## Understanding Einstein’s Mass-Energy Equivalence Equation (E=mc^{2}). Are Mass and Energy Equivalent?

Einstein’s mass-energy equivalence equation (E=mc^{2}) shows that mass and energy are equivalent (but not “exactly the same”) properties of a physical system. In more formal terms, the equation says: that for bodies in rest and motion, the total energy (**E**) in a physical system is equal to its mass (**m**) times the speed of light squared (**c ^{2}**).

### Mass-Energy Equivalence For Systems in Rest and Motion

Although it is true that mass and energy have equivalence, one has to consider the difference between systems in rest and systems in motion.

E=mc^{2} is a shorthand equation that works for bodies at rest as written, however it needs to be expanded on to work for bodies in motion (which makes sense, we need to consider another variable, so we need to expand the equation).

With the above in mind, Einstein’s mass-energy equivalence equation (E=mc^{2}) can be understood two ways:

**For systems at rest (with kinetic energy from momentum factored out)**: The total energy of a system (of one or more particles) is equal to energy stored in matter as potential energy (**rest-mass**, intrinsic mass, true mass of a system at rest) times the speed of light squared (why splitting in atom causes an explosion, all that potential energy is released in a moment).**For systems in motion (with both potential energy and kinetic energy from motion considered)**: In the words of special relativity, using a slightly more complex version of the equation E^{2}=(pc)^{2}+(mc^{2})^{2 }(which adds in “p” momentum, a type of motion, but for educational purposes E=mc^{2}can represent), kinetic energy (energy from motion) adds to the potential energy and mass of an object, because kinetic energy acts like mass at high speeds (**relativistic mass**, extra mass from motion).

**TIP**: Learn more about the different types of mass, learn about why some say this Energy–momentum relation is more correct than E=mc^{2}, or see a more technical version of the discussion here.)

Or, in Einstein’s own simple but essentially correct words, “the mass of a body is a measure of its energy content”

**The bottomline**: With mass and energy’s relation to “momentum” accounted for (see special relativity), we can safely say that mass and energy are equivalent, but not “exactly the same”, properties of a system that can be considered together as “**mass-energy**.” So mass-energy is mass, kinetic energy, and potential energy considered together.

So, in summary E=mc^{2} implies:

- Mass represents potential energy bound to a physical system as true intrinsic “rest” mass, and also relativistic mass from speed for bodies in motion.
- If a physical object speeds up, mass increases (adding to the intrinsic rest-mass of a system) due to kinetic energy acting as mass at high speeds (relativistic mass in special relativity).
- All measures of energy will always be proportional to the constant speed of light, because light
*is*energy (what we call “light” a visible portion of the electromagnetic spectrum). - Discussions on momentum aside, since mass and energy have equivalence (especially mathematically) we can consider mass and energy (potential and kinetic) together as mass-energy for most purposes.
- Even though mass and energy have equivalence, they aren’t “the exact same thing” (why it is mass-energy, and not “just energy”). Energy can be conserved as mass (and vice-versa) but not “converted”. Just like fat (energy stored in your body) and a candy bar can be described in terms of potential energy, and a candy bar can be conserved as fat, but one can’t magically convert a candy bar into body fat directly. A candy bar and body fat have equivalence in terms of energy, but they aren’t “exactly the same thing”.

These simple facts have many deep meanings. Below we will go into detail on each point we made to help clarify exactly how mass-energy equivalence does and doesn’t work. Try watching the following videos for a different frame of reference.

Einstein’s Proof of E=mc². Describing mass-energy equivalence and special relativity. The PBS video below is better, but this video and the one below are a little less heady.

E=mc² is Incomplete. Describing mass-energy equivalence for relativistic mass, i.e. with momentum “p” (a type of motion) added in.

**TIP**: E=mc^{2} describes the fact that mass is a measure of potential energy, a fuller relativistic energy equation E^{2}=(pc)^{2}+(mc^{2})^{2 }describes what happens when we increase “p” momentum. Generally all these equations are going to be “short hand” versions of the true full equations, and can be expressed a number of ways (some of which we describe below, see additional examples).

“We can explain everything about the universe (aside gravity and nuclear force) by understanding electromagnetic energy. Even the laugh of the audience when I say that statement can, if our theories are correct, likely be explained this way.” – Paraphrasing Richard Feynman speaking on Quantum Electrodynamics (the study of energy on the quantum level, and thus relevant to Einstein).

**FACT**:Einstein didn’t actually say E=mc^{2}, Einstein wrote down the equation (as roughly) m=E/c^{2} in his 1905 paper “DOES THE INERTIA OF A BODY DEPEND UPON ITS ENERGY-CONTENT?” Physics and math are languages that describe natural phenomena. There are many ways to say “the mass of a body is a measure of its energy content.” Einstein himself was more a thinker than a math whiz. He was, in some ways, only trying to solve special relativity as a paradox described by Henri Poincaré. Here we can say “although there are lone wolves and great men, most ideas, inventions, and discoveries are a collective effort, even if only indirectly”.

## Mass-Energy Equivalence, in Simple Terms

The above covers all the basics, but the concept is heady, so lets look at some other factors and examine the details from a few angles to tie everything together.

In very simple terms, [almost] everything that isn’t true empty space is energy in some form (including mass, motion, charge, light, heat, etc), thus it is no surprise that kinetic energy can act as mass and mass can be conserved as kinetic energy in physical systems (systems of one or more particle).

Our bodies are constantly conserving (not converting) mass as energy, and energy as mass, as we burn and intake calories. A star or another physical system isn’t much different. Einstein’s equation proved this phenomena with mathematics (building on the work of Newton, Maxwell, Lorentz, etc), and in doing so proved something fundamental about our universe.

The Kugelblitz: A Black Hole Made From Light. Since, on one level [almost] everything is energy and energy can act as mass, we should theoretically be able to create a black hole out of pure light. This “Kugelblitz” might not seem to have much to do with Einstein, but the concept actually demonstrates the basics of mass-energy equivalence in action in a unique and simple way.

### Mass-Energy Equivalence in Simple, but Formal Terms

In more formal terms, Einstein’s mass-energy equation is a simplified equation that shows that the potential energy of a system of one or more particles (mass) is equivalent in value to the total kinetic energy of the system (energy) using the speed of light (the max speed of energy) squared as a conversion factor.

This equation works because both mass and energy are measures of the same thing, “energy content”. This can be explained by realizing that energy, mass, and light all arise from the same fundamental forces, namely: electromagnetic energy.

Electromagnetic energy stored as potential energy is “essentially” what we call mass, electromagnetic energy used for acceleration of objects with mass is relativistic mass (special relativity), mass and motion curve spacetime (gravity, general relativity), and light speed is the constant and only speed pure energy can travel in a vacuum (light speed).

All known matter and energy can be described as quantum particles moving in wave-like fields at light speed exhibiting the measurable properties of mass-energy and motion, and all of that can be explained by understanding electromagnetic energy and using equations related to the shorthand equation E=mc^{2}.

That probably sounds more complicated than it is (working on a simple explainer over time here), try watching the following video for a visual and reading the clarifications below.

The Real Meaning of E=mc² | Space Time | PBS Digital Studios. This series is a must watch for anyone interested in physics. There is no simple way to explain mass-energy, but the concept is oddly simple once you get it (sort of like binary computing). No mathematics or physics background is needed to understand the basics, it just takes a little spacetime and mass-energy.

**TIP**: Einstein said, “if you can’t explain it simply, you don’t understand it well enough”. Feynman said, “you won’t understand it, and I don’t understand it either.” The goal isn’t perfection, it is a constant journey toward deeper truths.

**TIP**: When we consider the way motion affects things it is called special relativity, when we consider how gravity affects things it is called general relativity.

### The Physical Constants and Mass-Energy

We can prove mass-energy equivalence on many different mathematical and physical levels by comparing mass and energy to the universal physical constants (meaning things that don’t change in the physical universe). These constants are derived from the nature of “mass-energy” itself. This includes: light speed, the maximum speed of energy in the universe, the constant and only speed light can travel in a perfect vacuum; the Planck constant, the minimum measurement unit of the universe, including the minimum jump between “energy states”; and gravitational force, the constant push and pull between two bodies with mass.

## What Does Einstein’s Mass-Energy Equivalence Equation Mean?

To summarize the above again (this time in more detail), mass-energy equivalence has many meanings, including:

**Literally**, the equation says: For bodies in rest and motion the total energy (**E**) in a physical system is equal to its mass (**m**) times the speed of light squared (**c**). Total energy is equal to potential energy times the constant speed of pure energy.^{2}**Mathematically**, the equation shows that we canunits of mass into energy, or energy into mass, using the speed of light as a conversion factor.*convert***In Physics**, the equation shows that energy and mass can beas each other (not “converted”). In systems of one or more particles mass can be conserved as energy, either as kinetic, or unbound energy or it can be conserved as mass or as potential, bound energy. Given this, we can measure all of the kinetic and potential energy in a system together as “mass-energy”.*conserved***Also, in physics**, the equation shows that energy, despite having no “intrinsic” mass, can add to the total mass of the system (this is called**relativistic mass**, more on that below). Learn more about how massless particles become massive.**Lastly**, the fact that this is true results in understanding that time and space are relative to the frame of reference.

**FACT**: A single massless particle doesn’t have mass, but it does have momentum.

**TIP**: Although it comes up in pop-culture a lot, time dilation (the fact that time is relative) is probably the most difficult concept to grasp as it relies on grasping nearly all other basic Einstein concepts first.

**TIP**: Bound = part of a system. Unbound = not part of the system. Think of it as being like a membrane around a cell, everything in the cell is “bound” within the cell wall. Learn more about systems here.

### All Elementary Particles Have the Properties of Mass-energy and Motion

Einstein’s equation is very useful within the rule-set of our universe. The elementary particles (the building blocks of the universe) only have a few properties. All these properties can be described in terms of mass-energy and motion (and in some ways just motion).

Properties of elementary particles include: mass, charge, frequency, spin, etc. Those properties are what allow particles to exchange one of four forces: gravitational, electromagnetic, strong nuclear, and weak nuclear. Those forces are the only forces in the universe, so this is important.

When elementary particles bind in systems by exchanging forces, kinetic energy (from charge and motion) is “bound” in the system as “mass” (potential energy). In other words, the effects of all the interactions (forces) of elementary particles, and the systems they form can be measured in terms of mass-energy (E=m…) and motion (E…c^{2}, and m if you count the motion inside the system).

Other ways to calculate the energy content of a system include:

- E (total energy) = KE (kinetic energy) + PE (potential energy).
- If we consider momentum (p), a type of motion, this equation is also true: E
^{2}=m^{2}+p^{2}. - If we consider another type of motion, frequency (
*f*), we can calculate the energy content of a massless particle:*E*=*h**f*(where is Planck’s constant).

**TIP**: To think of it simply, when energy travels at light speed it’s pure energy (energy), when it does anything else, like say bounce off a Higgs Field, it’s potential energy (mass). Light speed is the speed of energy, the Planck length is the smallest distance energy quantizes to, and gravity is the constant affect of mass on larger scales. We can use Einstein’s equation, these constants, and a few relativity related equations (like Lorentz and Maxwell) to understand a large portion of the nature of the universe.

### Is Everything Motion?

In loose words (the thought experiment kind, and not cold hard fact kind) **we can describe this all (everything except empty space, gravity, nuclear force, although in some ways including these things) as motion**.

Something like: **total possible motion in a system = potential motion times the max speed of motion in the physical universe squared**.

The concept being that all particles are at their core are massless energy particles (what we call photons and gluons in the standard model), and mass, charge, spin (angular momentum), linear momentum, frequency, velocity, acceleration, and all other fundamental properties of these particles and their systems can be described in terms of motion. Motion doesn’t exactly describe “everything”, but like Feynman implies in the lecture series quoted at the type of the page, it comes pretty darn close. Physics is the basis for chemistry and there are only four fundamental forces, considering special and general relativity, the concept of “everything being motion” has just a little more explanation cooked in than simply saying “everything is mass-energy” (although the statements are pretty similar).

### Mass and Energy are Conserved, not Converted

Einstein’s mass-energy equivalence equation does *not* say mass and energy are the exact same thing or that they are “two sides of the same coin are one coin”, or that things are literally “converted” to become each other in real life. Mass-energy cannot be created or destroyed, and the term conversion implies creating or destroying. Rather mass and energy can only be conserved as forms of mass, energy, and motion in interactions. There is nothing “to convert.” Mass-energy is always mass-energy acting as either mass or energy.

Learn more about the laws of conservation.

## What Does E=mc^{2} Mean for Systems in Rest and motion? – Mass-Energy, In More Technical Terms

Above we brushed over the implications of the fact that the universe can be described in terms of mass-energy and motion. Below, we will discuss exactly how motions affect things in a little more detail, specifically we will now be considering **relativistic mass** and how that is different from **rest-mass**.

In more technical terms, E=mc^{2} is a shorthand version of a few different mass-energy equivalence equations that can calculate mass-energy equivalence for bodies at rest and in motion. So to be clear, this is a shorthand equation, but it is usable. It’s a little like saying “π” instead of saying explaining the math behind finding “the ratio of a circle’s circumference to its diameter.”

**For systems at rest (and in motion) E=mc ^{2} means**: the total energy (

**E**) in a physical system is equal to its mass (

**m**) times the speed of light squared (

**c**).

^{2}Understanding the difference between rest and motion is big understatement of the tricky part of the equation, so per Einstein’s request, let’s cover a system in “relative rest to an observer” first because “the laws of physics are the same in all inertial reference frames.”

**For systems at rest** the above terms can be described as a **Physical system** = any object in the universe consisting of one or more particles, **Mass** = the intrinsic mass of a system at rest (rest-mass), **Energy** = the total energy of a system at rest (rest-energy), and **Speed of Light** = Speed of light in a “vacuum” (the speed at which massless energy particles like photons travel when unimpeded by other particle fields, and the “speed” that “cause and effect relationships” actually happen at in the universe).

In other words, measured from a resting frame, we can calculate the total mass-energy of a body and show energy is equal in value to mass (times the speed of light squared). Or simply, we can use the equation to find the mass-energy of a system at rest, which can be thought of as its “true” mass-energy.

The true mass-energy of a “bound” system at rest never changes, but in real life mass changes due to speed and gravity. We call this relativistic mass-energy. So we have to address that.

**TIP**: If you are confused, forget about systems in motion for now and focus on the content above, especially the video. Alternatively you can learn about understanding the different types of mass here.

### Systems in Motion: An Introduction to Relativistic Mass

As a system’s speed increases, exponentially more energy is needed to accelerate the rest-mass of system. We know this from Newton’s F=ma (force equals mass times acceleration).

If we want to factor in momentum (particularly the acceleration of a system with rest-mass), we need to call our mass “relativistic mass” and our energy “relativistic energy.” In mathematical terms we have Kinetic Energy acting as mass = relativistic Mass x **c ^{2}** – intrinsic Mass x

**c**. All we are doing is subtracting the “real mass” from the “energy acting as mass”.

^{2}In other words, when we increase speed due to acceleration, we count the extra energy needed to accelerate the object as relativistic mass (which is just kinetic energy acting as mass. This only becomes apparent at very high speeds, but it has very real effects on physics).

**Since this relativistic-mass isn’t “bound” by the system, it’s not part of the systems “true” rest-mass**.

**Example**: If I make a snowball, and roll the snowball down the hill, the extra snow isn’t part of the original snowball. In the analogy, the extra snow is relativistic mass, and the original snowball is rest mass. We see the extra snow creates more mass, but this is really just extra kinetic energy from acceleration acting as mass.

Simply put, relativistic mass-energy increases with speed and gravity, but rest-mass never does (for this reason rest-mass is also called invariant mass). Both types of mass work with **E=mc ^{2}**, but extra math is needed depending on the type of equation.

The True Nature of Matter and Mass | Space Time | PBS Digital Studios. This video does an amazing job of explaining mass-energy and how it relates to matter in my opinion.

## Mass-Energy Equivalence Summary

To summarize this all up.

Quantum particles have properties of mass (potential energy) and energy (kinetic energy). When they bind in systems, kinetic and potential energy are used for binding. The “bound” system now has a greater mass (potential energy and bound kinetic energy). If we measure that system at rest, we get rest-mass. If we measure that system in motion, we get “relativistic” mass which counts the additional kinetic energy in the system used for the acceleration of the mass. If we factor out momentum, we are left with just rest-mass. Thinking about this, it shows that the rest-mass of the system’s parts is less than the rest-mass of the system as a whole. This is because when particles interact the energy “binding” them is potential energy, which curves spacetime, and can the affects of this can be measured as mass. Technically a system of two massless particles interacting creates a system with “rest-mass”, despite each part being massless.

That is just the surface of the doors that Einstein’s equation opens up. Check out the links below to take it to the next level.

**Spacetime**? In simple terms, a theoretical construct that puts time on a Y-axis and the three dimensions of space on an X-axis. Now imagine spacetime is a blanket and mass-energy is a bowling ball. Now imagine massless-light is shot across the surface of the blanket, notice how it would bend. Mass-energy curves spacetime, that creates the effect of gravity, and affects how massless particles like light travel. Learn more about gravity.

## Taking Mass-Energy to the Next Level

If I still have your attention, now is a great time to learn about the different types of mass, how systems of massless particles can be massless, the elementary particles, the forces, and time dilation and length contraction. All of these concepts are directly connected to mass-energy. Remember mass-energy is a core property of all physical systems, so there is plenty to learn about. You can also simply check out relativity in general (the study of how things like motion, time, and space are relative).

Lev Okun’s articles about this topic debunk Gabe Giz Perez. More YouTube nonsense. Mass and energy are not at all the same thing, as all particle physicists understand.

Good point, I didn’t realize it said this in the video (i’ll have to confirm).

On our page we don’t say “mass and energy are the same thing” (if they were then we wouldn’t need the Higgs field to explain how massless particles become massive for instance). We actually have a section that says “they aren’t the same thing”. They are two fundamental properties of systems and particles (not two sides of the same coin). From what I understand mass is a measure of potential energy mathematically, and equivalent physically in regards to energy content. Maybe like comparing heat and electricity or heat and motion, all measures of energy, but not really “the same thing”.

I’ll look up Lev Okun and see if I can glean any missing pieces. Always happy for more insight and clarity into subjects, especially when the reader has expertise and the subject is complex. Thanks!

Mike is correct! I agree with him and verify his statements which are supported by Lev Okun’s books

Gabe Perez goofed big time in the video. He even wrote the equation in the incorrect pop-science version…..

Einstein changed his view very significantly since his first papers. Poor choice…. Read what Einstein said post 1948. Einstein’s great discovery was the “rest energy.” Particle physicists today understand Eo=mc2 better than Einstein himself did.

In Eo=moc2 or Eo=mc2, (objects at rest, that subzero notation is essential and the way Einstein himself wrote it) the ONLY energy being put equal to mass times the velocity of light squared is the “rest-energy”. For objects in motion, energy is always greater than mc2. As Lev Okun stressed : There does N-O-T exist a *complete *equivalence between mass and energy.

I hope Gabe Perez-Giz rescinds his video. But I inow he will not due to his arrogance and ego.

Very interesting, I’ve certainly noted this in the article already (where I describe factoring motion in for the Energy–momentum relation). As for specifics on how this should all be phrased and to ensure i’m not missing something, I will have to look into it and check it against a few physicists who actively work in the field.

Maybe we could even get Gabe’s thoughts on it? You never know?

To clarify, you are all saying essentially what this article is, correct?: http://www.science20.com/hammock_physicist/whats_wrong_emc2

That “there is a mass-energy-momentum relation” and not “a direct mass-energy – period relation”? Not just that they are essentially describing the same thing, but that one is more detailed and accurate to the extent that the other isn’t?

That, without tons more research, doesn’t seem impossible off the bat. If momentum adds relativistic mass and the speed of energy is light speed, why wouldn’t momentum be a necessary key?

For me it almost seems like motion is the only thing actually happening in the universe. If the core of so much can be broken down to photons, and the only properties photons have are related to motion (very loosely speaking), why wouldn’t motion be at the heart of this?

But that is just musing, wouldn’t want to speak more on it without doing the additional research.

In the article I assume that, loosely speaking, E=mc^2 and E^2 = (pc)^2 + (m_0c^2)^2\ and Eo=mc2 are all ways of saying the same basic thing, only that one factors in motion and one speaks of systems at rest.

I’ll look to clarify this all for the article.

For now, to be clear, what we are saying is that mass and energy are equivalent, but since in motion energy acts as mass when an object increases its speed, momentum is an implicit factor.

Thanks for the insight.

The equation is half right and half wrong.

It is right when E is rest energy and m is rest mass.

It is wrong when E is total energy and m is relativistic mass.

In other words, the equation is correct if and only if velocity equals zero.

We agree (based on our research), and we think the article makes this clear.

That is, the equation is correct in a state of rest.

But in a state of motion one needs to consider the relation of motion.

This is why the term relativistic mass is used, it adds in the factor of motion (momentum; P).

That motion acts as mass. So the relation between energy and mass (E=mc2) is short hand for a fuller equation which considers motion for systems of more than one particle in states of motion.