Recall how we combined information about the interval between two events.

- the time between two events is measured as
**t**, a scalar, with units of seconds - the space between the two events is measured
as
**(x, y, z)**, a vector, with units of meters

In general, observers in different inertial frames
will NOT find the same values for **t** and for **(x, y, z)**.

However, if we merge the variables into a single 4-dimensional quantity

then it turns out that one particular combination of space and time
will be **invariant:**

Q: What are the units of this interval, as written above? Q: Is it possible to convert this interval into different units; for example, can one express it in seconds? If so, how?

This quantity, which has units of seconds, is also invariant.

What does this quantity *mean?*
We can compute its value, and we can assign it units,
but what does it really represent?

Well, it has something to do with two events. Let's consider a special case: suppose that these two events are

- Fred goes to sleep
- Fred wakes up

According to Fred, these events occurred in
exactly the same place.
Therefore, the distance interval
**Δx**
between
them is zero, and,
according to Fred,
the result simply represents the time between the two events.
Because Fred was present at the location of the two events,
his measurements yield the **proper time** between the events:
that is the shortest interval of time between them that
any inertial observer can measure.

Let's look at momentum and energy in the same way. In this case, we are considering just a single particle at a time, not two particles.

- the energy of a particle is measured as
**E**, a scalar, with units of Joules - the momentum of a particle is measured
as
**(px, py, pz)**, a vector, with units of kg*m/s

In general, observers in different inertial frames
will NOT find the same values for **E** and for **(px, py, pz)**.

However, if we merge the variables into a single 4-dimensional quantity

then it turns out that one particular combination of energy and
momentum
will be **invariant:**

Q: What are the units of this interval, as written above? Q: Is it possible to convert this interval into different units; for example, can one express it in kg*m/s? If so, how?

This quantity, which has units of momentum (kg*m/s), is also invariant.

One can even compute a quantity which has units of kg, if one wishes.

What does it *mean?*
This expression has something to do with the properties
of a single particle.
In one special frame -- the frame in which the particle
is at rest -- this quantity is equal to the energy of the
particle, because, in that frame, the particle has
no momentum (it's not moving).
We could call this the **rest energy** of the particle.
If we divide the invariant quantity **ME** by **c^2**,
to convert it to units of kg,
then we can call it
the **rest mass ** of the particle.

The Blue Man throws a ball (**m = 0.2 kg **)
at a speed of **v = 0.5 c** to the right.

Q: What is the momentum of the ball, according to the Blue Man? What is the total energy of the ball, according to the Blue Man? What is the momentum-energy of the ball, according to the Blue Man?

But the Blue Man and the ball are all travelling across
the landscape at a speed of **w = 0.8 c** to the right,
as measured by the Red Men.

Q: What is the momentum of the ball, as measured by the Red Men? What is the total energy of the ball, according to the Red Men? What is the momentum-energy of the ball, according to the Red Men?

The Green Man, inside a very small Green Room
which fits completely inside the Blue Room,
happens to be flying to the right
at exactly **0.5 c**
relative to the Blue Man.

Q: What is the momentum of the ball, as measured by the Green Man? What is the total energy of the ball, according to the Green Man? What is the momentum-energy of the ball, according to the Green Man?

If momentum-energy is a conserved quantity, why don't we use it in "normal" physics? Why didn't you learn it in high school?

Well, let's try using it, and see what happens.

Fred throws an ordinary ball of mass **m = 0.2 kg**
to the right at **v = 40 m/s**.

Q: What is the momentum of the ball, as measured by Fred? Try to use the relativistic expression. Multiply the momentum of the ball bycto form the expressionp c. Write it down, too. What is the total energy of the ball, according to Fred? Try to use the relativistic expression. What is the momentum-energy of the ball, according to Fred? Try to use the relativistic expression.

As you can see, at ordinary, everyday speeds,
the rest energy of an object is much, much, much
larger than its kinetic energy,
and also much, much, much larger
than its (momentum times **c**).
Two different low-speed observers
will compute the same momentum-energy combination,
but, for practical purposes,
they will each end up with just the rest energy.

In ordinary life, mass, energy, and momentum do NOT interact strongly. That is why you probably learned three different conservation laws in high school physics:

- Conservation of mass
- Conservation of energy
- Conservation of momentum

In the relativistic realm, these three laws merge into a single rule, in which one particular combination of the elements of a 4-dimensional vector is invariant to all inertial observers.

Jane sits on Earth while her sister Sally
flies away on a rocket at **v = 0.9c**.
Jane sends a message to her sister, encoded in a laser beam.
When Jane fires the laser, it produces photons
with the usual reddish wavelength of **lambda = 640 nm**.

Q: What is the energy of each photon? What is the momentum of each photon? What is the momentum-energy of each photon?

Sister Sally collects the laser beam with her telescope and decodes the message. Of course, she doesn't measure the same wavelength as her sister -- she measures a much LONGER wavelength.

Q: What is the wavelength of each photon, according to Sally? (hint) What is the energy of each photon? What is the momentum of each photon? What is the momentum-energy of each photon?

Is the momentum-energy of a photon invariant, too?

- See Taylor and Wheeler, Spacetime Physics . In Chapter 7, they discuss relativistic expressions for energy and momentum, which become entwined in a quantity they call "momenergy."

Copyright © Michael Richmond. This work is licensed under a Creative Commons License.