© 2005 Richard Conn Henry

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Where does E = mc2 come from?

Richard Conn Henry, The Johns Hopkins University, Baltimore MD

he location of an event⎯one beat of your heart, say⎯is given by its three space coordinates, and its time coordinate⎯but, if we want to use the same units for all of those 4 numbers we will require a units conversion factor: if there are c centimeters in one

second, the conversion factor is c cm/sec, and the location of a heartbeat is given by the set of numbers (x, y, z, ct) , where all four quantities are now in units of centimeters.

Of course we don’t yet know the value of c, but − we would like to find that out!

The famous Pythagorean Theorem states that ds2 = dx2 + dy2 for a right triangle, where dx designates a length in the x direction. Let us GUESS that for two heartbeats, that are separated in our (at least apparently) four-dimensional universe by dx, dy, dz, cdt( ) , the Pythagorean Theorem expands to:

ds2 = dx2 + dy2 + dz2 - c2dt2 where we assume (that is, guess!) that there is a fixed separation ds between the two events.

Well, that is Einstein’s Theory of Special Relativity! That is all there is to it.

The minus sign creates a distinction between space and time. The justification for this whole thing is that it works⎯it predicts the atomic bomb.

Let us discover what our equation means. Send your twin off on a trip, at constant speed v, directly away from you, in, say, the x direction.

Consider two beats of your twin’s heart: these are two “events” that, we have hypothesized, are separated in spacetime by a certain fixed separation ds. If this is so, you, and your twin, will agree on ds, in this equation:

dx2+ 0

2+ 0

2! c

2dt

2= ds

2= 0

2+ 0

2+ 0

2! c

2d"

2

where the left-hand part is YOUR application of our Pythagorean Theorem to your twin’s two heartbeats (remember that your twin is moving only in the x direction), while the right hand part is your twin’s application of the Pythagorean Theorem (remember that both heartbeats occur, of course, right at (in!) your twin). The separation in time of the two heartbeats is d! for your twin, but … is dt for you.

Until Einstein came along, everyone thought that dt = d! , i.e. that there is only one universal time. But, if our guess about the Pythagorean Theorem in our world is correct …. well, let’s see! Our equation reads

dx2! c

2dt

2= !c

2d"

2

T

2

But: what do you measure as the speed v of your twin? It is v =dx

dt, or dx = v dt .

Inserting this,

v dt2! c

2dt

2= !c

2d"

2 or v2 ! c2( )dt 2 = !c2d"

2

v2 ! c2

!c2=

d"dt

#$%

&'(2

or 1!v2

c2=

d"dt

#$%

&'(2

from which, dt

d!=

1

1"v2

c2

or, if we define ! "1

1#v2

c2

, dt = ! d"

So, if the time between those two heartbeats is one second (according to your twin), the

time YOU experience between those same two heartbeats is gamma !( ) seconds: if v = 0, you get one second (just like your twin), but, if v = c , you get infinite separation between the two heartbeats. For intermediate speeds, you get time separations that are greater than one second: this is the famous time dilation, which was predicted by Albert Einstein,1 and which had never previously been suggested even as a possibility by anyone.

How do we test our Pythagorean idea? Easily! Before she departs, simply instruct your twin to stop after a while⎯and then to return to you, at that same constant speed. The result, as

she returns, is identical: while it is true that, now, v = !dx

dt, or dx = !v dt , the only way that dx

enters our equation is as dx2 = !v dt( )2

= v2dt

2 which is exactly the same as for the outbound journey. Thus, on her return, when you ask your twin how many times her heart beat on the trip, she will say a number that is less than your count of your own heartbeats: she will be younger than you, because not just her heartbeat but of course her entire physiology has been slowed.

The result of our test? This has been checked and tested repeatedly, and is darned well true! In particular, as long ago as 1971 an atomic clock was flown around the world⎯and on its return, its time was compared with that of another atomic clock that had been left on the runway: both dt and d! had been measured. Knowing v, the speed of the airplane, the experimenters could (see our equation) work out c. The result was

c ! 3!10

10cm s

"1, which you recognize as

the speed at which light moves. So, light moves at the maximum speed that is allowed by our “expanded” Pythagorean Theorem. Not at the speed of light!

Notice that were someone to suggest to your twin that she travel at some speed v > c , ! becomes imaginary, and from dt = ! d" she would have to assert that a real number dt is equal to an imaginary number ! d" , which is not true. That is why your twin cannot go “faster than light.”

That also tells you why gravitational waves “travel at the speed of light.” The fact is, as we have already said, even light itself does not move “at the speed of light.” Light, gravitational waves, and indeed anything that is massless, moves at the maximum speed that is (as we have seen) permitted by what we perhaps should now call the “Einstein/Pythagorean Theorem.”

So, in general, the coordinates (x, y, z, ct) of an event (such as the beat of a heart), are different for observers that are in motion relative to one another, since separations in space x and

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time t are different (we demonstrated that, above, for time, and for space, x, it is almost equally easy).

Our set of four quantities is called (and defines) a vector. Any other set of four quantities that changes, depending on velocity of observer, in the same way, is, by definition, another vector.

Clearly, if you simply multiply all the four components of a vector by a fixed number, or invariant, the result must be another vector. And clearly (dx, dy, dz, cdt) is a vector (just move your origin of coordinates to the journey’s starting “point-moment” (event)⎯this change of origin cannot affect anything).

Since all vectors change the same way, if all four components of two vectors are equal in one frame, they will remain equal in another frame that is moving at velocity v, so, we will have a law of physics! That is why we care about vectors: we need them, if we want to have laws of physics.

Note, above, that ds2 = !c2d"

2 so, since c is a constant and ds is invariant (by our hypothesis), d! is an invariant. (So ds is imaginary: your last heartbeat is an imaginary spacetime interval away. Nothing wrong—in fact Einstein himself chose to use the alternate ds

2= !dx

2! dy

2! dz

2+ c

2dt

2 , which would make the Moon, at this moment, an imaginary spacetime interval away from the Earth!)

An invariant is a quantity that does not change as your twin moves. It is very easy to identify invariants experimentally, if you keep in mind that relativistic changes are real: they actually occur. Your twin is really younger. Now consider, for example, your twin’s temperature T. If T were not an invariant, say if it were to increase (or decrease) with velocity, your twin would really cook or freeze. Or, the electric charge !q of your twin’s electrons: if that were to change with velocity, your twin would instantly explode, since her electrons move much faster than her protons. Or her mass m : if that were to increase with speed, if she got near the limiting velocity, she would become a black hole, which can’t be undone … but in the frame of every photon of sunshine, your twin is moving at the limiting velocity! She does not become a black hole: mass m is an invariant.

So, we can construct a new vector,

mdx

d!, m

dy

d!, m

dz

d!, mc

dt

d!"#$

%&'= mux , muy , muz , mc(( )

where we have defined ux!dx

d" and it, and v

x=dx

dt, are both velocities (hey, two times—two

velocities) of your twin.

Momentum, p, we (obviously, and wisely) choose to define to be mass times the velocity u , not times the velocity v , so our new vector is: p

x, p

y, p

z, mc!( ) , our chosen definition of

momentum being crucial to ensuring that this is a vector (and hence in accord with the Einstein/Pythagorean Theorem, the validity of which we now accept). Now (drum roll!) what is the fourth component of our new vector? That is, what is mc! ?

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Let’s find out!

mc! = mc 1"v2

c2

#$%

&'(

"1

2

! mc 1+1

2

v2

c2

#$%

&'(=mc

2

c1+

1

2

v2

c2

#$%

&'(=1

cmc

2+1

2mv

2#$%

&'(

Since we recognize 1

2mv

2 as (traditional) kinetic energy of your moving twin, we will take

E = mc2+ K = mc

2+1

2mv

2+ ... to be the twin’s total energy: p

x, p

y, p

z,E

c

!"#

$%&

is our new

vector! Should your twin not be moving, then her total energy is

the most famous equation in all of physics. And, so easy!

(To really understand the above, copy it out, every bit of it. It works!)

While we now have the famous equation, we do not yet have the atomic bomb, for which we need two additional items: conservation of energy, plus the observed fact that a U235 nucleus spontaneously decays into two daughter nuclei, the sum of whose masses m = m

1+ m

2is

measured to be less than that (M) of the original nucleus. Our U235 nucleus sits at rest before us⎯we apply conservation of energy to its decay:

Mc2= (m

1c2+ K

1) + (m

2c2+ K

2) = mc

2+ K and so kinetic

energyK = M ! m( )c2 appears, and because c is so large, great destruction results (if you have enough such nuclei, and if you have found the means to induce them to decay together).

We have now obtained all of the most famous, important, and startling results of the Einstein/Pythagorean Theorem. Notice that to obtain these results required nothing but high school algebra and geometry. Every high school graduate in the world should therefore know Special Relativity, and should know it backwards!

But there is one additional result, not nearly so well known,2 that is of crucial importance in understanding the present crisis of physics: the search for a “theory of everything” that will combine relativity and quantum mechanics together with all the forces of nature.

To get that additional result, we will need the Lorentz Transformations, which give the relations between the coordinates (x, t) of an event in our stationary frame, and the coordinates ( !x , !t ) of that same event in a frame that is moving with respect to us with constant speed v in the +x direction. Since the LT was derived long ago, all we need to do is to verify their truth. They are:

! x = " x # vt( ) and !t = " t #v

c2x

$%&

'()

hey! our old friend, ! "1

1#v2

c2

.

To verify the LT, first note that, for v = 0, (i.e., for ! = 1) this reduces to ! x = x and !t = t, as required, and then note, by differentiation (v being constant), that

d !x = " dx # v dt( ) and d !t = " dt #v

c2dx

$%&

'()

and, finally,

E = mc2

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verify by substitution that dx2 ! c2 dt 2 = d "x2! c

2d "t

2 ( the Einstein/Pythagoras Theorem):

d !x 2" c2d !t 2= # dx " v dt( )$% &'

2

" c2 # dt "v

c2dx

()*

+,-

$

%.

&

'/

2

= ! 2dx

2 " 2vdxdt + v2dt 2( ) " c2! 2dt

2 "2v

c2dxdt +

v2

c4dx

2#$%

&'(

= ! 2dx

2 " 2vdxdt + v2dt 2 " c2dt 2 + 2vdxdt "v2

c2dx

2#$%

&'(=

1

1"v2

c2

dx21"

v2

c2

#$%

&'(" c2dt 2 1"

v2

c2

#$%

&'(

#

$%&

'(!

Since c is merely a conversion factor for units, agreed since 1983 to be exactly 299-79-2458 meters per second (sometimes known as God’s Social Security number), physicists often use a system of units in which c = 1. In such a system of units, note the beautiful symmetry of the LT:

!x = " x # v t( ) !t = " t # v x( )

Also note that our two vectors become simply: x, y, z, t( ) and px, p

y, p

z, E( )

To “warm up,” let’s use the first LT to show something yet untested: that a rod of length !0

in the moving frame is shorter in our stationary frame: !x1= " x

1# vt( ) and !x

2= " x

2# vt( )

where we are careful, since the rod is moving, to measure both ends x1and x

2at the same time t .

We obtain:

!x2" !x

1( ) # !0 = $ x2" x

1( ) # $ ! so, in our frame, the rod is of length ! = !

0! .

Roger Penrose, of Wadham College, Oxford, in his wonderful book3 “The Emperor’s New Mind,” calls attention to our final, critical, feature of Special Relativity: that, even if v is very small (e.g. a brisk trot), relativistic effects can still be very significant. In particular, suppose v = 10 km/hour (brisk trot!) Then ! can be set equal to one. Let’s check that:

! "1

1#v2

c2

= 1#v2

c2

$%&

'()

#1

2

* 1+1

2

v

c

$%&

'()2

= 1+1

2

10 +105

3600

3+1010

$

%

&&&

'

(

)))

2

= 1.000000000000000043 ! 1.0

Suppose that the Andromeda galaxy (which is L = 2,500,000 light years away) is just

rising above the eastern horizon. What time !t is it in that galaxy? Let’s calculate it, using the Lorentz transformation.

!t = " t #v

c2x

$%&

'()! t #

v

c2x is our formula, and we have two cases:

! t = t +v

c2

L case: I walk east (Andromeda is coming at me, i.e., v = !10 km hr-1)

! t = t "v

c2

L case: I walk west (Andromeda is receding from me at v = +10 km hr-1)

Now, one light year is 1018 cm, pretty closely.

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So, v

c2L =

10 !105cm

3600 s!

1

3!1010cm s

"1( )2! 2.5 !10

6!10

18cm = 771,600 seconds, or 9 days

When you trot toward Andromeda, it is 9 days from now in Andromeda. When you trot away from Andromeda, it is 9 days ago in Andromeda.

In both cases, it really is. So, you can walk away from Andromeda, and the meeting that

the aliens are planning (to decide if they will come to Earth and exterminate us) has not yet taken place. Walk toward Andromeda, and their meeting is over, and their decision has been made. (And, if we have cleverly placed a bomb in their meeting room, with a “radioactive-decay trigger” that gives a 50% chance of the bomb going off during their meeting, why, then, the bomb has gone off, or it has not.) But now, walk away from Andromeda. It is, again, really nine days ago! The meeting has really not yet occurred! Walk toward Andromeda for a second time. Now, the meeting is over, and the bomb has, or has not, gone off.

That is Roger Penrose’s example! Surely this is why Einstein never believed that quantum mechanics (which governs whether the bomb goes off, or does not) could possibly be a complete theory. For it seemed clear to Einstein that, since you could force the comedy to play as many times as you wanted, simply by walking back and forth, it must be a set piece, and it must be determined that the bomb either does, or does not, go off − contrary to what quantum mechanics asserts. But! Einstein was wrong about this!

At least, that is what most physicists believe. Both quantum mechanics and relativity have passed every experimental test, but in the end, something must give. Most physicists are convinced that quantum mechanics is the more fundamental of the two, but, we have learned that nature cannot be taken for granted, and that surprises are frequent. That is why physics is the most fun subject in the curriculum!

Acknowledgements This work was supported by Maryland Space Grant Consortium.

References 1. Albert Einstein, Ann. Phys. 17, 891 (1905).

2. Ralph Baierlein, Am. J. Phys. 74, 193 (2006).

3. Roger Penrose, The Emperor’s New Mind (Penguin Books, New York, 1991).

PACS codes: 01.70, 03.30, 11.00

Richard Conn Henry is a professor of physics and astronomy at The Johns Hopkins University, where he is also director, Maryland Space Grant Consortium (a NASA program for STEM workforce pipeline strengthening).

Henry A. Rowland Department of Physics and Astronomy

The Johns Hopkins University, Baltimore, MD 21218