Guided reading · About 12.5 minutes · No equations
The mass-energy paper in fifteen minutes
Einstein's three-page paper of 1905, read without equations: what it claims, how its reasoning works, and the test it proposed.
The times are estimates for reading at a steady pace, and you can take as long as you like.
Step 1 of 8 · about 3 minutes
Two accounts of the same flash
Two people write before-and-after accounts of the same flash of light. Comparing the accounts removes what neither of them knows about the body's insides.
First encounter · No algebra required
Can a body lose energy of motion without changing speed?
A body sends equal flashes of light in opposite directions. One observer stays beside it; a traveler keeps moving past. Both write a before-and-after energy account for the same event. The body does not recoil. Compare their accounts without guessing the energy inside the body.
Authored teaching example, calculated when the site was built. Each energy unit here is one joule (J). Numbers use the modern SI constant set, not an observation or a reviewed source transcription.
The traveler keeps the same speed before and after: 60 percent of light speed. Equal opposite emission prevents recoil in this model.
Beside the body
Before: unknown
After: that unknown amount minus 10 J
Light: 5 J in each of two opposite directions.
Total light: 10 J
The traveler’s account
Before: another unknown
After: that unknown amount minus 12.5 J
Light: 2.5 J one way; 10 J the other way.
Total light: 12.5 J
Use the button, or drag the “Beside the body” card here. Both actions perform the same alignment.
Compare the accounts
- Each account conserves energy. The body loses exactly what its departing light carries.
- Align the two before entries, then the two after entries. The unknown body energies remain unknown.
- Subtract the losses: 12.5 J minus 10 J leaves 2.5 J. This is a difference between accounts, not an absolute body energy.
- With the unchanged-offset premise, the body’s energy of motion decreases by 2.5 J at the same speed.
Read the full worked example without controls (also works without JavaScript)
The unchanged-offset premise is assumed in this static worked route.
60 percent of light speed
- Each account conserves energy. The body loses exactly what its departing light carries.
- Align the two before entries, then the two after entries. The unknown body energies remain unknown.
- Subtract the losses: 12.5 J minus 10 J leaves 2.5 J. This is a difference between accounts, not an absolute body energy.
- With the unchanged-offset premise, the body’s energy of motion decreases by 2.5 J at the same speed.
1 percent of light speed
- Each account conserves energy. The body loses exactly what its departing light carries.
- Align the two before entries, then the two after entries. The unknown body energies remain unknown.
- Subtract the losses: 10.00050004 J minus 10 J leaves 0.0005000375031 J. This is a difference between accounts, not an absolute body energy.
- With the unchanged-offset premise, the body’s energy of motion decreases by 0.0005000375031 J at the same speed.
Relaxing the unchanged-offset premise leaves the difference between accounts known but the energy-of-motion change underdetermined.
Why the slower traveler matters
Compare a traveler at sixty percent of light speed with one at one percent. The slow-speed approximation must be tested at slow speed, not certified by the easier large-number example.
Same emission, two speeds. The exact-within-model drop and its low-speed approximation are different quantities. Kinetic interpretation assumes the unchanged offset. Traveler speed Rest-frame light Moving-frame light Exact drop Low-speed approximation 60 percent of light speed 10 J 12.5 J 2.5 J 1.8 J 1 percent of light speed 10 J 10.00050004 J 0.0005000375031 J 0.0005 J Same emission, two speeds. The exact-within-model drop and its low-speed approximation are different quantities. Kinetic interpretation assumes the unchanged offset.
60 percent of light speed
- Rest-frame light
- 10 J
- Moving-frame light
- 12.5 J
- Exact drop
- 2.5 J
- Low-speed approximation
- 1.8 J
1 percent of light speed
- Rest-frame light
- 10 J
- Moving-frame light
- 10.00050004 J
- Exact drop
- 0.0005000375031 J
- Low-speed approximation
- 0.0005 J
The large-number example makes the subtraction easy to see. To identify inertia, compare at low speed: the energy of motion then follows the ordinary speed-squared rule. The high-speed result cannot make that approximation exact.
Why are the body energies unknown?
Only the amount transferred to the light is specified. The boxes are not hiding a mass-times-light-speed-squared formula; no absolute body energy has been supplied.
Why do the two observers disagree about the light?
The light-energy transformation is an input borrowed from relativity. It gives different energies in different frames. Each observer conserves energy within their own account.
What is an unchanged offset?
The difference between the two accounts is interpreted as energy of motion plus an offset. This route assumes that the offset is the same before and after emission. Without that assumption, subtraction alone cannot isolate the change in energy of motion.
From two accounts to inertia
The new skill: Subtract two accounts of the same event to eliminate what neither account determines.
Why it helps here: This isolates a change without guessing how much internal energy the body had. The slow-speed comparison then identifies the change in inertia.
More guidance: energy of motion and inertia →
Less guidance: open these exact two-ledger settings in ME-01 →
Continue with the slow traveler in the coefficient laboratory →
Step 2 of 8 · about 1 minute
The paper's question
The paper's title asks its question: does the inertia of a body depend on its energy content?
Einstein announces that his recent paper on the electrodynamics of moving bodies leads to an interesting consequence, which this short paper derives.
Step 3 of 8 · about 2 minutes
The total energy of two flashes
Light carries energy, and how much depends on who measures it. Predict what happens to the total.
The equation-free version of ME-01 is not built yet, so this step asks its question here. The laboratory itself shows equations.
A body at rest sends out two equal flashes of light in opposite directions, and someone moving past measures the total energy of the two flashes. If the pair is sent out at a slant to the traveler's motion instead of along it, what happens to that total?
Open the answer you expect. Nothing is scored.
It stays the same.
The total is the same whichever way the pair is sent. Each flash's measured energy does change with the direction, but what one flash gains the other loses. The traveler's total is larger than the body's own measurement, by an amount that grows with the traveler's speed: it is the traveler's measurement of the same flashes, not extra energy made from nothing, and the next step uses it.
It gets smaller.
Following one flash alone, its measured energy can fall as the direction changes. Its partner's rises by exactly as much, so the total stays the same.
It gets larger.
Slanting the pair adds no energy to it. The two flashes change by equal and opposite amounts, so their total stays the same.
Step 4 of 8 · about 2 minutes
Energy of motion at slow speeds
The traveler sees the body moving. After the flashes, at the same speed, the body carries less energy of motion. Predict what that drop looks like at slow speeds.
The equation-free version of ME-02 is not built yet, so this step asks its question here. The laboratory itself shows equations.
Compare the body's energy of motion before and after it sends out the flashes, at the same speed. As the speed gets smaller and smaller, what does that drop look like?
Open the answer you expect. Nothing is scored.
Like the energy of motion of a body that lost a little mass.
After the flashes, at the same speed, the body carries less energy of motion, just as a lighter body would. The mass it has lost is set by the energy it gave off, and it is the same at every slow speed.
It falls away faster than the speed does, so there is nothing left to compare.
The drop does shrink as the speed shrinks, as every energy of motion does. Set beside the energy of motion itself, though, it does not vanish: it settles on one fixed amount of lost mass.
It stays the same fraction of the energy that was given off.
That fraction is not fixed; it shrinks with the speed. What stays fixed is the mass the drop corresponds to, which is why the comparison is with a lighter body rather than with the energy given off.
Step 5 of 8 · about 2 minutes
A sealed box
Where you draw the line around a system decides what loses mass. Predict the sealed box.
The equation-free version of ME-03 is not built yet, so this step asks its question here. The laboratory itself shows equations.
A sealed box holds a battery-powered lamp and a mirror. When the lamp lights up inside, does the box's total mass change?
Open the answer you expect. Nothing is scored.
It decreases.
The light never leaves the box. Energy moves from the lamp to whatever absorbs the light inside, and the total mass stays the same.
It stays the same.
Energy that only moves around inside a sealed system leaves its total mass unchanged. The lamp alone would lose mass, and whatever absorbs the light inside would gain the same amount.
It increases.
Nothing enters the box from outside, so its total mass stays the same.
Step 6 of 8 · about 1 minute
The paper's conclusion
The paper's conclusion, and the general claim it draws from it.
So a body that gives off energy as radiation loses mass, by an amount set by the energy it gives off, and Einstein argues that it does not matter that the energy leaves as radiation.
The general conclusion: the mass of a body is a measure of its energy content. When the energy changes, the mass changes in the same direction, and the paper gives the size of that change in grams for an energy in ergs.
Step 7 of 8 · about 1 minute
The test the paper proposed
How the paper proposes the claim could be tested, with its condition kept.
Bodies whose energy content varies strongly, such as radium salts, might allow the theory to be tested.
If the theory agrees with the facts, radiation carries inertia from the body that emits it to the body that absorbs it.
Step 8 of 8 · about 0.5 minutes
The one step that matters
The one step that turns the accounts into a statement about mass.
Seen by an observer moving past, the body carries less energy of motion after giving off light than before, at the same speed, just as a slightly lighter body would when the motion is slow, so, within the paper's stated premises, giving off energy lowers the body's mass by an amount set by that energy.
What you can now say
You can now say what the paper claimed: a body that gives off energy becomes less massive by an amount set by that energy, whatever form the energy leaves in, so the mass of a body is a measure of its energy content.