of the propellant lost to space in the process, you would again find that it again precisely balances the net angular momentum imparted to the sphere.

[a1call]

If the sphere is free in space and you start spinning one half in one direction, the other half will inevitably start spinning in the other direction at exactly the same speed, leaving the total angular momentum at exactly zero, just as it was when it started.

Precisely, If the two halves are made to spin such that their circumferences rotate at a speed of say 84.6% of speed of light relative to each other then each half's circumference will have a relative speed to the initial frame of reference equal to 42.3% of speed of light. This system should have a greater mass than the initial system because parts of the matter spin at high speeds where initially they did not (or so I figure).

This increased mass should result in a net momentum gain.

[RussT]

the other half will inevitably start spinning in the other direction at exactly the same speed,

Isn't this an "Assumption"?

[Grey]

Precisely, If the two halves are made to spin such that their circumferences rotate at a speed of say 84.6% of speed of light relative to each other then each half's circumference will have a relative speed to the initial frame of reference equal to 42.3% of speed of light. This system should have a greater mass than the initial system because parts of the matter spin at high speeds where initially they did not (or so I figure).

This increased mass should result in a net momentum gain.

Ah, I understand your thought. It won't work that way, but the math is extremely complex, so I'll try to explain the results verbally. Instead of a half sphere rotating though, let's use a pair of rings connected by an axle, because it will be a little easier conceptually, and if we end up working out some of the math, it will be much easier. You're thinking that it's already moving at a high velocity, increasing it's relativistic mass*, and then when we start it spinning, that velocity adds to the initial linear velocity, increasing the mass still further. But velocity is a vector quantity, it doesn't just add magnitude; you have to consider direction as well. Let's look at each individual part of the spinning wheel moving away from us. One side of the rim is indeed moving away from us a little faster than it would be if it weren't rotating. We'll see that side as having an increased mass. But the opposite side is actually rotating toward us. Or, since the whole wheel is moving away from us much faster than it's rotating, better to say it's receding slightly slower than if it weren't rotating. Since it's moving more slowly relative to us, it's relativistic mass decreases from our perspective. And of course there are parts of the wheel that end up having the same velocity as before, but just in a different direction, so their effective mass is unchanged by the rotation of the wheel.

Yes, that means that as we follow a small section of wheel around the circle, it's relativistic mass fluctuates as it goes around, but that's exactly what we expect in relativity from an object which keeps changing its velocity relative to us. This kind of problem, where we have an extended object with internal motions, all at relativistic speeds, is one of the ugliest to work out in detail. In many cases, it's not really possible to work it out with special relativity alone. But I've worked on some simplifications of this sort of problem, and it's pretty impressive when you get to the end and see that it all works out. Momentum and energy remain conserved, and all the "strange" effects of relativistic motion end up balancing out to give a consistent solution.

* Generally speaking, physicists don't use the idea of increasing mass these days, preferring to use the term "mass" to mean rest mass, and just having more complex formulae for momentum and kinetic energy, but we can work with it here to try not to make this any more confusing than it already is.

[Grey]

Isn't this an "Assumption"?

Not really. It's extremely well founded, based on observations. It's true that we don't have such contraptions capable of spinning at relativistic rates. However, we can see that it works just this way for macroscopic objects moving at a variety of lower speeds (these kind of reactions are exactly what are needed to control spacecraft, for example, and so they've been tested quite well), and on the other end of the scale, subatomic particles in accelerators do move at these kinds of speeds, and still seem to obey conservation of momentum, both linear and angular. We simply have no observations of a case where momentum is not conserved.

[a1call]

Thank you for the reply Grey. I'm afraid it takes more gray matter than I have to comprehend the issue.

Let's break it down to simpler issues if we may:

Let's say I observe 2 objects each having a mass of 100 kg. They are both at rest relative to my observation post and are identical in every way. Then I blink and low an be hold, one of the objects is as before but the other has magically started spinning around itself at a relativistic speed. As I (probably mis-)understand it, this object has a now a mass more than 100 kg in my frame of reference.

Is that wrong?

[G O R T]

Wasn't this the justification against Goddard?

How does one push against a vacuum?

Perhaps just poorly worded, so I'll definately give you the benefit of the doubt, as I'm sure you're well aware of "for every action there must be an equal and opposite reaction" which works as well in space where there's nothing to push against.

The push of a rocket engine is against the ship itself.
The force of exhaust mass ejected at high velocity equals the force accelerating the mass of the spaceship.

Only light sails use an external force.

[a1call]

I think I understand now. The increased mass due to relativistic speeds will cause the system to decelerate preserving the net system momentum. Correct?

[G O R T]

I think I understand now. The increased mass due to relativistic speeds will cause the system to decelerate preserving the net system momentum. Correct?

No, momentum goes up with velocity. It actually goes up exponentially.

A theoretical disk spinning at relativistic velocity has increased mass. There is no cancelation due to "net" velocity. There is no conservation of momentum problem. When calculating the usefulness of objects spinning at relativistic velocities, you cannot ignore the immense amount of energy required to produce that spin. BTW, the only place you are likely to find such a condition is near the event horizon of a black hole.

Conservation of momentum is just what it says. The momentum of a mass stays the same if no force is applied. If force is applied the momentum changes by the amount of energy transfered . The total energy of a closed system is conserved.

This is observed in the action of billard balls - whether at relativistic velocity or not. When one ball hits another, energy is transfered but none is lost. The energies are just much higher at relativistic velocities.

[Grey]

Thank you for the reply Grey. I'm afraid it takes more gray matter than I have to comprehend the issue.

Let's break it down to simpler issues if we may:

Let's say I observe 2 objects each having a mass of 100 kg. They are both at rest relative to my observation post and are identical in every way. Then I blink and low an be hold, one of the objects is as before but the other has magically started spinning around itself at a relativistic speed. As I (probably mis-)understand it, this object has a now a mass more than 100 kg in my frame of reference.

Is that wrong?

No, that is correct. But let's look at how this situation is similar to the one where the two objects are not at rest relative to you, and how those situations are different. In a rotating object, different parts of the object are moving differently relative to you. The way you handle this kind of situation in physics is to look at each little piece of the object separately, and then add up all the results. Integral calculus is a set of mathematical tools to do that. In the case where the object is intially at rest, and then starts rotating, every piece of the object starts off with no velocity relative to you, and then after it's moving, it has a velocity relative to you. So yes, we'd expect it's relativistic mass to increase as a whole.

In the case of an object that is already moving relative to you, it's not quite that simple. Once it starts rotating, some parts of the object are moving faster relative to you than they were, but other parts are moving slower relative to you. It's like a receding galaxy. We can measure how fast the left side is moving away from us and how fast the right side is moving away from us, and one value will be smaller than the other. The difference tells us how fast the galaxy is rotating. So in this case, various parts of the object will show a mass increase, and others will show a decrease.

Frankly, it gets pretty complicated, and for a rapidly spinning object, it's not really possible to do it without a fell general relativistic treatment. That's because in a rotating object, things are accelerating (since the direction of motion is constantly changing) as well as just moving, and because there must necessarily be large stresses on the object to stay together when rotating at high speed. Neither of these effects is negligible. But actually, you don't have to work out specific cases to confirm that momentum and energy are conserved locally by relativity. That is, you can analyze the equations that govern motion directly, and show that no matter what kind of motion objects are involved in, momentum and energy will be conserved. So as long as general relativity is accurate, and experimental tests support it well, linear momentum, angular momentum, and energy are all conserved quantities.

[Grey]

Conservation of momentum is just what it says. The momentum of a mass stays the same if no force is applied. If force is applied the momentum changes by the amount of energy transfered . The total energy of a closed system is conserved.

This is a little misleading. Energy and momentum are separately conserved quantities. When you apply a force to a moving object in order to accelerate it, you provide both energy and momentum. The momentum does not change by the amount of energy transferred, it changes by the amount of momentum transferred.

[JohnD]

Thank you speedfreak and kenG.
WE had another thread where someone claimed that chemical reactions caused an increase in mass 'as the temperature and energy of the system was increased". I paraphrase the theory.

I found that only in a plasma could the atoms be moving at a fraction of c to have any significant effect but was ignored. I'm glad to have you support.

John the ignored

[Grey]

Thank you speedfreak and kenG.
WE had another thread where someone claimed that chemical reactions caused an increase in mass 'as the temperature and energy of the system was increased". I paraphrase the theory.

I found that only in a plasma could the atoms be moving at a fraction of c to have any significant effect but was ignored. I'm glad to have you support.

John the ignored

Actually several people participating in that thread, including Ken G, agreed that any kind of increase in energy increases the mass of a system. Now, you need to be a bit careful. For example, if I have an exothermic chemical reaction in a well insulated container, so that none of the heat escapes, all I've done is change chemical potential energy to thermal energy. Hence, no net change in energy, and therefore no change in mass. But if I allow that heat to escape, the total energy of the system will be lower than it was, so the total mass will be less. You also talk about a "significant" effect. It's very true that for typical chemical reactions, the amount of energy difference (and hence, the amount of mass difference) would be extremely hard to measure, if it's measurable at all, so it's reasonable to say the effect is "insignificant" for most purposes. Nevertheless, it's what relativity predicts, and in all cases where those predictions are measurable, they've been borne out.

[Ken G]

Indeed. And as well, there are physically relevant plasmas that do indeed move that fast-- a quark/gluon plasma in the early universe for example. You couldn't just say that the rest mass of the quarks will give you the rest mass of the baryons, because you also need the kinetic energy of the quarks, and a host of other possible ways to hide energy when you make baryons. All forms of energy contribute equally, it's just that in the vast majority of all astrophysical plasmas, the main energy source is in the rest mass of the baryons, which simplifies things a lot.