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If you simply say the name “Albert Einstein” and ask people what comes to mind, many of them will simply repeat his most famous equation, whether they known what it means or not: E = mc². In plain English, that equation states that the energy of an object is equal to that object’s mass multiplied by the speed of light squared. This simple-seeming mathematical relation holds an enormous amount of physics inside of it, including: The ability to spontaneously create new matter-antimatter pairs of particles as long as their rest mass is less than the amount of available energy you have to try and create them. Whenever a matter-antimatter pair of particles annihilates, that annihilation will produce a photon pair with a specific amount of energy given by the masses of the pair of particles that annihilated. And that every time you have a nuclear reaction, whether fusion or fission, if the mass of the products is less than the mass of the reactants, E = mc² tells you how much energy gets liberated by that reaction. This one equation, E = mc², describes how much energy is inherent to any massive particle at rest, including how much energy it takes to create it and how much energy gets released if you destroy it. But most particles in the Universe aren’t at rest, and plenty of particles don’t have any mass at all. In those cases, E = mc² isn’t the full equation needed to describe your system; it’s only half of the full equation. The other half holds a great number of insights, and shows that both halves are required to make physical sense of what’s going on. The production of matter/antimatter pairs (left) from two photons, or from any colliding pairs of matter and antimatter, is a completely reversible reaction (right), with matter/antimatter annihilating back to two photons in the majority of cases. This creation-and-annihilation process, which obeys E = mc², is the only known way to create and destroy matter or antimatter. If high-energy gamma-rays collide with other particles, there is a chance to produce electron-positron pairs, diminishing the gamma-ray flux observed at great distances. Credit: Dmitri Pogosyan/University of Alberta The reason “rest mass” is such an important concept is because motion — the rate of change of an object’s position over time — isn’t an “absolute” physical property in our Universe. Instead, the key lesson from Einstein’s relativity is that irrespective of what your position is or how your position is changing with time, the laws of physics and the constants of nature, including the all-important speed of light, are always going to appear identical in any inertial (non-accelerating) frame of reference. So if, for example, you have a clock where “one second” is defined by how long it takes light, moving at the speed of light, to: rise from the bottom of the clock to the top, reflect off of a mirror at the top, and come back down to the bottom once again, then two observers in relative motion to one another will experience the passage of time differently. From the perspective of one observer, they’ll be the ones at rest, and their definition of a “second” is the correct one: a round-trip for that light, to go from the bottom to the top to the bottom of the clock, that defines their passage through time. For anyone in motion relative to them, that additional motion means that those external, moving clocks appear to run slow, as their light will appear to require a longer path to rise to the top, reflect, and return to the bottom of their clocks. A light clock will appear to run differently for observers moving at different relative speeds, but this is due to the constancy of the speed of light. Einstein s law of special relativity governs how these time and distance transformations take place between different observers. However, each individual observer will see time pass at the same rate as long as they remain in their own reference frame: one second-per-second, even though when they bring their clocks together after the experiment, they’ll find that they no longer agree. Credit: John D. Norton/University of Pittsburgh The reason for this is that motion through space and time are connected and inextricable: woven together into a fabric known as spacetime. The maximum “motion through time” you can possess is what you experience when you’re at rest with respect to the Universe, which corresponds to when your motion through space is zero. If you do move through space, however, your motion through time must slow in response, which is why the closer you move to the speed of light, the less you age and the slower you experience the passage of time. This has led to a myriad of practical applications, from global positioning systems (GPS) to high-energy particle physics. It’s under these conditions that we’re compelled to look at another part of Einstein’s E = mc²: when you’re in motion, your energy isn’t just given by your rest-mass energy, which is the mc² contribution to your energy. Instead, you also have kinetic energy: the energy that arises due to motion. Whenever two objects collide, whether they stick together (inelastically) or bounce off of one another (elastically), it’s the kinetic energy that they possess, based on their motion relative to one another, that determines how fast they’ll each wind up moving after they crash into each other. This “energy of motion,” or kinetic energy, is essential to the physics of objects in motion, from billiard balls to automobiles to planetary systems. In Newton s theory of gravity, orbits make perfect ellipses when they occur around single, large masses. The presence of other masses, like the other planets, cause these elliptical orbits to precess. However, in general relativity, there is an additional precession effect due to both the curvature of spacetime and the fact that the planets are in motion with respect to the Sun, and this causes the orbit to shift over time, in a fashion that is sometimes measurable. Mercury exhibits the largest such effect within our Solar System, precessing at a rate of an extra 43 (where 1 is 1/3600th of one degree) per century due to this additional effect. Credit: dynamicdiagrams.com, 2011, now defunct However, if you look closely, you’ll notice that Einstein’s most famous equation, E = mc², has absolutely no dependence on motion at all! If energy is simply mass multiplied by the speed of light squared, then how does motion factor into this? Where does the energy of motion for “kinetic energy” even come from? It becomes quite clear that E = mc², alone, can’t do the job. Perhaps an even more compelling argument that there must be more to the story comes if we consider light: a quantum of energy that has no rest mass at all. Light, whether we treat it as a wave whose energy is defined by its wavelength or a particle whose energy is quantized into packets known as photons, has no rest mass, so the m in E = mc² has to equal zero. But light carries energy, so E = mc² can’t be all that there is, or E would equal zero, too, which simply isn’t true. There’s a hint toward the solution if you took physics in High School or college, and learned about the “standard” formula for kinetic energy: KE = ½mv², where v is the speed of the object in motion. This formula only applies at speeds that are low compared to the speed of light: where v is much smaller than c, the speed of light in a vacuum. (That’s the same “c” that appears in E = mc²: 299,792,458 m/s.) This 1934 photograph shows Einstein in front of a blackboard, deriving special relativity for a group of students and onlookers. Although special relativity is now taken for granted, it was revolutionary when Einstein first put it forth, and it doesn’t even describe his most famous equation, which is E = mc², or his most famous advance, which is our current theory of gravitation: general relativity. Credit: public domain The reason that “kinetic energy” offers such a useful hint is because it leads you one step closer to the real key concept in completing Einstein’s most famous equation: momentum. Momentum is the “quantity of motion” that an object has, and it’s well-defined whether the thing that’s in motion is massive or massless, and, if it is massive, whether it moves close to the speed of light or not. Momentum, labeled with a p for very Latin reasons — arising either from the verb “pellere” (to push forcefully) or “petere” (to go) — is basically a measure of how much “Ooomph!” an object has to its motion, and consequently, how difficult it is to bring that moving object to rest. For massive particles moving slow compared to the speed of light, momentum can be well-approximated by the simple formula p = mv. For massive particles moving at any speed, even at a substantial fraction of the speed of light, momentum is more precisely written p = mγv, where “γ” is the Lorentz factor: 1/√(1-(v/c)²). And for massless particles, like light, that move at the speed of light and have no rest mass at all, momentum can’t be written in terms of mass, but can be written in terms of energy very simply, as p = E/c. The particle tracks emanating from a high energy collision at the LHC in 2012 show the creation of many new particles. By building a sophisticated detector around the collision point of relativistic particles, the properties of what occurred and was created at the collision point can be reconstructed, but what’s created is limited by the available energy from the kinetic energy of the colliding particles, with new particles capable of being created from that available energy, limited by Einstein’s E = mc². The maximum LHC energies are nearly a factor of a trillion (10^12) lower than the energies present at the start of the hot Big Bang. Credit: Panos Charitos/Wikimedia Commons user PCharito If we want to give the true expression for the energy inherent to any particle, then, we need to include the effects of its quantity of motion on energy as well as its rest mass’s effects on energy. E = mc², as simple, compact, and notorious as it is, only applies to massive particles at rest: a useful quantity only in certain special cases. Fortunately, there’s an almost-as-simple formula that incorporates both the rest mass energy of a particle, when present, along with the contribution of its quantity of motion to energy as well. That formula for energy is as follows: E = √(m²c⁴ + p²c²). Think about what happens in all the different cases that apply here. If momentum (p) is zero, then that last term goes away entirely, and you simply get E = √(m²c⁴), which just becomes good old E = mc² once again: Einstein’s original rest mass-energy equivalence equation. This is our sanity check: in the original special case, we do indeed get Einstein’s original “most famous” equation back. When a meson, such as a charm-anticharm particle shown here, has its two constituent particles pulled apart by too great an amount, it becomes energetically favorable to rip a new (light) quark/antiquark pair out of the vacuum and create two mesons where there was one before. A strong enough electric field, for long-enough lived mesons, can cause this to occur, with the needed energy for creating more massive particles coming from the underlying electric field, and with the amount of energy required to create these new particles (or particle-antiparticle pairs) described by E = mc². Credit: The Particle Adventure/LBNL/Particle Data Group What if we’re moving slow compared to the speed of light, and we just put in p = mv for momentum? Then the equation becomes E = √(m²c⁴ + m²v²c²), or, if we pull out an mc² from inside the square root, E = mc² * √(1 + (v/c)²). This might not look particularly familiar to you, but consider the following: this equation only works for values of speed, or v, that are slow compared to the speed of light, or c in this equation. Therefore, the part of the equation that reads √(1 + (v/c)²) is only going to be a little bit greater than one, because the (v/c) term is small. Whenever you have, in mathematics, an expression that’s √(1 + x), wherever “x” is small compared to 1, it can be excellently approximated by 1 + ½*x, with the approximation being better for smaller values of “x.” If we apply that to our expression for energy, we turn √(1 + (v/c)²) into 1 + ½*(v/c)², which turns our expression for energy into E = mc² * (1 + ½*(v/c)²), which then becomes, when we multiply the terms out: E = mc² + ½mv². This should make us very happy; it tells us that the total energy is the rest mass energy (the mc² part) plus the kinetic energy (the ½mv² part): the same equations that they teach you in an introductory physics class! This image shows a hole that was made in the panel of NASA’s Solar Max satellite by a micrometeoroid impact. Although this hole likely arose from simply a piece of dust, the “v²” term in the equation for non-relativistic kinetic energy (½mv²) can become very large, very quickly. For particles that move close to the speed of light, the effects of kinetic energy, or energy-of-motion, become even more severe when relativistic effects are taken into account. Credit: NASA When we have a massive particle that moves close to the speed of light, however, the approximation that p = mv can no longer be made with any sort of reliability; that approximation would lead to large errors. Instead, you simply have to calculate the full quantity for yourself, using the full equation, E = √(m²c⁴ + p²c²). As you begin to consider very large momentums, which is precisely the case that we deal with in our largest and most powerful particle accelerators, the rest mass term winds up contributing very little to the overall energy. At 99.999%+ the speed of light, the m²c⁴ term will be much smaller than the p²c² term in the equation, which means we can justify neglecting it. If we do neglect the rest mass term, then we simply get E = √(p²c²), which becomes E = pc: the equation for the energy-momentum relationship for photons and other massless particles. We sometimes call this the ultra-relativistic approximation, as it’s useful wherever the rest mass energy of a system is small compared to the energy due to motion; we can neglect that first term — the m²c⁴ term — even if the object moving ultra-relativistically isn’t truly massless. The longer a photon s wavelength is, the lower in energy it is. But all photons, regardless of wavelength/energy, move at the same speed: the speed of light. This is, surprisingly, irrespective of the motion of the observer relative to light; the speed of all forms of light is measured to always be the same for all observers. Credit: NASA/Sonoma State University/Aurore Simonnet What’s kind of remarkable about this story is the timeline: a key test of Einstein’s general relativity would arrive in 1919, when a total solar eclipse occurred. According to Einstein’s theory, the presence of a large amount of energy, all in one location in spacetime (the Sun), would bend and distort the path of all objects that traveled close to it. This included the light from background stars, which, although massless, would still follow the path created by curved space: the brand-new important concept that came onto the scene with general relativity. But what would the older theory that general relativity was trying to supersede — Newton’s theory of universal gravitation — predict? Some people insisted that it would predict zero deflection, since light had no rest mass and Newton’s theory relied solely on mass for gravitational attraction. But others recognized that photons still carried energy in the form of E = pc, and therefore, if you used the energy that photons had in place of where you would’ve typically used mass (i.e., if you substituted a photon’s E/c² in place of the Newtonian mass, m), you could actually predict a deflection for Newtonian gravity, too, with an assist from Einstein’s 1905 work. The fact that Einstein’s theory of gravitation predicted twice the deflection of even the Einstein-aided Newtonian value, and that general relativity’s predictions were indeed borne out by the observations, provided the key test that enabled us to verify and validate Einstein’s theory, leading to a revolution in how we understood the Universe. Yet the very ability to conduct such a comparison relied on Einstein himself, and the profundity of that simplified version of his most famous equation: E = mc². The results of the 1919 Eddington eclipse expedition showed, conclusively, that the general theory of relativity described the bending of starlight around massive objects, overthrowing the Newtonian picture. This was the first observational confirmation of Einstein’s theory of gravity. Credit: London Illustrated News, 1919 When you think of Einstein’s most famous equation, you should still recognize how profound the simple statement, E = mc², actually is. It tells us that every massive particle has an inherent amount of energy inherent to it, even when it’s at rest, and that its energy can never drop below that key value: mc². If you want to create a particle like it, you require at least that much energy; if you must create that particle along with its antiparticle counterpart, you require at least double that energy. And if you destroy or annihilate any massive particle away, all of that rest mass energy, all mc² of it, will become part of the energy that all of the “daughter particles,” or particles produced in the annihilation, carry away. But you should also recognize that E = mc² is only half of the full story: because particles exist not merely at rest, but they also move through the Universe. The quantity of motion that they carry with them, momentum, leads to a certain amount of energy-of-motion being associated with that particle as well. For slow-moving, massive particles, you can approximate that energy-of-motion with E = ½mv². For massless particles and ultra-relativistic massive particles, you can approximate that energy of motion with E = pc. But if you want the general case, where rest mass and momentum both are included, you need the full equation for the energy of a particle: E = √(m²c⁴ + p²c²). As famous as it is, E = mc² is only half of the full equation that’s needed to describe a particle’s energy. To get the other half, you have to remember that you cannot simply describe the Universe by taking a snapshot of it. It has the kind of beauty — and energy — that can only be captured if motion is included as well. This article was first published in May of 2023. It was updated in June of 2026. This article Einstein’s E = mc² is only half of the full equation is featured on Big Think.

There’s more than a whiff of Taken in Kenji Tanigaki’s exhilarating martial-arts movie, in which a handyman goes after some evil people traffickers It keeps happening: every few years, usually during a run of lethargic Hollywood spectacles, the Overton window of screen violence gets recalibrated by a muscular wonder from the east. Thundering along in the bloody footsteps of the Raid films and the Hindi punch-’em-up Kill, this martial-arts showcase from Japanese-born, Hong Kong-based director Kenji Tanigaki opens in generic dadsploitation territory. “Somewhere in Southeast Asia”, as a caption has it, mute Chinese handyman Wang Wei (Miao Xie) tears off after the traffickers who have nabbed his daughter (Enyou Yang). Having Hulk-smashed its way out of the Taken box, though, The Furious starts to crank it up. Boy, does it crank it up; the closing half-hour achieves a pummelling intensity unlikely to be matched by any other release this year.There are one or two plot developments. Cribbing from John Woo’s buddy movies, Tanigaki has his hero run into an undercover journo (Danny Dyer lookalike Joe Taslim) with his own reasons for chasing the traffickers; this route-one approach bears out the advantages of keeping things simple while turning the dial to 11. The complexities are reserved for the frame itself: jaw-droppingly limber, seemingly boneless performers pull off bruising manoeuvres on concrete floors, with Tanigaki’s well-placed cameras capturing unexpected delicacies and flourishes amid otherwise crunching dustups. It’s as if someone has brought a crossbow and a ballpeen hammer to the dance, and they’re intent on using them. Continue reading...