# How is mass, energy and velocity defined?

1. Oct 22, 2009

### Nizzeberra

When I read discussions about physics, especially when it comes to relativity, things tend to become quite twisted, just because I have no clear definition of the most basic concepts.

Mass

There is inertial mass, gravitational mass, and "potential energy" (E = mc^2). My interpretation is that "mass" is different depending on what force you consider (gravity, electromagnetism, or the strong and weak forces). So my question is, can a particle's inertial mass differ from its gravitational mass? And what do we mean when we say that a photon have zero mass? It seems to have gravitational mass at least, doesn't it? And it does, obviously, have potential energy...!?

Another interesting question about potential energy... When charging a battery, we move electrons from one metal plate to another. No particles are added, but we have added potential energy. Does this mean that the mass of the battery has increased? Does it weigh more if you place it on a scale? (This is basically the same question as before. Is gravitational mass different from potential energy?)

Energy

"As a unit or work, it is defined as the work done by a force of one newton acting on an object to move it through a distance of one meter in the direction the force is applied."

This definition is obviously not correct (if my understanding of the english language is sufficient). Applying a force to an object already in motion, in the same direction as the motion, will result in different work depending on the initial speed of the object (since it takes different amount of time to travel that meter).

The definition should be something like "the work done by overcoming a force of one newton acting on an object...". However, how do we define one newton?

My point is... In order to define what "force" is, you need a definition of "mass". But in order to define "mass", you need "force". And round it goes. As in math, you need some basic axioms, but they can't be based on each other using circular arguments. So are there any sensible definitions of mass and energy, that does not refer to each other?

Velocity

In my opinion, velocity is defined as "the difference in distance between two objects over time". And we all know the fact that "nothing can travel with a speed faster than the speed of light". But if the space between two objects is expanding, it seems as if it is totally ok that these objects fly apart at speeds greater than the speed of light, since they are not travelling faster than the speed of light relative to the, eh... Spacetime?

Now, what did Einstein say about reference frames? You can't have the spacetime as your reference frame, or can you? Because if you can, we can suddenly speak about things as "absolute velocity". Does the term "aether" ring a bell?

Last edited by a moderator: Apr 24, 2017
2. Oct 22, 2009

### mikeph

Experiments have been done to verify that inertial mass and gravitational mass are the same in the beginning of the 20th century, and I believe are still being done- as of yet no discrepancy has been found and the accuracies of these experiments are pretty good. In theory a stretched spring will weigh more than a loose spring because the mass/energy of it is larger- they are sort of equivalent, yes, but do not think of the 'mass' as increasing, there is still the same amount of matter, just think of the energy contained by the spring as also being coupled to the gravitational field.

For force- I think you are nit-picking. Implicit in the wording of the definition is that the movement done is purely as a result of the force acting. They could add the disclaimer "in the rest frame of the mass", but that isn't really necessary in my opinion. I am not sure about your circular definition argument, I'll have to think about that one.

Velocity- that is not such a good definition, velocity is the instantaneous time change in displacement of an object. Distance is a scalar so cannot be used like you have in the definition, and also it needs to be specified that the velocity is the difference in the limit of t->0.

The second point with regards to this is using special relativity to say that for an object with mass, |v|< c, but in general relativity the spacetime can itself change over time, giving the idea that two distant objects occupying stationary comoving points in a fast expanding spacetime can still be receding at a velocity faster than c. (but if you use a ruler to measure the recession speed, does the ruler not also expand? )

The solution is that objects cannot travel through space faster than c, locally, where the spacetime is flat and special relativity becomes an approximation, but over large distances the spacetime can itself expand faster, which explains how regions of causally connected spacetime can become disconnected through expansion faster than the speed of light.

3. Oct 22, 2009

### Staff: Mentor

Please search in the relativity forum here, where there have been many threads discussing the concept of mass in relativity, the mass of the photon, etc. In fact, there are a couple such threads going on right now!

4. Oct 22, 2009

### Nizzeberra

Interesting! How about all potential energy between all stars in our galaxy? Is it reasonable to think about potential energy between, say, the earth and the sun, as in potential kinetic energy if the earth would fall down into the sun? If it is, doesn't that explain dark matter?

Well, my definition can of course be improved. "Distance" was the wrong word. "Relative position" would probably be better. But "instantaneous time change in displacement" is, for me, just a mathemtical construct that works nice on a macroscopic scale, but obviously not as well at microscopic scales (as in quantum mechanics). Because of that, I think a more statistical approach to defining velocity is more appropriate, where t > e ("epsilon", a small value still large enough to make measurements meaningful), and velocity simply is the mean distance over that time.

The question about the ruler is interesting. If the ruler expands, we can't measure the expansion, which we can (the universe would be static). But at the same time, it has to expand... Think of a ruler 13 billion light years long. In order to follow the same physical laws as all galaxies that we see, it has to join them in the expansion. So the only reasonable solution to this is that the ruler breaks, if we assume that the universe really is expanding (there are other theories on this).

Anyway, this doesn't answer anything. Actually, I can accept the whole thing about motion relative to spacetime, so there doesn't have to be an answer. But in that case, why is the old theory about aether taboo in these days? Isn't aether and spacetime exactly the same thing, an expanding medium through which waves travel? Instead of talking about "warps in spacetime" we could say "density differences at different positions in the aether", and it would explain gravity lensing and all that in the exact same way.

5. Oct 22, 2009

### mikeph

In quantum mechanics, velocity is found a totally different way by operating on the wavefunction, so you do not need to worry about actually making infinitely small measurements. The definition is the derivative of the displacement with respect to time.

No, not motion "relative" to spacetime- I said "through" spacetime. The difference is that your wording implies the 'aether' way of thinking, that one inertial frame is favoured. In my "through spacetime", the motion occurs in a Minkowski limit where the region of space and length of time are so small compared to the curvature, that the motion is unaffected by any sort of stretching effects. This local approximation of an, in general, curved and evolving spacetime, is what I meant when a particle cannot move faster than the speed of light- and that is exactly special relativity. That's what I think I'm trying to say.

6. Oct 22, 2009

### Staff: Mentor

Hi Nizzeberra, your OP is way too lengthy and involved. Please pick a single topic that is most important and ask it in a single thread. I will just touch on one that jumped out at me.
The definition is correct. The amount of work is the same regardless of the time that it takes. The power is the amount of work per unit time, which is probably what you are thinking of.

Last edited by a moderator: Apr 24, 2017
7. Oct 23, 2009

### Nizzeberra

I think I've got all the answers I need soon, so I won't keep this alive for much longer. Some of my questions are actually mostly retorical, because I actually don't think there is an answer to them. For instance, the circular definition of mass and energy (through the definition of force). If there really are no better definitions, someone should really start looking into this (but not me, since I am not a scientist), because I think all proposed theories that try to unite relativity with quantum mechanics will fail if they are based on definitions like these.

I finally did the required calculations, and yes, it seems like you are right. I based my arguments on my intuition, which was wrong.

Last edited: Oct 23, 2009
8. Oct 23, 2009

### Staff: Mentor

The definitions are not circular. In the SI system, mass, length, and time are defined operationally, then force and energy are defined in terms of those fundamental quantities. In other systems of units you can define other quantities fundamentally and then mass, length, or time may be a derived quantity. That would be a different but equivalent definition, and neither would be circular.

9. Oct 27, 2009

### Nizzeberra

To keep things more focused on a single subject, I would like to focus on the following topic... (So let's forget about mass and energy).

About the definition - "the derivative of the displacement with respect to time"... This "displacement" has to be measured using a frame of reference (let's say earth). Using the common-sense-definition of velocity (the one I gave you, with or without infinitely small measurements), there are galaxies that by this definition are moving at speeds greater than light!!! And the only way (that I know of) to get around this little delicate issue is to say that things can't travel faster than light "through spacetime".

By saying "through" spacetime, you are implicitly using the spacetime, or some abstract "coordinate system" that has no physical origin, as your frame of reference. By doing so, you can likewise say "relative to spacetime", and we can start talk about the aether-theory all over again. Alternatively, you are (kind of) invalidating objects at large distances as reference frames - only objects local enough can be used as reference frames.

Now... Were am I wrong?

Notes:

The distance beyond which you can't see the galaxies anymore (since they are moving faster than the speed of light) has been called an event horizon in some articles I have read.

There are people trying to simulate black holes and event horizons using optical effects only (refraction index). I have personally noticed the same thing as these scientists, namely that light being refracted by gravity behaves in the exact same way as if there was a medium (the "aether") with continuously varying density around a massive body.

Personal theory:

I believe particles consists of wavelike energy "trapped" in a tiny spot. How the waves get trapped is beyond my knowledge, but these waves constantly move at the speed of light, making the particle "vibrate". That's why you can't reach absolute zero temperature. Motion is caused by the net velocity of these waves - which causes all exotic time effects, like time dilation. Spending all "energy" (or whatever we should call it) moving in a single direction gives less of this "energy" left for perpendicular motion. (This "energy" I talk about is simply the length of the velocity vector, which is constant). Thus, the particle's vibrations will go slower regarding to an external observer, and he/she thinks that time goes slower. Think about a clock... As the entire clock starts moving in one direction, the entire machinery of the clock has to slow down in order for the waves (the particles) that constitutes the clock to maintain a constant speed of light.

Last edited: Oct 27, 2009
10. Oct 27, 2009

### Staff: Mentor

One of the nice things about general relativity is that it gives you some interesting and valuable insights into these kinds of ideas. In GR you can use pretty much any coordinate system that you want. The coordinates don't have to have any physical significance whatsoever. All physical laws and quantities are then expressed in terms of coordinate-independent geometric entities.

In this geometric sense the relative speed of two particles is represented by the angle between two worldlines. Since there is no physical significance in the coordinate system chosen there is also no sense in which there is an angle through or wrt the spacetime itself. There is no "velocity through spacetime", there is only the angle or relative speed between two worldlines.

However, in curved spacetimes the concept of this angle is a strictly local concept. To see why, consider geometry on the surface of a sphere. Let's say that you have two vectors pointing North from the equator, one at 0º longitude and the other at 180º longitude. If you bring them together by parallel transport along the equator then they will wind up parallel, but if you bring them together by parallel transport along the lattitude line they will wind up anti-parallel.

So there is no consideration of speed through spacetime itself, only between two worldlines, and even that can only be defined locally.

This is not the proper forum for personal theories.

11. Oct 28, 2009

### Nizzeberra

I knew this already, and this is where things doesn't seem to fit anymore. Your analogy with the sphere only gives that "velocity is only well defined in flat spacetime", thus in curved spacetime, velocity is not well defined. Correct? So my question about how velocity is defined has no good answer in curved spacetime, does it?

So back to the issue with the galaxies. I assume that the universe is overall quite flat. This means that we don't have any major problems defining the velocity of distant galaxies, since we all agree on the coordinate system. Using this flat and well defined coordinate system, galaxies seem to fly away from us at speeds greater than c (even if we can't see them). This is where my head starts to ache. The distance is obviously increasing at a magnitude greater than c, but still we don't call this their "speed". What does the world lines look like?

12. Oct 28, 2009

### Staff: Mentor

That is a bad assumption, and not supported by current data. On cosmological scales it is quite clear that spacetime is curved, even if the spatial "slices" of the FLRW metric are flat.

13. Nov 2, 2009

### Nizzeberra

To sum things up... It seems as if this thread won't give me the answers I hoped for. We have discussed motion "through spacetime", as if one can use the spacetime as a reference frame (and what are those speedy galaxies moving through, if not spacetime?). Introducing the concept of worldlines instead moved the focus back to using physical objects as reference frames. But using worldlines can't explain how these galaxies move at superluminous speeds.

My conclusion is that you either have to believe in the aether, or in the fact that you can travel at speeds greater than c, depending on how you define speed. There are actually people saying that it is totally valid to travel at speeds greater than c, but that you just can't accelerate beyond this speed limit using conventional engines. (Naturally, one would like to know why.)

By starting this thread, I hoped that I could start a discussion that questions things like this. It is quite interesting that, by accelerating, everything behaves just as it should using classical newtonian physics, with one exception - how an observer measures your speed.

Can the speedlimit of c just be an illusion caused by the probing particle (the photon) you use for the measurement? And can the fact that we always get c when measuring the speed of light, be caused by the fact that every particle lives in a bubble of spacetime, created by the particle itself? (Analogy: Measure the speed of sound from an ambulance passing by, inside a parked car - "the spacetime bubble". You will always get 340.29 m/s, no matter the relative velocity between the car and the ambulance).

14. Nov 2, 2009

### mikeph

When you think that a single coordinate system is implied when I say "spacetime". The physics are invariant under any position or velocity translation in spacetime, so there is no special frame of reference implied.

This isn't a forum for discussing your own theories.

15. Nov 2, 2009

### Staff: Mentor

I am sorry that the Universe doesn't conform to your hopes, but I was not consulted either.
They don't move at superluminous speeds in any coordinate-independent sense.
The second option is correct.
No, there is lots of evidence that this limit is built into the geometry of the universe. The muon decay observation is one example not based on the speed of photons. See the FAQ "experimental basis of special relativity" for other examples.

16. Nov 2, 2009

### Bob S

Relativistic total energy, mass, and velocity are related by

E2 = (γm0c2)2 = (βγm0c2)2 + (m0c2)2

where momentum in energy units is pc = βγm0c2

β and γ are the usual relativistic parameters; β = v/c and γ = 1/sqrt(1-β2).

Mass is generally considered to be the rest mass, i.e., the count of electrons, protons and neutrons, less binding energies, times their rest masses.

Bob S

17. Nov 4, 2009

### Nizzeberra

There is nothing wrong with the universe. :) It is just the fact that I didn't get any answer on how something can move at speeds greater than c, while not using the spacetime as a reference frame. But I think I have figured it out, finally.

The expansion of the universe is as if the number of "grid lines" (think of a coordinate system on a paper) is increasing between two galaxies, right? Any object in this coordinate system has to overcome the speed of this expansion if it should have any hope reaching its target. Every tiny bit of space between you and your target is expanding. How annoying...

Using my newly found insights, I don't have to question this further. But still, it is good to question things. What if everything is made of the same "stuff"? What if all particles are electromagnetic waves trapped in a tiny spot? (This assumption is almost what string theory is saying). Wouldn't muons, electrons, protons etc, all follow the same rules? Of course they would! Thus, wether the probing particle is a photon or a muon doesn't make any difference. And it seems as if my personal layman theory still holds, since it doesn't contradict any existing theory as far as I can see. (Btw, it is not a "theory" per se, it is just another way to think about existing theories).

What I find most interesting is how a single property - the speed of electromagnetic waves - defines how space, time, and gravity behaves, since this was the starting point for Einstein and his theories. But that's another topic I suppose. By the way, I would be really interested in a link to some text that explains how GR predicts the expansion of the universe.

Last edited: Nov 5, 2009
18. Nov 5, 2009

### Nizzeberra

In what context did that comment belong?

19. Nov 5, 2009

### Nizzeberra

Search!? I would love to, but where is the Search-field where I can enter my search term?

20. Nov 18, 2009

### Nizzeberra

One more thing about this... Who's theories are ok to discuss here? Only those worked out by employees at some academy? How about officially accepted theories, proved to be wrong? It may not take long until general relativity turns out to be wrong, and just another approximation of reality. There is no proof that black holes exist at all - they may be what is known as "black stars".

What does it take to make a "personal theory" not being personal anymore? Does it have to be published in a newspaper? It actually happens that for every new article I read, it seems like professional scientists are starting to look into theories that more and more looks like my own. The only difference is that they have worked out a more robust mathematical framework. So if these look-alike theories turns out to be true, is it suddenly ok to discuss them!?