# Astronomy Without A Telescope – Mass Is Energy

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Some say that the reason you can’t travel faster than light is that your mass will increase as your speed approaches light speed – so, regardless of how much energy your star drive can generate, you reach a point where no amount of energy can further accelerate your spacecraft because its mass is approaching infinite.

This line of thinking is at best an incomplete description of what’s really going on and is not a particularly effective way of explaining why you can’t move faster than light (even though you really can’t). However, the story does offer useful insight into why mass is equivalent to energy, in accordance with the relationship e=mc2.

Firstly, here’s why the story isn’t complete. Although someone back on Earth might see your spacecraft’s mass increase as you move near light speed – you the pilot aren’t going notice your mass change at all. Within your spacecraft, you would still be able to climb stairs, jump rope – and if you had a set of bathroom scales along for the ride you would still weigh just the same as you did back on Earth (assuming your ship is equipped with the latest in artificial gravity technology that mimics conditions back on Earth’s surface).

The change perceived by an Earth observer is just relativistic mass. If you hit the brakes and returned to a more conventional velocity, all the relativistic mass would go away and an Earth observer would just see you retaining with same proper (or rest) mass that the spacecraft and you had before you left Earth.

The Earth observer would be more correct to consider your situation in terms of momentum energy, which is a product of your mass and your speed. So as you pump more energy in to your star drive system, someone on Earth really sees your momentum increase – but interprets it as a mass increase, since your speed doesn’t seem to increase much at all once it is up around 99% of the speed of light. Then when you slow down again, although you might seem to be losing mass you are really offloading energy – perhaps by converting your kinetic energy of motion into heat (assuming your spacecraft is equipped with the latest in relativistic braking technology).

From the perspective of the Earth-based observer, you can formulate that the relativistic mass gain observed when travelling near light speed is the sum of the spacecraft’s rest mass/energy plus the kinetic energy of its motion – all divided by c2. From that you can (stepping around some moderately complex math) derive that e=mc2. This is a useful finding, but it has little to do with why the spacecraft’s speed cannot exceed light speed.

The phenomenon of relativistic mass follows a similar, though inverse, asymptotic relationship to your speed. So as you approach light speed, your relativistic time approaches zero (clocks slow), your relativistic spatial dimensions approach zero (lengths contract) – but your relativistic mass grows towards infinite.

But as we’ve covered already, on the spacecraft you do not experience your spacecraft gaining mass (nor does it seem to shrink, nor its clocks slow down). So you must interpret your increase in momentum energy as a genuine speed increase – at least with respect to a new understanding you have developed about speed.

For you, the pilot, when you approach light speed and keep pumping more energy into your drive system, what you find is that you keep reaching your destination faster – not so much because you are moving faster, but because the time you estimated it would take you to cross the distance from point A to Point B becomes perceivably much less, indeed the distance between point A to Point B also becomes perceivably much less. So you never break light speed because the distance over time parameters of your speed keep changing in a way that ensures that you can’t.

In any case, consideration of relativistic mass is probably the best way to derive the relationship e=mc2 since the relativistic mass is a direct result of the kinetic energy of motion. The relationship does not easily fall out of consideration of (say) a nuclear explosion – since much of the energy of the blast derives from the release of the binding energy which holds a heavy atom together. A nuclear blast is more about energy transformation than about matter converting to energy, although at a system level it still represents genuine mass to energy conversion.

Similarly you might consider that your cup of coffee is more massive when it’s hot – and gets measurably less massive when it cools down. Matter, in terms of protons, neutrons, electrons …and coffee, is largely conserved throughout this process. But, for a while, the heat energy really does add to the mass of the system – although since it’s a mass of m=e/c2, it is a very tiny amount of mass.

## 67 Replies to “Astronomy Without A Telescope – Mass Is Energy”

1. Gore Gogore says:

somebody is mixing mass with impulse or kinetic energy here. While Einstein`s formula was refering strictly to mass . Am i wrong ?

1. Robert Gleisinger says:

Remember that Kinetic Energy = 1/2 mass times velocity squared and in this case velocity equals the speed of light. I am not sure how the half factors out but this translates into Einstein’s equation

1. actually it’s a tad more complex. i mean, the derivation stems in calculus methods (where there’s dm {mass is also a variable}) and culminates in the equation which is actually, Delta(E) = Delta(m) x c^2. we’re talking about the change in mass and the corresponding change in energy. 🙂

2. Anonymous says:

We als have this thing like energy of a photon which has no mass.

2. Steve Nerlich says:

It’s convention to assume that when someone says ‘mass’ they mean proper (or rest) mass, but relativistic mass is a real phenomenon.

Einstein`s various formulae actually refer to both – and the relationship e=mc2 is valid for either proper mass or relativistic mass (and I understand that he derived it from a consideration of relativistic mass).

3. as said before, Mass IS Energy. No difference between the two. 🙂 Impulse, kinetic, potential, dark energy – none of it matters. Mass and energy are same. only by a factor of c(squared) 😛

Mass derives from many sources.

– Elementary particles have intrinsic mass that is believed to derive from interactions with a Higgs field (or fields).

– Composite particles (say, protons) have energy additions from the quantum color force acting on quarks, which we measure as particle mass.

– Atomic nuclei have energy additions from the strong force acting on composite particles (so results from the color force), which we measure as nuclear mass.

– Atoms have energy additions from the electromagnetic force acting on electron clouds, which goes into atomic mass.

– Molecules have energy additions from diverse forms of bonding energies of electron clouds, which goes into molecular mass and chemical energy.

– On that special relativity adds energy that we can measure as mass.

– Finally general relativity adds the equivalence between kinetic inertia and gravitational mass.

It is better to say what Steve Nerlich said in his comment.

2. Charlie says:

I love your articles and I really try to comprehend them, BUT YOU MAKE MY HEAD HURT!

1. Steve Nerlich says:

Thanks Charlie – I am sure someone else could describe all this more neatly and concisely, but was best I could do 😉

1. William Sparrow says:

You did a fine job for this layman Steve!

3. Anonymous says:

Hmm. ‘Nuclear wessel.’

4. IVAN3MAN_AT_LARGE says:

Although someone back on Earth might see your spacecraft’s mass increase as you move near light speed – you certainly aren’t going notice your spacecraft’s, or your own, mass change at all.

Well, I have certainly noticed that the mass of most American motorists seems to increase every time they pass through a McDonald’s drive-thru!

5. Anonymous says:

It is necessary to clear away some confusion. This requires some ABCs of special relativity. The most important aspect of special relativity is that the distance a particle travels is invariant, or does not change under transformations of reference frame, that this is equal to the distance in spacetime the particle travels, and that this is also the time the clock on that particle or moving frame “ticks off.” The distance d is given by a coordinate time t according to some observer and the spatial distance x so that

d^2 = (ct)^2 – x^2.

If the particle is moving along the x direction at a velocity v then x = vt. Substitute this in and we get

d^2 = (ct)^2 – (vt)^2 = c^2(1 – (v/c)^2)t^2.

So the time marked off on the particle frame of reference is d = t’ and this is related to the coordinate stationary observer’s time t by this factor sqrt(1 – (v/c)^2) = 1/? and the time dilaton is t’ = t/?. This leads to the Lorentz factor ?, which is a cornerstone of special relativity and coordinate transformations. On your coordinate system if you watch a system with a regular clock rate moving at a velocity v you then observe its clock rate slowed down by 1/? relative to your clock rate.

We have a similar interval with momentum and energy. Space and time have a conjugate relationship with momentum and energy. In wave mechanics this is given by Fourier transformations, and in classical mechanics by Poisson brackets and so forth. The same holds with special relativity. The invariant momentum-energy interval is

(mc^2)^2 = E^2 – (pc)^2,

for p the momentum, E energy and m the rest mass of a particle. If the momentum p = 0 it is easy to see that E = mc^2 obtains. What if the particle is moving much slower than light? We compute the energy

E = sqrt{(pc)^2 + mc^2}.

Slow motion means that mc^2 >> pc, so we factor mc^2 out

E = mc^2 sqrt{1 + (pc)^2/(mc^2)^2},

And now use the binomial theorem

E =~ mc^2(1 + p^2/2(mc)^2) = mc^2 + p^2/2m

Then for the momentum p = mv we have the result

E = mc^2 + mv^2/2

The total energy is then equal to the famous mc^2 result for the mass-energy equivalent plus the standard Newtonian kinetic energy K = mv^2/2. This then connects special relativity to classical mechanics for low velocities v << c. In standard classical mechanics the mc^2 is absent and in the case v <> p_x^2 and p_y^2. So we factor this out of the square root

E = p_z sqrt{1 + (p_x^2 + p_y^2)/p_z^2 + m^2/p_z^2}

We then use the same binomial theorem trick to write

E =~ p_z + (p_x^2 + p_y^2)/2p_z + m^2/2p_z

To the energy of the part we are interested in, along the x and y axis perpendicular to the direction of the motion of this system, is then

E’ = (p_x^2 + p_y^2)/2p_z + m^2/2p_z.

And for E’ — > E’p_z, multiply through by p_z the energy is

E’ = (p_x^2 + p_y^2)/2 + m^2/2,

which is curiously very classical mechanical in form. The term p_z serves as a Lorentz factor. If one further writes the p_x and p_y in quantum mechanical form parametrized along a string length and includes a potential that is quadratic in a similar fashion you derive easily the energy operator (Hamiltonian) for basic “101 string theory.” String theory is formulated on this extremely boosted frame, sometimes called the light cone frame or infinite momentum frame, which ties in with the extreme Lorentz transformation of a string observed near a black hole and the holographic principle.

LC

1. Steve Nerlich says:

Not getting the invariant distance idea. Don’t spatial dimensions change relativistically, just a time changes relativistically?

1. Anonymous says:

The invariant interval is a distance d = c? which marks the distance of a path in spacetime as measured by a clock on that path. The ? is the time measured by the clock on that frame. Now suppose you the observer have assigned coordinates (t, x, y, z), then you observe this clock mark of this path according to

(c?)^2 = (ct)^2 – x^2 – y^2 – z^2.

If you change the coordinates you observe this from, say by changing position or moving with some velocity, to (t’, x’, y’, z’) then the c? remains the same so that

(c?)^2 = (ct’)^2 – x’^2 – y’^2 – z’^2.

From there one can derive the Lorentz transformations.

To make this simpler, consider the case of a vector on the x and y plane in Euclidean space. It is easier to do this for the derivation of the Lorentz transformations will take up a bit of space here. The vector we look at is v = (x, y), with the length

|v| = sqrt{x^2 + y^2}.

The standard rotation by an angle ? to new coordinates is a simple matrix M =

|cos(?) sin(?)|
|-sin(?) cos(?)|

so that the components of the vector transform as v’ = Mv with

x’ = x cos(?) + y sin(?), y’ = y cos(?) – x sin(?).

Now compute the length of this vector

|v’| = sqrt{x’^2 + y’^2}

= sqrt{(x^2 + y^2) (cos^2 ? + sin^2 ?)}

where the trigonometric identity cos^2 ? + sin^2 ? = 1 tells us that |v| = |v’|. The length of the vector remains the same no matter how we rotate our coordinate system by any angle.

The Lorentz transformation is similar to this, but the rotation is not on a circle, but is hyperbolic.

LC

6. Peter G Brooks says:

What intrigues me is the question of whether the equation holds true for dark matter/energy, and I wonder whether it would be possible to push an envelope of spacetime (if such a thing can be created) through spacetime at a speed faster than light.

Since it is posited that dark energy is accelerating the expansion of the universe in such a way that at some point that expansion will exceed the speed of light, it suggests (to me, anyway) that faster-than-light speeds are possible – just not for “regular” matter.

1. Anonymous says:

It holds for dark matter. The prime candidate for dark matter is the neutralino, which may be a Majorana fermion. A Majorana fermion is its own anti-particle and so they annihilate each other. The PAMELA data indicates a spike in the TeV range for gamma rays, which corresponds to around the expected mass of the neutralino. This data is not conclusive as yet, but it does suggest that dark matter is annihilating and converting its mass into energy.

LC

1. Peter G Brooks says:

I hadn’t read that dark matter is thought to be annihilating – thanks for the info. That would answer another of my questions (not posted) about a prediction that ultimately all objects in the universe will be so far apart that we (if “we” still exist) would not be able to view any other object in the universe at all.

If dark matter is holding “regular” or non-dark matter together while dark energy is creating an accelerating expansion of the universe, I had figured that ultimately even if such constrained structures (i.e., galaxies) became too far apart to be visible to each other, we’d still be able to view other objects in our galaxy, and that seemed inconsistent with the prediction.

2. Anonymous says:

Peter, as to your second paragraph and musing, the salient point there is that space is expanding, not just the distance between objects. So AFAIU, speed limits will not be broken. When not even information transmission can exceed the speed of light, space time “envelopes” will not either. The only way to circumvent the speed boundary is AROUND space time, or THROUGH it, not along it.

1. Peter G Brooks says:

So wormholes, then?

– There are results such as the Alcubierre “warp drive” which have seemingly consistent solutions of spacetime bubbles traveling faster than the local light speed in vacuum.

However, besides that they demand more energy to create than the entire observable universe contains and so called “exotic” never observed matter, they haven’t solved the problem. You have to populate them with matter to actually do some work on it.

And since you can’t get particles to move faster than light in vacuum to catch a ride in and hence populate such a bubble, you can’t get particles to move faster than light in vacuum that way.

There is only one catch and that is Catch-22, which specifies that a concern for one’s special relativity is the process of special relativity.

– The local expansion of spacetime is exceedingly slow, some parts per million per year or so IIRC. Hence the local acceleration from dark energy doesn’t make it anywhere a local expansion exceeding the speed of light in vacuum.

It is only when you integrate over vast volumes that you can see an apparent expansion exceeding vacuum light speed. But that has nothing to do with relativity, which is concerned with local velocities.

However during inflation and possibly in the far future of the universe you will have expansion speeds faster than light I think. Even so, as they don’t affect particles but spacetime, no relativistic laws are broken. Relativity concerns particles in relativistic frames, while such frames are descriptions of precisely spacetime. Those different categories of objects should not be confused.

1. Peter G Brooks says:

I’d forgotten about Alcubierre – thanks for reminding me.

4. Steve Nerlich says:

Einstein’s relativity requires mass/energy conservation. Dark matter could just be an exotic form of matter we are yet identify and possess the property of mass, so no problem there. The standard Lambda CDM model of cosmology requires that it be ‘cold’ – i.e. DM particles are not whizzing around at relativistic speeds ( so for e.g. are probably not neutrinos).

Dark energy is a bafflingly strange proposition that does not obviously follow the principle of energy conservation, nor does it obviously have a mass equivalence. Very much a place marker concept, it is not energy in a conventional sense – and it seems doubtful it could be ‘tamed’ to do useful work.

Otherwise, see the Alcubierre drive for 2nd half of your question: http://en.wikipedia.org/wiki/Alcubierre_drive

1. Anonymous says:

As much as we have all been trained up in physics to think energy conservation is an absolute bedrock, it turns out that general relativity does not imply energy conservation. A spacetime only gives energy conservation if there is some isometry that preserves a time-like vector. Spacetimes with symmetries have Killing vectors K_a which when projected onto a momentum vector give K_aP^a = constant then conserve P^a. If the index a is time then the P^t = E is a constant along the parallel transport by the Killing vector. Not all spacetimes have this. Black holes do, which permits us to do analyses of their ADM mass and so forth. Cosmologies do not. The cosmological metric in the FLRW theory is

d?^2 = dt^2 – a(t)[dx^2 + dy^2 + dz^2]

the fact that the scale factor a(t) is time dependent means there is no Killing vector which preserve a pure timelike vector along geodesics.

The de Sitter metric is such that the time variation in the density of energy ? in the spacetime is d?/dt = 3(? + p) and is zero. We then have that p = -?, so there is this negative pressure. We may then appeal to the first law of thermodynamics

dQ = TdS = dE + dU.

The dE = ?dV and U = pdV and so the theory is adiabatic and the universe has no net input of thermal energy. This is then something we can hang our hat on. However, one must be a bit careful, for pressure really does work if there is a spatial difference in pressure. This negative pressure is uniform throughout spacetime. So this argument for some energy conservation is a bit awkward.

The de Sitter universe is a time dependent conformal spacetime. This is conformally equivalent to a flat Minkowski spacetime which has zero energy. However, there was the production of particles, which seems to suggest that maybe our universe is some sort of quantum fluctuation which has been “frozen” by some sort of phase transition. This then means that mass-energy was generated out of nothing by a curious confluence of general relativity which does not have a universal energy conservation and quantum mechanics which has these uncertainty fluctuations in energy.

LC

This again?

Dark energy is a bafflingly strange proposition that does not obviously follow the principle of energy conservation,

How do you substantiate the claim that dark energy does not follow energy conservation?

– Locally dark energy follows energy conservation:

— Dark energy has a constant energy density, so any local system has a conserved total energy.
— Dark energy is presumably the vacuum energy, which in turn is constituted of zero point energies of particle fields and hence is conserved as:

— fields are conservative.
— zero point energies have constant energy.

– Globally dark energy follows energy conservation:

— The total energy of the universe, which is exactly zero, is conserved.
— This result ultimately follows from the thermodynamical definition of the universe as closed.

So if dark energy would violate energy conservation, something else must too.

It is only if you start to consider general relativity you will have problems, since energy is not conserved in general relativity:

“When the space through which particles move is changing, the total energy of those particles is not conserved.”

For these reasons there isn’t a global energy measure of general relativity.

“This bothers some people, but it’s nothing newfangled that has been pushed in our face by the idea of dark energy. It’s just as true for “radiation” — particles like photons that move at or near the speed of light. The thing about photons is that they redshift, losing energy as space expands.”

By this logic ‘photons is a bafflingly strange proposition that does not obviously follow the principle of energy conservation’.

I am constantly amazed that people start to question dark energy, when it is the expansion of space that is the root of the seeming problem.

Why is gravity and particles acceptable in their unfamiliar behavior, but not dark energy when it obeys precisely the same process of space change? It looks like “energy bigotry”. :-/

And FWIW, a strict “does not … follow the principle of energy conservation” claim has a whiff of crackpotism, see example below. I mean, does anyone think that the cosmologists haven’t checked the physics?

Very much a place marker concept,

This is mainly semantics, but a predictive theory has no place markers for the properties that goes into the predictions.

Even the more tangible placeholders are devices that are intended to be replaced later, but that isn’t the case here. If dark energy is elaborated we have a new more predictive theory instead.

So the semantics here is erroneously used to imply doubt or problems where none exist. I think the term originated with crackpots; it is certainly used by creationists (“The Big Bang is just a placeholder for an extreme violation of this fundamental physical principle.” [energy conservation]); it has no place in discussing science IMO.

it is not energy in a conventional sense – and it seems doubtful it could be ‘tamed’ to do useful work.

It is energy in the conventional sense of thermodynamics (universe total energy), general relativity (standard cosmology mass-energy) from which it derives its name, and likely in the sense of vacuum energy and particle field energy.

It obviously does work, it accelerates the expansion of space volume against other mass-energy that would keep the rate of expansion constant.* In fact, it is all it is known to do at the current time!

This is a step up if it is indeed zero point energy, because it usually can’t do work as it is the effectively lowest energy of the field. So if your definition of “useful” is to expand space quicker, it does that. It indirectly allows the entropy to increase as well.

But to do more useful things with vacuum energy we have already instead demos based on emergent properties like the static and dynamic Casimir effect (nanotechnology repulsion, which does mechanical work, respectively light source).

——————
* Since dQ = 0 (no exchange of heat with the environment) but entropy increases, we have an adiabatic irreversible expansion. The 0 energy result applies strictly for the cosmological horizon, a result of the system behavior over time, so I assume there is no constraint of U = 0 ⇒ dU = 0. But instead Lambda-CDM universe makes an adiabatic free expansion.

Then we can’t really define thermodynamical work with respect to different parts of the system. Maybe if you look at the idealized Lambda-CDM FRW model you can, but I don’t know how to do that.

Hence I have used a mechanical sense of work.

3. Wezley Jackson says:

quoting Torbjorn:
I am constantly amazed that people start to question dark energy, when it is the expansion of space that is the root of the seeming problem.

Hi Torbjorn – Got your name spelled right this time 😉

Yes people (myself included) do question the existence of a “Phantom Energy” field. We are no crackpots.

Quite the opposite – After falling for a few snake oil pitches I wised up – Show us the proof! If this phenomena does exist we will need some convincing. We are not exactly flat-earthers either – we are just sceptics. Any I believe scepticism is a healthy and necessary part of the scientific method.

After searching for minute variations in equipment designed to detect minute distance changes in a beam of light projected along a long metal arm, Michelson-Morley experiment failed to detect the variations of distance of beam travel predicted if there was an “Aether wind.” After dozens of “non-hits” to detecting this phenomena over decades, many prominent scientists realized perhaps it was the understanding that was wrong not the instruments and this led to the birth of some theories any here hold dear (Einstein and others).

Quoting Wikipedia:
http://en.wikipedia.org/wiki/Michelson%E2%80%93Morley_experiment
The Michelson–Morley experiment was performed in 1887 by Albert Michelson and Edward Morley at what is now Case Western Reserve University in Cleveland, Ohio. Its results are generally considered to be the first strong evidence against the theory of a luminiferous ether and in favor of special relativity

Also I remember you chided me saying Schrodinger and Einstein were not really connected on this Aether issue and that it wasn’t so defining as I am making out…
Quoting Schrödinger, life and thought
By Walter J. Moore –

In 1921 Dayton C Miller, who had been a junior colleague of Michelson at the Case Institute in Cleveland, performed a Michelson-Morley type experiment at the top of Mount Wilson, and reported a positive result, a small but significant effect of the earth’s motion on the velocity of light *39. He theorized that at sea level the aether is carried along by the earth, while at higher altitudes a small aether wind becomes evident. Einstein was visiting Princeton when he heard of these results and made his famous statement ‘The Lord God is subtle but he is not malicious…’ The Miller result was received with delight by the enemies of the Einstein theory, but, according to Schrodinger, surprisingly Leonard did not believe it.

So it looks like the (subsequently disproven) Luminiferous Aether was a place-holder and defining moment in modern physics and science. So to be fair dark energy is an important place holder but it is not a phenomena. It is not real (yet?)

Now that I have seen satellite photos I know the earth is (an oblate spheroid). Now that I have seen the photos of the rover tracks and other equipment, I know the moon landings were real and I didn’t even need to be punched in the face by Buzz Aldrin 😉

Show us your proof Torbjorn 🙂

ps. just set up my Disqus profile – It’s me Wezley Jackson 😉

7. Anonymous says:

Stupid layperson question, here, but if neurtrinos have a tiny mass, yet are found to truly exceed the speed of light, then won’t those mass/energy/momentum equasions contain a negative somehow? Is the law of cause and effect then violated?

– Most neutrinos have been measured as having a real, positive mass.

To go faster than light they would have to be tachyons in the relativistic version, which as lcrowell derived the other day destabilizes the vacuum. (Increasing radiation as the tachyons have to accelerate towards infinite velocity.)

Tachyons have imaginary mass.

– Yes, but people imagine loopholes around that one.

For a naive classical first order estimate, neutrinos doesn’t interact very much. Most cause and effect would be “safe”. However, see above.

More realistic loopholes comes from string (aka M) theory and how the strings building particles interact with M theory branes.

1. Gore Gogore says:

it is known that we move about 1,000 km/s thowards The Great Atractor. Remember the LHC neutrinos experiment now. What if we launch beams of neutrinos exactly when the direction is thowards TGA? The earth revolves around in 24 hours so what if we launch another beam exactly in 12 hours , in the exact oposite direction ? Also what if we use aditional beams of laser on 2 spots on the ground above the same points (LHC and that Italy receiver) ? Will the light travel slower and than faster with those ~700 miles/sec ? What will be the real mass of the photons and the neutrinos?

With neutrinos having real mass, those travels follows from relativity. Light will travel the same velocity whatever the frame of the sender; real mass particles will have a Lorentz factor modifying the classical velocity.

For neutrinos, if the LHC result is valid and applies to them (obviously not all neutrinos, since we have the supernova results of other neutrinos traveling slightly slower than light in vacuum), that is the open question depending on which physics they would have.

This is still a bit confused.

– The article makes a good heuristic of how each observer would see their own situation but doesn’t clarify why.

In general terms, relativity is introduced in order to make physical theories universal, i.e. independent of the reference frame of the local observer. A reference frame describes spacetime.

That is why the local observer tend to see the local physical properties as little changed as possible. Instead it is the perception of spacetime by different observers that gives away to a relative characterization.

– There seems to be a concerted effort to abandon the old concepts of “proper” and “relativistic” mass for invariant mass alone.

– To derive the relativistic energy from first principles instead of consistency is quite simple and explains the factor of 2. I believe Einstein did the same derivation in his works.

We look at two particles with invariant mass m = m(0) that collides inelastically, i.e. stick after collision. The particles have the relative velocities v and -v before collision.

In the stationary frame we can add moving masses with the help of the Lorentz transformation that lcrowell derived: M(0) = m(v) + m(-v) = 2m(0)*γ = 2m(0)*√(1 – (v/c)^2). To derive the non-relativistic classic limit we expand the Taylor series:

M(0) = 2m(0)*[1 + 1/2*v^2/c^2 + …] ~ 2m(0) + m(0)*v^2/c^2.

Before the collision each particle had the kinetic energy 1/2m(0)v^2 in the classic limit. After the inelastic collision that kinetic energy has been transformed to heat Q = m(0)v^2.

By looking at the expression above, we see that the relativistic kinetic energy results in a factor of 1/c^2 to the heat energy Q. Hence the conservation of mass gives that the total energy of a particle scales with m and c^2: m = E/c^2 ⇒ E = m*c^2.

9. Anonymous says:

The third paragraph starts out by saying,”Firstly, here’s why the story isn’t complete.” It’s still incomplete since actually E = mc^2 is an incomplete version of the equation. It relates only to stationary objects. The complete version is E = ?mc^2, where gamma is a factor that has to do with motion.

Thank you, an excellent complement to my derivation of the stationary frame version!

1. Anonymous says:

I don’t know if there is any need for pointing out the following elementary consequences of the fact that we have two versions of the formula.

It so happens, then, that the common version everybody will run across sooner or later, in other words, the one applicable to the condition of being stationary, is a special case of the general case, and in that particular case motion is absent, so that the additional factor that takes motion into account is still there, really, but it has been assigned a value of one (to show that there is no motion), which nullifies it and eliminates it since it is a factor. (If it were an addend then it would have to be assigned a value of zero in order to get rid of it, of course.)

I’m at a loss as concerns the consequences of the fact that everything is in motion in relation to something else, so that being stationary is an imaginary condition. Everything is rushing through space at extremely high speeds. That condition, then, is relative and unreal, whereas its opposite –motion–, is real…but also relative. This would seem to be suggesting that there are two kinds of relativity there. What are the theoretical and practical consequences of working with imaginary states, if any?

No, stationary isn’t “imaginary”, relativity says that stationarity is relative.

3. Wezley Jackson says:

Please correct me if I am wrong here but the gamma factor daniel mentions is the basis of the theorom of proof of a cosmological constant – that is that is can be reduced to a single frame of constant motion and velocity. So even more complete would be 1=^y(EMC^2) Lets not leave the fact that its a balanced binomial equation out.. we might confuse our audience 😉 Or am I out on a limb here?

10. Bill Holder says:

My mass keeps increasing over time. 🙂

11. Baris Bicer says:

My tiny brain couldn’t comprehend much of the post, but what really concerns me is that fact that US history seems to have happily neglected to teach me about the fact that the US had a nuclear powered aircraft carrier that sailed around the world. I literally had no idea about it until now. How the crap do you skip over explaining something so awesome?

1. Anonymous says:

Ummm, the Nimitz class Nuclear Powered Aircraft Carriers have been in service since 1975. You would nearly have to have lived under a rock for the last 35 years to not know the US was using nuclear fission powered aircraft carriers. FYI 10 of these bad boyz are patrolling the seven seas!

1. Anonymous says:

And their potential speeds are still classified.

All sorts of crafts are or have been nuclear powered, some are useful (large ships or submarines, or boats for tugging them respectively break the ice for them), most are not.

1. Anonymous says:

A nuclear powered aircraft is not a terribly good idea. The problem is there is a always a chance for an accident or crash which releases the contents of the nuclear reactor. The Wiki site briefly mentions the B-36 tests which was designed to work out how to control a nuclear reactor in an aircraft. The reactor was held in the bomb bay of the aircraft, but did not power anything on the craft. What is not mentioned is that the final test saw the reactor dropped from the bomb bay and it crashed on the ground. This was probably one reason the program was cancelled. The exact point of impact was never revealed, but it is not terribly far from where I live. So there is some place north east of Bernalillo NM that is probably radioactively contaminated.

The idea has occurred to me that a nuclear powered robotic aircraft could be deployed in the upper atmosphere of Jupiter. This could be a way of doing long term studies of the atmosphere from within it. The air breathing nuclear turbines would certainly work in this environment, at least I should think. There is minimal risk of this crashing on Earth during operations so as to cause health and environmental concerns.

LC

12. Ewan Fallon says:

There is only one possibility that has an answer to black holes, dark matter, dark energy, gravity and the big bang etc, etc. and that is a dynamic ether. ‘Branes, Strings and populated vacuums are all heading that way, but seem to be totally unaware of it. If the ether travels with the earth Michelson’s efforts were like trying to find the wind speed with a pitot tube in a balloon. “The Dynamic Ether” from Kunaki

No, any “ether” is out since the aether was rejected by observation some century ago.

1. Wezley Jackson says:

I think Ewan is aware lumeniferous aether does not exist.. He has presented his alternate placeholder name (dynamic aether). I would agree with dynamic equilibrium of a non-zero point vacuum acting on a white hole singularity. Everyone can have their theories. What we need obviously is some proof 😉

I hear many people talking about Dark Energy as though it existed. I’ve never heard anyone who believes in Lumeniferous Aether… TV isn’t that bad today I hope 😉

2. Wezley Jackson says:

I would have to agree that there is room to reserve judgement that our universe is stranger and more wonderful than both you and I imagine. We humans can only scratch the surface…

The Aether is a good example of how we humans can only advance our best theories and knowledge at the time. Yes it is limited and yes when we look back decades later many of our old ideas may seem antiquated and almost comical but we are progressing as a species and science is a wonderful tool for descovery. Internet, satellites, gps, binary, communications, intersteller travel (voyager 1/2) are all happening thanks to that science.

I am not sure what you mean by a dynamic ether but I do believe a plain old (non) zero point vacuum pulling on normal baryonic matter could be the answer. To date there have been zero observations of any ether or “Aether” However there has also been zero observations of Dark Energy and only anectodal evidence of Dark Matter.

Since I have waded in this deep.. Those bluish ghost like halo like features of the bullet cluster could be exited halos of the interacting host galaxies caused by gamma radation exciting netural hydrogen atoms.

Why do I need to accept that because a room full of scientist feel that they have a dark matter map that they do?
There needs to be a heck of a lot better correlation to other experimental data before making that leap darn it.. [plasma vent sequence completed] [utod.com forum pot thouroughly stirred]

Until we some person or team finds the right experiments inferring/proving/showing otherwise, its anyone’s guess 🙂

Wezley

13. Ewan Fallon says:

When observations don’t fit their explanations the experts fall back on lawyer like tactics like “after the big bang the expanding universe did not exceed the speed limit because time did not exist yet” or ” space itself stretches taking the galaxies with it” or “gravity must leak into another universe” because it is the weakest of forces that somehow or other keeps the universe in its place.
When a duplicate galaxy shows up it is credited to gravitational lensing when it has all the earmarks of a mirror image.
They are so obsessed with so called space /time they keep announcing that telescopes look back in time instead of likening it to looking at last weeks newspaper.
Is there nobody willing to admit that the better the observations become, instead of simplifying or solving the cosmic problems, they make matters worse.?

1. Anonymous says:

It’s not that they make matters worse, but that both time and space are paradoxical and thus incomprehensible notions. None of us is to blame. We’ve been discussing this for about 2,500-3,000 years. Any attempt at grasping those two matters will run into contradictions. Philosophers know how to handle the matter more adequately.

Take space, for example. You simply can’t have something spatial (all that exists) surrounded by something that is nonspatial (what science pretends is the “nothingness” beyond everything that exists). Science insists on telling us that there is no “beyond”, that space does not expand into anything, that it invents itself as it expands. This makes no sense and they must acknowledge it instead of explaining away the problem as though it weren’t there. We’re being forced to believe that existence is a Blob immersed in nothingness, but nothingness, by definition, is nonexistent.

They subject time to the same absurd treatment. They claim that when the Big Expansion started this magically invented time and space and that the question about the Before makes no sense. This explains nothing and also runs into a contradiction since there can be no instant without a previous instant. You can’t even begin to talk about time without realizing first that the concept implies eternity in both directions.

So, there logically has to be both a Before and a Beyond. Not only that, but that a singularity (infinite point density) was the starting point is not a necessity. The expansion could be the manifestation of a pulsating motion like the beating of the heart, and the contraction doesn’t have to end in a singularity. It could stop long before achieving infinite point density, then there would be another expansion. This would be the Heart Universe.

A Greek thinker of Antiquity came up with this thought experiment: suppose I were to walk all over to the end of the Universe and poke my walking stick through the limit with nothingness. What would happen to the tip of my stick? Would it vanish? Two thousand years later we’ve made no progress as concerns the answer to that because human reason is unable to understand infinity, space and time.

1. Anonymous says:

What is harder to grasp:
A morning without a yesterday, or something without a beginning?

Honestly, I can’t imagine both. This may be a contradiction, but if you are really honest to yourself, I guess most will agree with me.

Physics is not meant for us to comprehend. Nature doesn’t care about us understanding it. There are things in the universe that we can’t explain properly with words, and yet they exist. I cannot imagine the uncertainty principle of quantum mechanics. And yet it is a fundamental property of nature. The same holds for space, time (it is actually “spacetime” for a good reason) and the universe.

2. Your comment reminded me of this.
It’s only 2 minutes. Enjoy! (I hope)

3. Anonymous says:

” There are things in the universe that we can’t explain properly with words, and yet they exist.”

What science sometimes does is explain with words (and equations) and then ASCRIBE THE CONDITION OF EXISTENCE to whatever the words (and numbers) refer to, as though words (and mathematics) could magically conjure up whatever they’re trying to describe. To what extent can we assure that a theory does not acquire a life of its own, in other words, that it is not used to obtain certain results to which then a certain phenomenon is assigned? Modern physics and cosmology stand accused of this crime before the Court of Logical Thought.

It looks like the way to proceed, in order to avoid having to argue about these matters for another 2,000 years, is to take a mystical qualitative leap and arrive at a level of reality that they call the Eternal Now, where time and space don’t exist and everything (what we call “past, present and future”) is simultaneous. There’s no reason why this has to make any sense, either, in our paradoxical spacetime.

4. Wezley Jackson says:

awesome post mr rey_m !! Quantum physics does my head in 😉 Reminds me of those movies the meaning of life and hitchikers guide to the universe- to paraphrase perhaps we aren’ t asking the right questions if we have a hard time figuring out the answers.

At the risk of anthropmorphising; Our quantum universe has its own just humour for we hominid experimenters… As we make progress we are humbled by the majesty of nature.

5. Anonymous says:

“(…)perhaps we aren’ t asking the right questions if we have a hard time figuring out the answers.”

Yes, that might explain it. Whatever the case, the logical contradictions indicate that there is something fundamentally mistaken in mainstream cosmology. The discovery of the expansion of the Universe is given an inadequate interpretation. The only rational conclusion is that the Universe is both infinite in extension and eternal, as pointed out by Fred Hoyle and others who came up with the derogatory name “Big Bang” in order to make fun of their contradictors. They realized that “Zero Time”, when the clock starts ticking, and an end to space, are meaningless notions. Even so, infinitude and eternity are just as impossible to grasp. How can space go on forever? How is it that there was no beginning and there can be no end to time? At our level of reality this will remain forever incomprehensible.

Thanks for the flattering remark!

6. Anonymous says:

My response to the apparent contradiction of existence is that all of life is paradoxical. Existence is based on a paradox, which is absurd. Absurdity is the foundation of existence which means that many many things that do not seem possible may be possible. Perhaps we can even influence our quantum reality with our thoughts. I realize scientists don’t like this type of mystical thinking, but I believe in the mystery and absurdity. It leaves more room for imagination and possibility while we continue to catalog the facts. A beautiful universe would keep its deepest mysteries lest we discover all the machinery only to find we have nothing left to wonder about. Mystery is beautiful. Even when it frustrates.

7. Anonymous says:

Actually science make predictable theories that can be tested when the technologies starts to exists.

But tell me, what is your evidence that absurdity is the foundation of existence? Do you actually have real physically testable way to prove/disprove this? Or is it only based on personal experience and anecdotes?

8. Anonymous says:

Neither of us can prove we actually exist. Isn’t that absurd enough for you? 🙂

9. Anonymous says:

I’ll bet if I punched you in the nose, you’d swear I exist 😀

10. Anonymous says:

Good luck. I’m also invisible.

Not lawyer like tactics, but judgmental tactics: disprovable evidences for or against, and if not the question is open and goes free.

14. Bharath Purtipli says:

“pre Zefram Cochrane era” ! Awesome ! xD

15. Anonymous says:

Classification of matter & Einstein’s energy equation:
The internal mechanism to radiate energy from matter or absorption fact is not known to us till now. Because we know E = mc^2. But in what way matter can produce energy or what is the internal function of matter during emission of energy by the effect of c^2, we do not know this. The matter is normally defined as on the properties of matter like it has mass, volume, can occupy space, acceleration, etc. There are the simple definitions. But from the view of unification of physics, all matters inversely proportional to Curie constant with effect of velocity of light or photon and directly proportional to square of Avogadro number with multiplication of square root of mass of populated photon. We can formulate the equation taking the equation (K) & (L), ( not given details here), we get:

APPLICATION OF EQUATION OF UNIFICATION OF PHYSICS TO THE DIFFERENT FIELD OF PHYSICS.
matter (m) = Nvo. No.^2 x root of mass of populated photons / Cic ——– (M). (This is the definition of matter)

So, the Einstein’s famous energy equation E = mc^2 will turn to:

Energy = E = Nvo. No.^2x root of mass of populated photons x c / Ci ————— (N)

This equation is the internal functions of m of Einstein equation that in what way energy can liberate. The meaning of populated photo is the mass of bunch of photon, where the accurate mass of a photon is 1.659×10^-54 gm. Complete Unified Theory is single theory, with the help of this theory, we can determine many things from particle to the universe, it can find the birth of particles, stars. galaxies,, universe, we can find the max. mass of the universe, structure of electron, EMR, quantum circulation of Black Hole, Black matter properties, its mass, structure, mass of graviton particle etc. There is no variation between the experimental results and calculated results.This book is very unique and wonder of scientific world.

Reference : Nirmalendu Das , COMPLETE UNIFIED THEORY , Bani Prakash ( P ) LTD., Panbazar , Guwahati, Assam, India 1998. ISBN-81-7643-000-5, Page-223.

Residential Address: *Nirmalendu Das, MUKUL DEEP, Saratpally, W.No.- 40, 74/48, Meghlal Roy Road, Haiderpara, Siliguri – 734006, Dt: Jalpaiguri, West Bengal. Email: [email protected] , [email protected] , [email protected] , Mob: INDIA 9475089337
Dated: 22-11-2011

16. Spike says:

I wonder what all those sailors thought of of E=mc2 while crammed together on the flight deck of the Enterprise for that photo op?

17. Wezley Jackson says:

I was in a rush and just wanted to add that I think negative repulsion of a near zero point vacuum acting on a white hole singularity fits in perfect with standard cosmology observations and the inflation parameters observed. Exponential expansion with a high z log function / order can be identified in datasets and this is I beleive a signpost for this theory.

Also testable could be that light as a constant (C) should phase-shift upwards as the future progresses in this scenario (this phase-shift of photon velocity would be at sub-plank scales so very hard to detect but nonetheless theoretically detectable – we would obviously need a way to measure the speed a beam travels with exquisit accuracy and repeat the experiment a few decades later and compare the numbers basically in a nutshell).

No fudge necessary. I am not berating Dark Energy as a fudge but history I believe will show it to be an important “fudge in numbers” physicists need to explain inflation, expansion and velocities of rotational rates of galaxies observed.

On a personal note my baloney detector goes off when someone presents me with a theory which is caused by a non-observable agent. I don’t mean to be dismissive and do value the work of physicists researching DM/DE to get us here…

Hopefully some day soon we can understand what those forces/phenomena are and can fully reconcile Cosmology and physics achieving a Unity Theorem. Not for the faint of heart and I don’t pretend to have the answers but I have offered an alternative explanation without needing to invoke a non-observable energy field phenomena.

18. Wezley Jackson says:

Hello Steve,

Thanks for some great articles. Its great when you stir the pot a little with DM/DE – It nice to stir up a lively debate and I enjoy reading the informed posts here… It sure beats the intellectual wasteland of FTA tv/radio… Anyhow, Thanks again for great articles.. I’ll be keeping an eye out for those star trek references in future 😉

19. Roger Overcash says:

The Lorentz contraction is a governor on your carburetor.

20. kebilfree says:

your perceptions of time would never change, nor could they, you and your perceptions are all based entirely on the workings of a neurophysiology of your local frame of reference. In somebody elses frame you might be seen as going fast or slower or whatever through time, but you yourself always experience time at your regulary time like speed. The heating of your cup of coffee does create an increase in mass, but that is not necessarily equated with the production of matter. In fact interchanges between matter into energy are generally termed nuclear, fissile decay, radioactive decay, etc. Please let me know if I am incorred

21. kebilfree says:

Also, so what is it about nuclear explosions that releases so much energy. All those bonds that we break immediately reform with each other, this is not a chemical explosion where bonds are broken atoms j(i.e. nitroxide bonds, or is oxynitrile, whatever explosive you feel is your poison today) nor is it a combustion reaction, but in a nuclear reaction, the bonds that are broken are quickly remade. The only energy release is the conversion of matter to energy. 1 gram transformed into energy would be 900,000,000,000,000,000,000.0 ergs of energy, which can be converted into many other types of energy. Other than the insignificant contributions of the triggering explosives, all the rest of the energy is solely produced by fission, or fusion, both of which result in matter being turned into pure energy