I Real life example of the energy contained at E=γmc^2

[Orodruin]

The big problem is that "energy" in many layman conversations refer to "usable energy" in the sense of energy available to do work on something I need to do work on.

To add to this, I believe this is exactly what the OP has run into based on this:

But where exactly is this energy present in real life? For example the keyboard I am currently typing this post with has a huge amount of energy, according to this equation, but how is it usable?

In the colloquial use, energy is what is "produced by power plants" and it therefore becomes a barrier when laymen are faced with energy as being a property of things and certainly not being something that can be produced.

 

[Herman Trivilino]

Matter is just an informal word meaning stuff that has mass, isn't it?

If so then a pair of photons could be called matter in some reference frames but not in others. A single photon would not be matter in any reference frame.

In any case, the Einstein mass-energy equivalence teaches us that mass is not a measure of the quantity of matter.

 

[Orodruin]

If so then a pair of photons could be called matter in some reference frames but not in others.

The square of the total 4-momentum of the photons is invariant.

 

[Herman Trivilino]

The square of the total 4-momentum of the photons is invariant.

Ahhh... yes. I didn't explain my thoughts correctly. If the pair of photons are co-moving their mass is zero (in all frames) but if not then they have a nonzero mass (which has the same value in all frames). So what I should have said is that a pair of photons may or may not have mass, depending on how they are moving. Not on the reference frame.

 

[gnnmartin]

I think the equation e=mc^2 should be interpreted as telling us that energy has mass, not that mass is energy. Mass is the unambiguous term, defined by Newton's laws. Energy is ambiguous, since it implies something usable, but its usability depends on the state of our technology and our intentions, and anyhow the usability is limited by the laws of thermodynamics.

 

[weirdoguy]

Mass is the unambiguous term, defined by Newton's laws.

You are mixing non-relativistic and relativistic mechanics. Mass (squared) in relativity is defined as a norm of momentum 4-vector. It has nothing to do with Newton's laws.

Energy is ambiguous

No it's not.

 

[SiennaTheGr8]

Energy is ambiguous, since it implies something usable

As @Orodruin pointed out earlier in this thread, this is a common misconception. "Usability" is a red herring. Conservation is the ticket.

 

[Philip Koeck]

We've just begun studying about relativity, and I find it amazing that bodies have the energy of E=γmc^2. Even at rest they have E=mc^2.
But where exactly is this energy present in real life? For example the keyboard I am currently typing this post with has a huge amount of energy, according to this equation, but how is it usable?

Hi Foruer. I'm not sure if anyone has given similar answers since I haven't looked through the list. Anyway: One example is something I calculated in a University course. We were given some information about the total radiation from the sun and asked to calculate how much lighter the sun gets per year. This is quite an everyday thing I would say. All the energy radiated off by the sun leads to a tremendous mass loss. Another example is of course particle-antiparticle annihilation. There the energy of the photons produced is equal to the sum of relativistiv masses times c squared.Hope that helps.

 

[Ibix]

Mass is the unambiguous term, defined by Newton's laws.

Defining the concepts of a more precise theory in terms of the concepts of a less general, less precise, theory seems back to front. Relating the terms is fine (e.g. $(\gamma-1) mc^2\simeq mv^2/2$ implying that this is the correct expression for KE) is fine, but don't mistake that for definition.

I think that $E=mc^2$ is effectively the definition of mass in relativistic terms. It's the modulus of the energy-momentum four-vector, which is a conserved quantity you can relate directly to experiment. It happens to be interpretable as a generalisation of what Newton would call mass.

 

[Ibix]

There the energy of the photons produced is equal to the sum of relativistiv masses times c squared.

...or equal to the sum of the total energies of the particles, if you are trying to avoid the term "relativistic mass" (which is the modern convention).

 

[Orodruin]

I think the equation e=mc^2 should be interpreted as telling us that energy has mass, not that mass is energy. Mass is the unambiguous term, defined by Newton's laws. Energy is ambiguous, since it implies something usable, but its usability depends on the state of our technology and our intentions, and anyhow the usability is limited by the laws of thermodynamics.

As has already been pointed out in this thread, mass in relativity is not defined through Newton’s laws (which are not Lorentz invariant!), but through the square of the 4-momenta of which energy is a component. As such, mass is nothing but the rest energy of a system. The fact that this corresponds to the inertia in the rest frame is the mass-energy equivalence.

Energy in no way implies ”usefulness” - this I have also already pointed out.

 

[DrGreg]

$(\gamma-1) mv\simeq mv^2/2$

I think you meant to say
$$(\gamma-1) mc^2\simeq mv^2/2$$

 

[Ibix]

I think you meant to say
$$(\gamma-1) mc^2\simeq mv^2/2$$

Indeed - now corrected above. Thanks.

 

[Herman Trivilino]

I think the equation e=mc^2 should be interpreted as telling us that energy has mass, not that mass is energy.

Mass is equivalent to rest energy. Two names for the same thing.

 

[Dadface]

We've just begun studying about relativity, and I find it amazing that bodies have the energy of E=γmc^2. Even at rest they have E=mc^2.
But where exactly is this energy present in real life? For example the keyboard I am currently typing this post with has a huge amount of energy, according to this equation, but how is it usable?

I think some of the best places to find the presence of "this energy in real life" is to observe nuclear reactions including electron positron annihilation and pair production. There are numerous other reactions which result in mass/energy changes but there are problems in making the necessary observations. For example if we burn methane with oxygen we can easily measure the resulting heat energy and we can equate this to the mass difference between the reactants and the products. But we can't (yet) measure this mass difference accurately by mass measurement techniques. With annihilation/pair production we can measure the energy of gamma rays, for example by using gamma ray spectroscopy techniques and we can measure the mass of electrons and positrons using mass spectoscopy/penning trap techniques.