What is the difference between mass and energy?

• jnorman
In summary: If you're in an accelerated frame, all the laws of physics still apply, but you can't use them to measure anything. Instead, you have to use the laws of an (as of yet) unaccelerated frame to make measurements.#3 is the interesting case. In general relativity, spacetime is curved. The geometry of spacetime determines the laws of physics that can be applied to objects in that spacetime. If you're in an accelerated frame, the geometry of spacetime will determine the laws of physics that can be used to measure things.
jnorman
since the equation, E=MC2, indicates that mass and energy are equivalent, what exactly is the difference between them? ie, how do you explain that energy can travel at C, but not matter? or is there some significant difference between matter and an object which has mass?

also, if mass and energy are interchangeable as implied by the equation, where do all the other properties of matter come from, such as charge?

thanks.

Hi. The relation of mass, energy and momentum of a particle is m^2 = E^2/c^4 - p^2/c^2. Mass and energy are proportional only when momentum is zero.
Regards.

sweet springs said:
Hi. The relation of mass, energy and momentum of a particle is m^2 = E^2/c^4 - p^2/c^2. Mass and energy are proportional only when momentum is zero.

This only works when m is the *rest* mass, and it also doesn't work in all cases in general relativity. There is no satisfactory definition of mass or energy in GR that works in all cases. For example, GR does not have any sensible way to write down a local energy density of an electromagnetic wave.

Hi.
bcrowell said:
There is no satisfactory definition of mass or energy in GR that works in all cases. For example, GR does not have any sensible way to write down a local energy density of an electromagnetic wave.
Mass of a single particle is scalar even in GR. RHS of Einstein equation has energy-momentum tensor T which could include electromagnetic energy-momentum. You say we cannot write down the equation? I know that gravitational energy-momentum is pseudo- or non-tensor, but it is another topic.
Regards.

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sweet springs said:
Mass of a single particle is scalar even in GR. RHS of Einstein equation has energy-momentum tensor T which could include electromagnetic energy-momentum.
The energy-momentum tensor isn't the same thing as an integrated mass. There is no general way of writing down an integrated mass that transforms like a tensor.

sweet springs said:
You say we cannot write down the equation? I know that gravitational energy-momentum is pseudo- or non-tensor, but it is another topic.
Which version do you have in mind? None of them work in general. For instance, ADM and Bondi mass require asymptotic flatness.

These are not minor technical issues. For a good discussion of why conservation laws basically don't exist in GR, see MTW, p. 457.

Hi.
bcrowell said:
The energy-momentum tensor isn't the same thing as an integrated mass. There is no general way of writing down an integrated mass that transforms like a tensor.
I see stress energy tensor and you see integrated mass. There is no crossing argument, isn't it?
Coming back to jnorman's original interest, it maybe helpful to show whether your integrated "mass" and integrated "energy" are independent or not.
I say my answer to jnorman. Energy(-mementum) is expressed in tensor. Mass is its invariant.
bcrowell said:
Which version do you have in mind?
I learned Einstein way and Landau-Lifgarbagez way many years ago. Your teaching on recent progress will be appreciated.
Regards.PS By the way, why people call gravitational energy momentum a "PSEUDO"tensor ?. I understand pseudotensor is one that change sign for inversion of axis. It does not apply this GR case. It should be called "NON"tensor, shouldn't be?

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Mass is a type of energy. Energy is conserved in inertial frames including in special relativity, but general relativity deals with accelerating frames.

ikos9lives said:
Mass is a type of energy. Energy is conserved in inertial frames including in special relativity, but general relativity deals with accelerating frames.

FAQ: Does special relativity apply when things are accelerating?

Yes. There are three things you might want to do using relativity: (1) describe an object that's accelerating in flat spacetime; (2) adopt a frame of reference, in flat spacetime, that's accelerating; (3) describe curved spacetime. General relativity is only needed for #3.

A prohibition on #1 is particularly silly. It would make SR into a trivial theory incapable of describing interactions. If you believed this, you would have to stop believing, for example, in the special-relativistic description of the Compton effect and fine structure in hydrogen; these phenomena would have to be described by some as yet undiscovered theory of quantum gravity.

#1 often comes up in discussions of the twin paradox. A good way to see that general relativity is totally unnecessary for understanding the twin paradox is to pose a version in which the four-vector equation a=b+c represents the unaccelerated twin's world-line a and the accelerated twin's world-line consisting of displacements b and c. The accelerated twin is subjected to (theoretically) infinite accelerations at the vertices of the triangle. The triangle inequality for flat spacetime is reversed compared to the one in flat Euclidean space, so proper time |a| is greater than proper time |b|+|c|.

#2, accelerated *frames*, is less trivial. It's for historical reasons that you'll see statements that SR can't handle accelerated frames. Einstein published special relativity in 1905, general relativity in 1915. During that ten-year period in between, nobody really knew what the boundaries of applicability of special relativity were. This uncertainty made its way into textbooks and lectures, and because of the conservative nature of education, some students are still hearing, a century later, incorrect assertions about it.

In an accelerating frame, the equivalence principle tells us that measurements will come out the same as if there were a gravitational field. But if the spacetime is flat, describing it in an accelerating frame doesn't make it curved. (Curvature is invariant under any smooth coordinate transformation.) Thus relativity allows us to have gravitational fields in flat space --- but only for certain special configurations like uniform fields. SR is capable of operating just fine in this context. For example, Chung et al. did a high-precision test of SR in 2009 using a matter interferometer in a vertical plane, specifically in order to test whether there was any violation of Lorentz invariance in a uniform gravitational field. Their experiment is interpreted purely as a test of SR, not GR.

Chung -- http://arxiv.org/abs/0905.1929

"Difference...and... how do you explain that energy can travel at C, but not matter?"

fundamentally, nobody knows...we have some math, and via some rules, we can describe observational results...but WHY things are they way they are is generally beyond our reach so far. As Crowell implies we can say that particles with rest mass can't travel at c...becuse it would take infinite energy to get anything moving that fast...but why things are that way we don't know.

We don't even know exactly what mass nor energy are nor how they originate.

bcrowell said:
#2, accelerated *frames*, is less trivial. It's for historical reasons that you'll see statements that SR can't handle accelerated frames. Einstein published special relativity in 1905, general relativity in 1915. During that ten-year period in between, nobody really knew what the boundaries of applicability of special relativity were. This uncertainty made its way into textbooks and lectures, and because of the conservative nature of education, some students are still hearing, a century later, incorrect assertions about it.
I still hold that SR cannot handle accelerated frames, by definition.
From Einstein's book:
"The basal principle, which was the pivot of all our previous considerations, was the special principle of relativity, i.e. the principle of the physical relativity of all uniform motion. "
So if we deal with
-special covariance (uniform motion, inertial frames): special relativity
-general covariance (arbitrary motion, arbitrary frames): general relativity

AFAIK this is still the "official" definition of the theories: by their structure.
The definition by "physical content", aka flat vs curved spacetime, is rather informal and not supported by the literature, IIRC.

The LL pseudotensor also requires asymptotically flat space-time, at least in order to have any physical significance other than some number that results from a specific choice of coordinates.

Wald, "General Relativity" refers to the following paper:

which demonstrates that the LL pseudotensor approach is equivalent to the Bondi mass.

abstract said:
..., then the Landau-Lifgarbagez complex appropriately formulated from g _μv and h_μv gives a covariant definition of the total 4-momentum of the gravitational field equivalent to the Bondi prescription.

pervect said:
Wald, "General Relativity" refers to the following paper:
which demonstrates that the LL pseudotensor approach is equivalent to the Bondi mass.

Thank you for information. I am a little happy to know that LL pseudotentor that I learned long time ago still survives in theoretical physics.

Regards.

Ich said:
I still hold that SR cannot handle accelerated frames, by definition.
From Einstein's book:
"The basal principle, which was the pivot of all our previous considerations, was the special principle of relativity, i.e. the principle of the physical relativity of all uniform motion. "
So if we deal with
-special covariance (uniform motion, inertial frames): special relativity
-general covariance (arbitrary motion, arbitrary frames): general relativity

AFAIK this is still the "official" definition of the theories: by their structure.
The definition by "physical content", aka flat vs curved spacetime, is rather informal and not supported by the literature, IIRC.
The quote only supports the principle of relativity, not that non-inertial coordinate systems must be thrown out. What we get from the principle of relativity (along with some other stuff) is that spacetime is either Galilean spacetime or Minkowski spacetime. The invariance of the speed of light rules out the former. So we're left with Minkowski spacetime, which can be defined mathematically in at least three different ways: as a vector space, an affine space, or a manifold. But regardless of which of these options we choose, there's nothing that forces us to throw out non-inertial coordinate systems. In fact, it's quite unnatural to do so.

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Ich said:
I still hold that SR cannot handle accelerated frames, by definition.
From Einstein's book:
"The basal principle, which was the pivot of all our previous considerations, was the special principle of relativity, i.e. the principle of the physical relativity of all uniform motion. "
So if we deal with
-special covariance (uniform motion, inertial frames): special relativity
-general covariance (arbitrary motion, arbitrary frames): general relativity

AFAIK this is still the "official" definition of the theories: by their structure.
The definition by "physical content", aka flat vs curved spacetime, is rather informal and not supported by the literature, IIRC.
I completely disagree.

What a theory can handle or not can be established by experiments. It has nothing to do with human made definitions.

For instance if someone develops a theory about neutron stars and it turns out to be valid for all stars then it silly to say that such a theory is only valid for neutron stars because the inventor says so.

Ich said:
I still hold that SR cannot handle accelerated frames, by definition.
From Einstein's book:
"The basal principle, which was the pivot of all our previous considerations, was the special principle of relativity, i.e. the principle of the physical relativity of all uniform motion. "
So if we deal with
-special covariance (uniform motion, inertial frames): special relativity
-general covariance (arbitrary motion, arbitrary frames): general relativity

AFAIK this is still the "official" definition of the theories: by their structure.
The definition by "physical content", aka flat vs curved spacetime, is rather informal and not supported by the literature, IIRC.

From a literary standpoint, why not interpret the quote to mean that special relativity cannot deal with accelerated motion?

Hmm, reading carefully, I see you have indeed classed the twin paradox as a general relativistic problem ... really?

General covariance (of the equations of motion) isn't the key principle of general relativity is it? It's "no prior geometry".

Also, structurally, isn't it just that inertial frames exist? The moment we know the relationship between the laws in all inertial frames, then we automatically know the laws in accelerated frames too.

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Ich said:
I still hold that SR cannot handle accelerated frames, by definition.
From Einstein's book:
"The basal principle, which was the pivot of all our previous considerations, was the special principle of relativity, i.e. the principle of the physical relativity of all uniform motion. "
Just because SR grants privileged status to certain frames, that doesn't mean that those are the only frames SR can describe.

Newtonian mechanics considers certain frames to be privileged, e.g., a frame fixed to the Earth's surface is (approximately) inertial, so it's a privileged frame. The frame of an accelerating elevator is not a privileged frame in Newtonian mechanics. The laws of Newtonian mechanics have a certain simple form in the privileged frames. Their form is different in the non-privileged frames, e.g., Newton's third law is violated by fictitious forces. Nevertheless, we can and often do use the non-privileged frames in Newtonian mechanics.

SR classifies frames as privileged and non-privileged by exactly the same criteria as Newtonian mechanics. Exactly as in Newtonian mechanics, the laws of physics are form-invariant in different privileged frames, but have a different and more complicated form in the non-privileged frames. Exactly as in Newtonian mechanics, we can choose to use the non-privileged frames if we wish.

GR also has privileged and non-privileged frames, but the classification is different than in Newtonian mechanics and SR. A frame fixed to the Earth's surface is not a privileged frame. The privileged frames are the free-falling frames, which would have been *non*-privileged frames in Newtonian mechanics and SR. Not only is the classification different, but we have two other differences as well: (1) the laws of physics (Einstein field equations) are form-invariant across all frames, not just privileged frames; (2) frames lose much of their importance because they can no longer be defiend globally.

Ich said:
AFAIK this is still the "official" definition of the theories: by their structure.
The definition by "physical content", aka flat vs curved spacetime, is rather informal and not supported by the literature, IIRC.
I could certainly dig up authorities (probably mostly post-1950) to support my view, and I'm sure you could dig up authorities (probably mostly pre-1950) to support yours.

from Jnorman

also, if mass and energy are interchangeable as implied by the equation, where do all the other properties of matter come from, such as charge?

Nobody knows that either. The Standard Model of particle physics is the best we have for strong, weak and electromagnetic forces and particles...but values are measured, experimentally determined, because nobody has a theory from which to derive such values. Why the electron? nobody knows. It appears the three forces possibly emerge from a single initial high energy state at the big bang, but why a unified force would break into the three we observe nobody really knows. And gravity remains outside that theory so far; the search to combine all forces is called grand unification and so far eludes science.

String theory is another theory and there different vibrational patterns of fundamental strings, one dimensional particle extensions, dictated by hidden geometries of space, determine characteristics...so one vibrational pattern confers mass, another charge, yet another the graviton...and the more energetic the "mass vibration", the more mass is observed.

Ok, so we're tied: Ich, me, myself, and I vs Fredrik, Passionflower, atyy, and bcrowell.

I'll open a new thread for this topic, hopefully tomorrow. You may well convince me, especially if you show me a definition in a respected GR textbook that the difference between GR and SR is exactly the inclusion of ("real"=spacetime curvature) gravity.

if spacetime is flat, energy is conserved, but once you have to go to another frame of reference (because of curved spacetime), energy will not be conserved — it is not invariant between frames.:zzz:

jnorman said:
since the equation, E=MC2, indicates that mass and energy are equivalent, what exactly is the difference between them? ie, how do you explain that energy can travel at C, but not matter? or is there some significant difference between matter and an object which has mass?

also, if mass and energy are interchangeable as implied by the equation, where do all the other properties of matter come from, such as charge?

thanks.

If you think carefully, E=mc2 is actually a unit, another unit of mass (like Kg) in terms of energy (J). We are converting neither mass into energy nor energy into mass, which at this point in time seem impossible.

In all fissions, fusions or breaking up nuclei we are releasing trapped energy, we are not converting mass into energy.

Neandethal00 said:
If you think carefully, E=mc2 is actually a unit, another unit of mass (like Kg) in terms of energy (J). We are converting neither mass into energy nor energy into mass, which at this point in time seem impossible.

In all fissions, fusions or breaking up nuclei we are releasing trapped energy, we are not converting mass into energy.

So, how do you explain http://en.wikipedia.org/wiki/Pair_production" and electron/positron annihilation that produces gammas of the exact energy given by E=mc2?

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Ich said:
I still hold that SR cannot handle accelerated frames, by definition.
The two postulates of SR do refer to inertial frames, so I can understand your assertion here. But I don't think that implies that you cannot do a transformation to a non-inertial frame and work there, as long as there exists a transformation to an inertial frame and the conditions of the postulates are satisfied there. That is essentially how non-inertial frames are handled in Newtonian mechanics: Newton's laws don't hold in non-inertial frames but there exists a transformation to an inertial frame where they do hold.

I don't have an "official" reference to support either position, and it probably is not horribly important to draw the line anyway.

ikos9lives said:
if spacetime is flat, energy is conserved, but once you have to go to another frame of reference (because of curved spacetime), energy will not be conserved — it is not invariant between frames.:zzz:

This is incorrect. Energy is not frame-invariant in Newtonian mechanics, but it is conserved. Conservation and frame-invariance are two different things.

The issues involved in curved spacetime are both qualitatively different from and much more complicated than what you're saying. We had some discussion about that recently: https://www.physicsforums.com/showthread.php?t=426479

DaleSpam said:
ich said:
I still hold that SR cannot handle accelerated frames, by definition.]/quote]
The two postulates of SR do refer to inertial frames, so I can understand your assertion here.
Einstein wrote the postulates in 1905, before GR existed, so they can't be taken as defining the boundary between SR and GR. There has been a historical evolution of our understanding of the best way to define the distinction. The definition in terms of curved versus flat spacetime has been widely accepted for decades, but it wasn't understood in 1905, or even in 1915.

Misner, Thorne, and Wheeler, Gravitation, 1973, p. 163, "Accelerated motion and accelerated observers can be analyzed using special relativity."

Penrose, The Road to Reality, 2004, p. 422, "It used to be frequently argued that it would be necessary to pass to Einstein's general relativity in order to handle acceleration, but this is completely wrong. [...] We are working in special relativity provided that [the] metric is the flat metric of Minkowski Geometry M."

bcrowell said:
Misner, Thorne, and Wheeler, Gravitation, 1973, p. 163, "Accelerated motion and accelerated observers can be analyzed using special relativity."

Penrose, The Road to Reality, 2004, p. 422, "It used to be frequently argued that it would be necessary to pass to Einstein's general relativity in order to handle acceleration, but this is completely wrong. [...] We are working in special relativity provided that [the] metric is the flat metric of Minkowski Geometry M."
Neither of these quotes refer to SR handling a non-inertial frame. They talk about acceleration from the perspective of an inertial frame. The second quote explicitly so. Using the numbering from your post #9 these both address situation (1) which I think nobody disputes, it is situation (2) that is the gray area.

As I said above, my inclination is to say (2)->SR, but I do understand the (2)->GR position which is usually tied to the two postulates or other very early formulations.

bcrowell said:
This is incorrect. Energy is not frame-invariant in Newtonian mechanics, but it is conserved. Conservation and frame-invariance are two different things.

The issues involved in curved spacetime are both qualitatively different from and much more complicated than what you're saying. We had some discussion about that recently: https://www.physicsforums.com/showthread.php?t=426479
Yes, they are different things. I meant energy conservation does not apply between frames or with multiple frames, since it is not an invariant quantity.

DaleSpam said:
Neither of these quotes refer to SR handling a non-inertial frame. They talk about acceleration from the perspective of an inertial frame. The second quote explicitly so.

(1) "Accelerated motion and accelerated observers can be analyzed using special relativity."

Surely accelerated observer = non-inertial chart / coordinate system / reference frame?

(2) "We are working in special relativity provided that [the] metric is the flat metric of Minkowski Geometry M."

Should we interpret Penrose's "metric" here as coefficient matrix of the pseudo-metric tensor field in some coordinate basis field, and "flat" as + or - diag(-1,1,1,1)? The context suggests otherwise. Elsewhere in The Road to Reality, Penrose uses "(pseudo)metric" to mean the tensor or tensor field itself, without reference to a basis (field):

"Under appropriate circumstances, a symmetric, non-singular (0,2)-tensor gab is called a metric--or sometimes a pseudometric when g is not positive definite" (13.8).

"Recall from 13.8 that a metric (or pseudometric) is a non-singular symmetric (0,2)-valent tensor. We require that g be a smooth tensor field" (14.7).

The index notation is ambiguous, but the fact that he uses boldface italic for g actually make it explicit that he's not referring to a coefficient matrix, but to the tensor itself (as explained in the section Notation at the beginning.)

I can't find "flat" in the index, but that quote goes on "this means that gravitational fields can be neglected"; a couple of pages later, he says of Minkowski space, "its intrinsic metric is indeed flat" (18.4), again suggesting that he has a basis-independent quantity in mind.

Hi.
ikos9lives said:
I meant energy conservation does not apply between frames or with multiple frames, since it is not an invariant quantity.

In GR, 4-divergence, or contract of covariant derivative, of energy stress tensor is zero at ANYWHERE in space-time and in ANY FRAME OF REFERENCE which means energy and momentum is locally conserved.
Regards.

1. What is mass?

Mass is a measure of the amount of matter in an object. It is a property of physical objects and is typically measured in units such as kilograms (kg) or grams (g).

2. What is energy?

Energy is the ability to do work or cause change. It is a property of physical systems and is typically measured in units such as joules (J) or kilowatt-hours (kWh).

3. How are mass and energy related?

Mass and energy are two fundamental properties of matter and they are related by Einstein's famous equation, E = mc^2, where E represents energy, m represents mass, and c represents the speed of light. This equation shows that mass and energy are interchangeable and can be converted into one another.

4. What is the main difference between mass and energy?

The main difference between mass and energy is that mass is a measure of the amount of matter in an object, while energy is a measure of the ability of an object to do work or cause change. Mass is a property of physical objects, whereas energy is a property of physical systems.

5. How do mass and energy affect each other?

Mass and energy are closely related and affect each other through their interconversion. When mass is converted into energy, it can release a tremendous amount of energy, as shown by nuclear reactions. On the other hand, energy can also be converted into mass, as demonstrated by particle accelerators that can create new particles from high-energy collisions.

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