View Full Version : E=MC2: why?

JohnL

2009-Jul-06, 05:32 PM

In a recent AstronomyCast, Pam explained a way of visualising E=MC2, and a question hit me: why is there a multiplier at all? If matter is converted into energy, don't I get the same amount of energy back because of the law of conservation? That is shouldn't the equation be E=M?

Of course, that's not true, or that would be the equation, right? So where does this C2 multiplier come from? Is this just a clash of units, maybe a difference in how we measure matter and energy? Or is there something more freaky going on?

Or am I barking up the wrong tree? (this strikes me as the most likely case)

Thanks

gzhpcu

2009-Jul-06, 05:48 PM

Energy is measured in Joules (=1 kilogram x meters2 x seconds-2).

Mass is measured in kilograms.

c2 is the conversion factor.

JohnL

2009-Jul-06, 05:56 PM

Wow that was fast :)

So it is just the units used, then - E=MC2 is just a coincidence that how we measure energy happens to give us a value E joules and how we measure mass happens to give us M kilograms and C2 is just the relationship between those two values?

So could we up up with new units which would give us the same numbers on both sides of the equation?

Huh... I mean, if we decided to use pounds or ounces rather than kilograms, E wouldn't equal MC2 because the multiplier would be different?

Seems odd since it seems that the equation suddenly doesn't seem so important anymore...

Ken G

2009-Jul-06, 06:02 PM

Or am I barking up the wrong tree? (this strikes me as the most likely case)

You are barking up the right tree. The presence of the c2 is really just a convention, about how we chose to define energy and mass. It has no physical meaning at all, except it can make the point that on our familiar scales of mass and energy, a little mass translates to a lot of energy because c2 is so large. But physicists who routinely use that formula generally choose units where c=1, so they use exactly the formula you cite: E = m. That is more convenient, and preserves the philosophical content: it's not that mass can be converted into energy, it is that mass and energy are two forms of the same thing.

Tobin Dax

2009-Jul-06, 06:05 PM

gzhpcu said it. They are the same amounts, but we have to convert between them.

An analogy would be the length of a room. Measure the length in feet and in meters, and you have two different numbers, but the length is the same. A conversion factor can change one measurement value into the other. The conversion from kilograms to Joules also requires a conversion factor since are two different "measuring sticks."

John Jaksich

2009-Jul-06, 06:20 PM

Dear JohnL

Welcome to BAUT and it is a very good question that could take awhile to answer...in short there are many popular to intermediate level expositions on the meaning of Relativity and Einstein's famous equation.

Google-Books has quite a few of them...

and I will list three:

The Meaning of Relatvity--by A. Einstein

Special Relativity -- by A. P. French

The Evolution of Physics -- by A. Einstein and L. Infeld

Delvo

2009-Jul-06, 06:21 PM

As far as the universe is concerned, c=1, so when you multiply by c˛, you're multiplying by 1˛, which is multiplying by 1, which is not multiplying at all.

Also, multiplying by even a very large number wouldn't create a problem anyway, because it doesn't mean there's more energy than matter, because to convert back, you just divide by that same large number again, and get exactly the same thing you started with.

thoth II

2009-Jul-06, 06:24 PM

Some convenient non-SI units for mass are for example:

m in "MeV/c2" . For example, 1 atomic mass unit is approx. 1 GeV/c2

JohnL

2009-Jul-06, 07:06 PM

Thanks for the explanations guys.

It seems like a less than satisfying explanation because E=MC2 is often held up as the pinnacle of modern science, when the equation is the least important part of it - the important part of it is that there can be an equation at all.

That said, I pretty much expected this to be the case...

Thanks again for explanation and the welcome.

Ken G

2009-Jul-06, 10:01 PM

It seems like a less than satisfying explanation because E=MC2 is often held up as the pinnacle of modern science, when the equation is the least important part of it - the important part of it is that there can be an equation at all.Right. In fact, I'd say part of the "popular success" of E = mc2 is that it "rolls off the tongue", certainly far better than E = m does (which sounds trivial, but of course is anything but). What's more, there is "coolness in complexity", and most people who do not know much science consider a formula with squares in it to be much "cooler" by virtue of that mystifying arithmetic operation. They don't realize that is it not the complicated formulae that are most important, it is the simple ones.

DrRocket

2009-Jul-06, 10:30 PM

You are barking up the right tree. The presence of the c2 is really just a convention, about how we chose to define energy and mass. It has no physical meaning at all, except it can make the point that on our familiar scales of mass and energy, a little mass translates to a lot of energy because c2 is so large. But physicists who routinely use that formula generally choose units where c=1, so they use exactly the formula you cite: E = m. That is more convenient, and preserves the philosophical content: it's not that mass can be converted into energy, it is that mass and energy are two forms of the same thing.

Precisely. Furthermore this very fundamental idea seems to be lost on some people who think that they are being "modern" and "hip" and restricting their definition of "mass" to mean strictly rest mass. This results in the equivalent statement that E^2 = (mc^2) + (pc)^2, which is correct but leads to the unfortunate idea that mass and energy are two different things.

ShinAce

2009-Jul-06, 10:58 PM

I don't see how the relativistic version of E=mc2 implies that mass is not a form of energy.

Simple arithmetic will show that m = (total energy - kinetic energy) / c2.

I like it better, as it's more complete. Rest mass is a form of energy and so is momentum. I like it.

AndrewJ

2009-Jul-07, 12:34 AM

Descriptions of energy and momentum as being invariant to space and time (or perhaps it's the other way round) allow c to convert both Joules to kilos and metres to seconds.

Ken G

2009-Jul-07, 02:48 AM

One way to think of it is that rest mass is essentially "proper energy", i.e., it is the energy in the frame of an observer moving with the object, or the invariant concept of energy to anyone else. Note that momentum counts negatively against the proper energy, and the coordinate energy is always more than the rest mass/energy, just like distance counts negatively against the proper time, and the coordinate time is always more (for an inertial observer). So there is some reason to think of "mass" as rest mass-- it is the invariant version of energy.

DrRocket

2009-Jul-07, 03:23 AM

One way to think of it is that rest mass is essentially "proper energy", i.e., it is the energy in the frame of an observer moving with the object, or the invariant concept of energy to anyone else. Note that momentum counts negatively against the proper energy, and the coordinate energy is always more than the rest mass/energy, just like distance counts negatively against the proper time, and the coordinate time is always more (for an inertial observer). So there is some reason to think of "mass" as rest mass-- it is the invariant version of energy.

That works OK for simple particles.

But consider the case of ordinary bulk matter. It contains particles and you can add up the rest mass. But those particles are also whizzing around due to internal energy in a more or less random way. So there is no reference frame in which all of the particles are at rest. Nevertheless the mass, as would be measured on a laboratory scale, consists of both rest mass of the particles and the additional thermal energy. In short, for macroscopic objects made of real molecules, there is no reference frame in which the mass is simply the sum of the rest mass of the individual particles comprising it.

Ken G

2009-Jul-07, 04:15 AM

But consider the case of ordinary bulk matter. It contains particles and you can add up the rest mass. But those particles are also whizzing around due to internal energy in a more or less random way. So there is no reference frame in which all of the particles are at rest.That's the purpose of invariants-- they also work in other frames. They just aren't their "proper" versions in those other frames. In other words, even in bulk matter, all you have is rest energy (mass)-- what we call "kinetic energy" is a modification due to our perception of momentum. When we correct for the momentum, we end up back with rest energy. Granted, the gravity is more than just what you get from the rest mass, but that's because gravity does not just depend on the invariant energy-- it has to be different in different frames to be generally covariant.

Nevertheless the mass, as would be measured on a laboratory scale, consists of both rest mass of the particles and the additional thermal energy. In short, for macroscopic objects made of real molecules, there is no reference frame in which the mass is simply the sum of the rest mass of the individual particles comprising it.But that is because gravity depends on stress as well as on energy, it's not because there is more energy there. There really isn't-- the invariant energy is what is "really there", and that's all rest mass. Interestingly, the coordinate energy is still a conserved quantity-- within one reference frame. So it's very useful as a bookkeeping tool, but it's not "real", it's a coordinate effect.

DrRocket

2009-Jul-07, 04:47 AM

That's the purpose of invariants-- they also work in other frames. They just aren't their "proper" versions in those other frames. In other words, even in bulk matter, all you have is rest energy (mass)-- what we call "kinetic energy" is a modification due to our perception of momentum. When we correct for the momentum, we end up back with rest energy. Granted, the gravity is more than just what you get from the rest mass, but that's because gravity does not just depend on the invariant energy-- it has to be different in different frames to be generally covariant.

You lost me here. In this bulk matter there is no net momentum. And if you idealize this to a continuum approximation for matter there is no momentum density either.

As far as I can tell, energy is not an invariant at all. It is wildly coordinate dependent, and so is momentum.

I am also confused by your statement with respect to gravity. Publius correctly pointed out that the stress eneregy tensor is invariant (though I think I need to understand this a bit more from the physical perspective) which is must be since it is essentially the curvature tensor which is more clearly invariant. So gravity, in the form of curvature is invariant, but energy is not. As I see it this is because the off-diagonal terms (in a selected local coordinate system) that account for momentum maintain the tensorial property and "compensate" for the change in mass-energy that comes with a selection of a coordinate system in which the mass is not at rest.

But that is because gravity depends on stress as well as on energy, it's not because there is more energy there. There really isn't-- the invariant energy is what is "really there", and that's all rest mass. Interestingly, the coordinate energy is still a conserved quantity-- within one reference frame. So it's very useful as a bookkeeping tool, but it's not "real", it's a coordinate effect.

As far as I can tell mass, energy, time, and space are all coordinate effects. The only way I can understand this stuff is to work completely coordinate free until pressed against the wall and forced to use coordinates to produce specific measurable numbers. What is invariant is the geometry -- the Levi-Civita connection and things that it determines ( the metric, the curvature, etc.). Everything else is a coordinate effect.

But if I interpret your statement correctly, you are telling me that although a hot container of gas weighs more than a cold one, all that is really there is the rest energy. So what is all that additional inertia that is coming from the thermal energy ? Is it not real ? What is real ? If I can weigh it it seems pretty real to me.

mugaliens

2009-Jul-07, 06:56 AM

Energy is measured in Joules (=1 kilogram x meters2 x seconds-2).

Mass is measured in kilograms.

c2 is the conversion factor.

Actually, the coefficient of light's velocity measured in meters per second is the conversion factor.

The reason this works has to to with the interesting interaction between various units of measurement, which, for science, is embodied in The International System of Units (SI) (http://www.bipm.org/en/si/).

The seven SI base units (http://en.wikipedia.org/wiki/SI_base_unit)define length, mass, time, electric current, thermodynamic temperature, amount of a substance, and luminous intensity. From thes base units, and various derived combinations thereof (http://en.wikipedia.org/wiki/SI_derived_unit), all other physical phenomena can be described.

Each of these units does not exist by itself, however. Although the "yardsticks" for each, such as a meter's length, for example, were arbitrarily chosen, it's the relationship between these units which reveals the underlying fundamental constants of the universe, such as the speed of light.

Thus, regardless of whether one is measuring c in m/s, ft/s, or mph, provided one uses the same system of units for all terms in Einstein's equation, the equation holds.

Ken G

2009-Jul-07, 07:01 AM

You lost me here. In this bulk matter there is no net momentum.It is not net momentum that goes into m2 = E2 - p2, nor the momentum contribution to gravity.

As far as I can tell, energy is not an invariant at all. It is wildly coordinate dependent, and so is momentum.m2 is the invariant energy2, it is not coordinate dependent. That's the beauty of rest mass.

As far as I can tell mass, energy, time, and space are all coordinate effects. Not rest mass. The rest mass of every particle is an invariant, it is what is "real" about that particle's energy.

The only way I can understand this stuff is to work completely coordinate free until pressed against the wall and forced to use coordinates to produce specific measurable numbers. What is invariant is the geometry -- the Levi-Civita connection and things that it determines ( the metric, the curvature, etc.). Everything else is a coordinate effect.That all sounds reasonable to me, in respect to motion, but particles must have additional properties independent of motion-- like rest mass.

But if I interpret your statement correctly, you are telling me that although a hot container of gas weighs more than a cold one, all that is really there is the rest energy. So what is all that additional inertia that is coming from the thermal energy ? Is it not real ? What is real ? If I can weigh it it seems pretty real to me.It's a coordinate effect, like you say. I can't answer the details, they are well beyond my expertise, but I see it as similar to the twin paradox. When you have motion involved, you can have coordinate effects (like time dilation) become "annealed" in some sense into the final invariant reality. There's a very strange relation between what is real and what is just our language about what is real. The philosophers have known that for a long time-- physics is just catching up.

publius

2009-Jul-07, 07:52 AM

Doc,

Consider the four momentum. That merges the notion of energy and momentum in one invariant "thing", the 4-momentum 4-vector. Energy is the time-like component of that and the regular 3-momentum is the space-like part. Those coordinate component change with frame, but the vector is the same. Stress-energy is a tensor version of thing (and as a density that would be integrated over a surface to get the net momentum in a volume).

The norm of the 4-momentum is simply m*c, where m is this business we call the rest mass. That's the result of the fact the every particle moves along its worldine at 'c' -- everything moves through its own proper time at the speed of light. The norm of 4-momentum of light, a photon, is 0! That's the nullness of light in space-time.

Energy and momentum are simply splitting that into space and time components. Energy can be thought of as the time-like component of momentum. In the rest frame, everything is in the time component, with the space-like components zero.

The famous relation fall right out of this:

We know momentum is some mv = m dr/dt, where r is the position vector in 3-space. Well, extend that to four vectors, and define a 4-momentum as

mV = m*dR/dt, where R is the four-position, some (t, r). Actually, we need to paramertize that by the proper time along the world line, m*dr/dtau. This is the invariant way of parameterizing the tangent vector of a curve according to its arc-lenght rather than one of the coordinates, and 't' is a coordinate.

The magnitude of the four-velocity is always c, so we can write the 4-momentum as m*c*T, where T is the unit tangent 4-vector to the world line.

In components that works out to be

mV = my*(c, v), where y = gamma. You can trivially see the Lorentz norm of that (times^2 - space^2) is just mc.

Now, we can also write that as(E/c, p). Explicitly writing the norm,

norm^2 = (E/c)^2 - p^2 = constant = (mc)^2.

Multiply by c^2 and we have:

E^2 - (pc)^2 = (mc^2)^2

Which is our famous relation. Note this is simply a statement that the 4-momentum is invariant.

Also not that the since the magnitude is constant, changes in 4-momentum are in direction only. That is P dot dP = 0. The 4-acceleration is always "perpendicular" to the 4-velocity.

Energy and momentum are just the temporal and spatial components of an invariant whose magnitude is (mc)^2.

-Richard

publius

2009-Jul-07, 08:13 AM

You may also find it interesting that, in GR under certain nice conditions, an invariant gravitational mass may be defined. That is just the volume integral of the trace of the stress-energy tensor, T, over c^2.

Newton can be expressed in differential form as simply div g = -4piG*rho, where rho is the mass density. In GR, the term on the right becomes

-4piG*trace_T = -4piG*(rho + 3p/c^2), where p is the scalar pressure, which would be the equal lower 3 diagonal terms of T in the rest frame of bulk matter. In GR, the "g", in the strong field is best thought of as just some mathematical entity that integrates to the gravitational mass via Gauss' law equivalent. The actual coordinate accelerations are something else more complicated.

Thus, momentum, expressed as pressure contributes to the gravity just like energy does.

Note that is the "rest frame" of bulk matter with pressure. In a frame where the mass is actually moving, gravity gets more complicated because we've got gravitomagnetic and other higher order things afoot. Gravity is a tensor field, after all, not a vector.

-Richard

astromark

2009-Jul-07, 08:59 AM

E = MC squared. Look at what you are doing...

Are you trying to impress each other ? its not working.

Albert Einstein said this so it is true. Understanding this equation took me some time as most mathematics closely resembles a ant trail to my thick head. To see you fellows discussing this as if it were just a equation. Its not. Its Art. A statement of facts. The explanation of matter over energy. The understanding is unquestionable. Its the definitive statement. What is what is. So what is it you are doing to the epic... E = MC squared.? have some respect. leave this alone.

Frog march

2009-Jul-07, 10:49 AM

I have wondered whether mass is a form of space-time, so that where you have a planet, what you really have is compressed space-time, so things around a planet are just traveling in a straight line, as if all that space were still there, but it looks curved.

Extravoice

2009-Jul-07, 11:58 AM

Right. In fact, I'd say part of the "popular success" of E = mc2 is that it "rolls off the tongue", certainly far better than E = m does (which sounds trivial, but of course is anything but). What's more, there is "coolness in complexity", and most people who do not know much science consider a formula with squares in it to be much "cooler" by virtue of that mystifying arithmetic operation. They don't realize that is it not the complicated formulae that are most important, it is the simple ones.

Unrelated to the OP, but related to the "coolness factor"...

Our local community college, named Mercer County Community College, uses the advertising slogan:

Education=MC3

After all, cubed is even more cool than squared :)

Ken G

2009-Jul-07, 12:43 PM

Education=MC3

Reminds me of the Far Side cartoon that shows Einstein in front of a blackboard, scratching his head, with E = mc crossed out, and E = mc3 crossed out, and E = mc2 circled, or some such thing.

Ken G

2009-Jul-07, 12:44 PM

So what is it you are doing to the epic... E = MC squared.? have some respect. leave this alone.I would say that to leave a scientific relation alone would be the height of disrespect.

aguerami

2009-Jul-07, 01:49 PM

Nice forum, I am a new member.

The main problem I see with E=mC^2 is this.

Mass is a zero dimensional structure. C is a scalar. E is a field.

You cannot multiply a zero dimensional system by a scalar and get a field.

I explain this in a paper.

Disproof of Gravity (http://www.wbabin.net/science/guerami.pdf)

This also extends to all equations with Mass in them. The universe is not zero dimensional.

F=ma : Force a 3 dimensional field is not equal to a zero dimensional mass * a 2 dimensional vector.

Thanks

Aaron Guerami

http://aaronsreality.blogspot.com

Jetlack

2009-Jul-07, 01:51 PM

wait a minute, i was under the impression that we have measured or verified e=mc squared. Is this an exact equation which is falsifiable or not?

I'm reading this thread and becoming somewhat confused. So i looked ta the wiki page and it uses some rather dubious and uncertain language to describe the equation.

as in

"Expressed in words: energy equals mass multiplied by the speed of light squared. Because the speed of light is a very large number in everyday units, the formula implies that any small amount of matter contains a very large amount of energy. Some of this energy may be released as heat and light by nuclear transformations."

implies? if its a solid fact then what is implied?

aguerami

2009-Jul-07, 02:03 PM

Hi, I am new to this forum. It looks like a interesting topic. I don't think my quick reply posted. I apologize if this posts twice.

Mass is a zero dimensional non point.

E=mC^2

Energy a 3 dimensional field does not equal mass a zero dimensional non point * a scalar.

I write about this problem in a paper Disproof of Gravity (http://www.wbabin.net/science/guerami.pdf)

Other papers on this topic can be found at http://aaronsreality.blogspot.com or http://wbabin.net

So a zero dimensional non points do not exist in nature. A density cannot be reduced to a zero dimensional non point and then returned to a 3 dimensional density.

Aaron Guerami

Moderator's note: Yep, you posted twice. As a spam control measure, newcomers' posts are held in a queue for moderator approval. Once you get a few more posts under your belt, you'll be past that.

nauthiz

2009-Jul-07, 02:07 PM

The solid fact is what is implied. Imply just means "say indirectly."

Consider the logical form P->Q, "P implies Q". The idea is, if P is true then we know that Q is true. Nobody has to explicitly state that Q is true, because we can infer that fact from P's truth - P implies it.

Spaceman Spiff

2009-Jul-07, 02:35 PM

Reminds me of the Far Side cartoon that shows Einstein in front of a blackboard, scratching his head, with E = mc crossed out, and E = mc3 crossed out, and E = mc2 circled, or some such thing.

For an interesting story of crack pots who don't like the findings of Einstein, along with the Far Side cartoon that Ken G refers to, go here (http://www.sunclipse.org/?p=332). :lol:

DrRocket

2009-Jul-07, 02:44 PM

Doc,

Consider the four momentum. ......

Energy and momentum are just the temporal and spatial components of an invariant whose magnitude is (mc)^2.

-Richard

OK, so in short, neither energy nor momentum are invariant, but are wildly coordinate dependent, as I said.

What is invariant is the energy-momentum 4-vector along a world line, which again, does not require coordinates. Correct ?

I have a bit of a problem with the description of rest mass itself as an invariant. I recognize that it is the same for all observers, but only by fiat. There is only one class of observer that actually sees rest mass -- the co-moving observer with the particle. Making something invariant, by requiring the observer to change to a unnatural state just doesn't seem right, and flies in the face of what is really meant by "invariant".

Maybe this is closer to the truth. There is this thing we call "mass" which is not at all invariant and is wildly dependent on the observer. But as a parameter it has a minimum value taken across all possible observers. That minimum is "rest mass". Rest mass is not to be confused with "how much stuff is there" which is not invariant is just plain old mass. ???

I think the problem arises by trying to put the description in terms of classical matter, energy, and momentum in 3 dimensions when the proper description is in terms of quantities describable only in terms of 4-vectors in the tangent space of a Lorentzian 4-manifold. We have a terminology problem more than a physics problem. One should probably not talk of mass or energy but only of mass-energy and in fact perhaps only of mass-energy-momentum in terms of a 4-vector field.

loglo

2009-Jul-07, 02:49 PM

OK, so in short, neither energy nor momentum are invariant, but are wildly coordinate dependent, as I said.

What is invariant is the energy-momentum 4-vector along a world line, which again, does not require coordinates. Correct ?

I have a bit of a problem with the description of rest mass itself as an invariant. I recognize that it is the same for all observers, but only by fiat. There is only one class of observer that actually sees rest mass -- the co-moving observer with the particle. Making something invariant, by requiring the observer to change to a unnatural state just doesn't seem right, and flies in the face of what is really meant by "invariant".

I think the problem arises by trying to put the description in terms of classical matter, energy, and momentum in 3 dimensions when the proper description is in terms of quantities describable only in terms of 4-vectors in the tangent space of a Lorentzian 4-manifold. We have a terminology problem more than a physics problem. One should probably not talk of mass or energy but only of mass-energy and in fact perhaps only of mass-energy-momentum in terms of a 4-vector field.

Taylor/Wheeler called it "momenergy", which I like as a concept but hate as a word! :)

dwnielsen

2009-Jul-07, 02:52 PM

I remember reading somewhere once that it is more consistent and elegant to speak of relativistic momentum, but not to say "relativistic mass", and that this is a valid interpretation. Is this true? Would this mean that momentum and energy are well related - but not necessarily mass - in the equations? Am I completely off-base?

Spaceman Spiff

2009-Jul-07, 03:06 PM

Cosmic Variance comments on the original question (http://blogs.discovermagazine.com/cosmicvariance/2007/02/23/why-does-emc2/).

nokton

2009-Jul-07, 03:27 PM

In a recent AstronomyCast, Pam explained a way of visualising E=MC2, and a question hit me: why is there a multiplier at all? If matter is converted into energy, don't I get the same amount of energy back because of the law of conservation? That is shouldn't the equation be E=M?

Of course, that's not true, or that would be the equation, right? So where does this C2 multiplier come from? Is this just a clash of units, maybe a difference in how we measure matter and energy? Or is there something more freaky going on?

Or am I barking up the wrong tree? (this strikes me as the most likely case)

Thanks

As Albert said, it all depends on your point of view. If I may, Albert not wrong.

In a different scenario, time makes a huge difference. Nevermind Newtonion

physics, time is dynamic and related to gravity in a way we not understand, yet.

Nokton

Sam5

2009-Jul-07, 05:10 PM

I remember reading somewhere once that it is more consistent and elegant to speak of relativistic momentum, but not to say "relativistic mass", and that this is a valid interpretation.

Hmm, I’ve been wondering..... is this thing you call “relativistic momentum”, and what others have called “relativistic mass”, nothing more than the fact that a ball hitting someone at a slow speed carries less force or kinetic energy than the same ball hitting someone at a fast speed?

Or, let’s say we weigh a ball on a scale and the scale registers half a pound, then we throw the ball down hard on the scale and the scale temporarily registers 5 pounds, even though the ball does not “weigh” 5 pounds?

ShinAce

2009-Jul-07, 06:53 PM

I dare not answer your question since a pound is a unit of force, and not a unit of mass.

publius

2009-Jul-07, 07:12 PM

OK, so in short, neither energy nor momentum are invariant, but are wildly coordinate dependent, as I said.

What is invariant is the energy-momentum 4-vector along a world line, which again, does not require coordinates. Correct ?

I have a bit of a problem with the description of rest mass itself as an invariant. I recognize that it is the same for all observers, but only by fiat.

The thing about a proper invariant is the calculation is the same in all frames. The norm of the 4-momentum is mc, and all frames calculate that value. We could multiply it by another factor of c, making P = (E, pc), and having units of energy, thus mc^2 would be the invariant.

Or forget that dimensional silliness and go to geometric units where

c = 1. Then the norm is just m, and that can be mass, momentum, or energy. All the same thing. In that view "rest mass" is just another name for the invariant 4-momentum.

It happens to be that scalar thing we call energy in the rest frame. So we're not really imposing a frame, as all frames "see" that same invariant norm.

-Richard

Ken G

2009-Jul-07, 07:18 PM

wait a minute, i was under the impression that we have measured or verified e=mc squared. Is this an exact equation which is falsifiable or not?No one is disputing the value of that equation (except for ATM posters that I will not comment on), we are talking about what it means, and whether or not it really requires the c2. It really doesn't-- we can just redefine the units of mass to be that of energy.

"Expressed in words: energy equals mass multiplied by the speed of light squared. Because the speed of light is a very large number in everyday units, the formula implies that any small amount of matter contains a very large amount of energy. Some of this energy may be released as heat and light by nuclear transformations."What this "c2 is a big number" really means is that if we choose a scale of mass that we are used to (say the mass of a bread box) and a scale of energy that we are used to (say the energy of dropping a bread box on the floor), then the mass is huge compared to the energy. That's all it really says, the reference to c2 to make that point is actually quite a roundabout way to do it if you track the full argument.

Ken G

2009-Jul-07, 07:25 PM

OK, so in short, neither energy nor momentum are invariant, but are wildly coordinate dependent, as I said.But rest mass, in energy units, is invariant, and is an energy.

What is invariant is the energy-momentum 4-vector along a world line, which again, does not require coordinates. Correct ?Correct, but don't forget that the norm of the 4-vector is also invariant-- and is the rest mass2. That is what I'm saying, that's the reason it makes sense to elevate rest mass to the generic meaning of "mass". It is the energy that is really there, that is not coordinate dependent.

I have a bit of a problem with the description of rest mass itself as an invariant. I recognize that it is the same for all observers, but only by fiat. It is the norm of the 4-vector, it has the same invariance as all such norms.

Making something invariant, by requiring the observer to change to a unnatural state just doesn't seem right, and flies in the face of what is really meant by "invariant". Actually, I would argue that it makes perfect sense to require that change, because the observer who is continuously "on the scene" is the only true witness for what is "actually happening" there. But in more generality, an invariant does not mean "what everyone will observe", a key point of relativity is that not everyone will observe the invariants (proper time and distance, for example, are invariants, but few can measure them between any given events). The reason we need invariants is to convert from what people observe, which is subjective, to what is "real", which is objective (i.e., invariant). Note the way to make any observation an invariant is to include the observer-- i.e., it is always an invariant that I will observe what I observe, but my observation is itself not an invariant. Rest mass conforms perfectly to that meaning.

DrRocket

2009-Jul-07, 07:44 PM

The thing about a proper invariant is the calculation is the same in all frames. The norm of the 4-momentum is mc, and all frames calculate that value. We could multiply it by another factor of c, making P = (E, pc), and having units of energy, thus mc^2 would be the invariant.

Or forget that dimensional silliness and go to geometric units where

c = 1. Then the norm is just m, and that can be mass, momentum, or energy. All the same thing. In that view "rest mass" is just another name for the invariant 4-momentum.

It happens to be that scalar thing we call energy in the rest frame. So we're not really imposing a frame, as all frames "see" that same invariant norm.

-Richard

OK.

My only quibble is the statement that "all frames see the same invariant norm". This is true, but it makes more sense to me to simply note that the norm and the momentum-energy vector can be defined without ever invoking a frame.

To me that is what you really mean by an invariant -- it has a clear defninition and existence with no reference to any coordinate system. You only need to invoke coordinate systems when you need to produce a specific number to be measure by specific equipment in a specific laboratory. Analogy -- linear transformations are real things, not matrices, and you only need matrices when you need a specific expression in terms of specific basis vectors.

I think that perhaps the key, at least to me, is to stop thinking of mass and energy entirely and just look at the 4-vector of momentum-energy. In that regard the typical treatment of special relativity is getting the way in that the 4-vector is the fundamental object and not just conventient notation.

snowflakeuniverse

2009-Jul-08, 01:52 AM

Just a quick clarification.

Referring to c or c^2 as = to 1 or constant can be misleading.

For example, one of the first checks an engineer does in evaluating the validity of a relationship is a dimensional check. Since c = dimensions of distance / time, stating that c = 1 fails a dimensional check, distance/ distance = 1 and Time/time = 1 but distance per time does not appear to = 1.

How ever, distance and time conform to a geometric relationship described by the speed of light. In one second light travels 3 x 10^8 meters. Knowing the nature of this geometry allows a substitution for one term for the other.

The following link is somewhat helpful.

http://en.wikipedia.org/wiki/Minkowski_metric

Snowflake

dwnielsen

2009-Jul-08, 02:08 AM

Just a quick clarification.

Referring to c or c^2 as = to 1 or constant can be misleading.

For example, one of the first checks an engineer does in evaluating the validity of a relationship is a dimensional check. Since c = dimensions of distance / time, stating that c = 1 fails a dimensional check, distance/ distance = 1 and Time/time = 1 but distance per time does not appear to = 1.

How ever, distance and time conform to a geometric relationship described by the speed of light. In one second light travels 3 x 10^8 meters. Knowing the nature of this geometry allows a substitution for one term for the other.

The following link is somewhat helpful.

http://en.wikipedia.org/wiki/Minkowski_metric

Snowflake

Thank you for stating that, snowflakeuniverse. I have seen that substitution made many times with practically no explanation - and it is truly baffling the first time it is witnessed. Special relativity has a few of those initially baffling but finally obvious pedagogies de rigeur.

Ken G

2009-Jul-08, 02:36 AM

Just a quick clarification.

Referring to c or c^2 as = to 1 or constant can be misleading.

For example, one of the first checks an engineer does in evaluating the validity of a relationship is a dimensional check.But there is an important, sublte, and often misunderstood connection between "dimensions" and "units". c does have dimensions of distance over time, but one may still set c=1 with impunity, simply say c = 1 d/t, where d indicates the dimension of distance, and t indicates the dimension of time. Then one can use the value c=1, and still do all the dimensionality checks to which you refer. It is true that dimensions are useful in checking for mistakes, but units are not-- units go a step farther and give quantities to the dimensions, and that is completely arbitrary, and often not terribly sensibly done.

A good example of insensible units is using meters and seconds with the speed of light, as those choices come from our daily experience of other things, but have nothing to do with light. What one really means when one says c = 3 times 108 meters per second is that the ratio of c to an arbitrary velocity "unit" of 1 meter per second is 3 times 108. The number comes entirely from the pointless choice of unit. It is all shorthand for saying we are making a comparison between the speed of light and some other arbitrarily chosen speed. But if one is going to define speeds in terms of comparisons, it makes more sense to choose the speed of light itself for the comparison, after all we have just as much access to light as we do to meter sticks and clocks. It is the latter comparison that gives c = 1, and it has just as much meaning as c = 3 times 108, if one simply tacks on a d/t to track the dimensions, rather than tacking on the meters per second which really doesn't bring that much insight when we are just as familiar with light as with meters (we have, for example, all experienced the short lag that comes when TV signals have to be bounced of satellites during interviews, and so forth).

Tobin Dax

2009-Jul-08, 03:18 AM

Just a quick clarification.

Referring to c or c^2 as = to 1 or constant can be misleading.

For example, one of the first checks an engineer does in evaluating the validity of a relationship is a dimensional check. Since c = dimensions of distance / time, stating that c = 1 fails a dimensional check, distance/ distance = 1 and Time/time = 1 but distance per time does not appear to = 1.

As Ken G said, setting c=1 is a redefinition of units. (This is also pointed out early in this thread where it's said that E=M really is true.) There is a very old convention used every day where we've defined a derived value to be 1. The density of water is 1g/cm3. Mass/mass = 1, volume/volume = 1, and mass/volume = 1 in the case of these specific units.

Setting c=1 does the same thing: it sets length/time = 1, which is specifically true if lengths are measured in light-seconds. So our distance rulers now measure light-seconds. Energy still has dimensions of mass*distance2/time2, but the units are something other than ergs or Joules or BTUs. Setting c=1 isn't misleading, it is leading, since it defines two units.

Frog march

2009-Jul-08, 03:50 AM

If c=1, then time must be measure by change in space? I suppose that for time to exist, things must change, therefor with no matter, there would be no time...I've reached that conclusion before.

Tobin Dax

2009-Jul-08, 04:36 AM

If c=1, then time must be measure by change in space?

That's a leap from this topic (as is the rest of your post). We are leaving off units, but we can say that c = 1 in a few ways. c = 1 lightyear/year or c = 1 lightsecond/second or c = 1 cm/(3.33x10-11 s). In each case, time and distance are measured in the standard way, but the marks on our rulers and the ticks of our clocks match the units given instead of the length of a rod in France or a number of vibrations of a Cesium atom. The units we choose don't have any physical significance. The dimensions do, as discussed above.

mugaliens

2009-Jul-08, 07:17 AM

I write about this problem in a paper Disproof of Gravity (http://www.wbabin.net/science/guerami.pdf)

Sounds ripe for our ATM section. Give it a whirl!

- Mugs

robross

2009-Jul-23, 02:32 AM

wait a minute, i was under the impression that we have measured or verified e=mc squared. Is this an exact equation which is falsifiable or not?

I'm reading this thread and becoming somewhat confused. So i looked ta the wiki page and it uses some rather dubious and uncertain language to describe the equation.

as in

"Expressed in words: energy equals mass multiplied by the speed of light squared. Because the speed of light is a very large number in everyday units, the formula implies that any small amount of matter contains a very large amount of energy. Some of this energy may be released as heat and light by nuclear transformations."

implies? if its a solid fact then what is implied?

I didn't see this addressed by anyone else yet, so let me point out the word "implies" has a specific meaning in mathematics. Just as certain words have very precise meanings in a certain field of study, say physics, that may differ subtly from their colloquial use, the word "implies" in math just means that based on the truth of one statement, the next statement follows logically from it. This would be the "=>" symbol in a proof, which is pronounced "implies."

Rob

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