When Noether died, Einstein wrote in the New
York Times: “Noether was the most significant
creative mathematical genius thus far produced
since the higher education of women began.”
It’s a hearty compliment. But Einstein’s praise
alluded to Noether’s gender instead of recognizing that she also stood out among her male
colleagues. Likewise, several mathematicians who
eulogized her remarked on her “heavy build,” and
one even commented on her sex life. Even those
who admired Noether judged her by different
standards than they judged men.
Symmetry leads the way
There’s something inherently appealing about
symmetry (SN Online: 4/12/07). Some studies
report that humans find symmetrical faces more
beautiful than asymmetrical ones. The two halves
of a face are nearly mirror images of each other, a
property known as reflection symmetry. Art often
exhibits symmetry, especially mosaics, textiles and
stained-glass windows. Nature does, too: A typical
snowflake, when rotated by 60 degrees, looks the
same. Similar rotational symmetries appear in
flowers, spider webs and sea urchins, to name a few.
But Noether’s theorem doesn’t directly apply to
these familiar examples. That’s because the symmetries we see and admire around us are discrete;
they hold only for certain values, for example,
rotation by exactly 60 degrees for a snowflake.
The symmetries relevant for Noether’s theorem,
on the other hand, are continuous: They hold no
matter how far you move in space or time.
One kind of continuous symmetry, known as
translation symmetry, means that the laws of physics remain the same as we move about the cosmos.
The conservation laws that relate to each
continuous symmetry are basic tools of physics.
In physics classes, students are taught that energy
is always conserved. When a billiard ball thwacks
another, the energy of that first ball’s motion
is divvied up. Some goes into the second ball’s
motion, some generates sound or heat, and some
energy remains with the first ball. But the total
amount of energy remains the same — no matter
what. Same goes for momentum.
These rules are taught as rote facts, but there’s
a mathematical reason behind their existence.
Energy conservation, according to Noether,
comes from translation symmetry in time.
Similarly, momentum conservation is due to
translation symmetry in space. And conservation of angular momentum, the property that
allows ice skaters to speed up their spins by hugging their arms close to their bodies, emerges from
rotational symmetry, the idea that physics stays
the same as we spin around in space.
In Einstein’s general theory of relativity, there
is no absolute sense of time or space, and conservation laws become more difficult to comprehend.
It’s that complexity that brought Noether to the
topic in the first place.
Gravity gets Noether’d
In 1915, general relativity was a fascinating new
theory. German mathematicians David Hilbert and
Felix Klein, both at the University of Göttingen,
were immersed in the new theory’s quirks. Hilbert
had been competing with Einstein to develop the
mathematically complex theory, which describes
gravity as the result of matter curving spacetime
(SN: 10/17/15, p. 16).
But Hilbert and Klein stumbled on a puzzle.
Attempts to use the framework of general relativity
The laws of physics are
symmetric in space, time
and rotation. According
to Noether’s theorem,
those symmetries suggest that momentum,
energy and angular momentum are conserved.
Conservation of momentum Conservation of energy Conservation of angular momentum
A rocket launch converts chemical energy in
fuel into kinetic energy and potential energy.
Total energy remains constant because of
symmetry of time.
Spinning ice skater
A skater’s twirl speeds up when she pulls
in her arms. That’s because total angular
momentum must stay the same, thanks to
symmetry of rotation.
When one ball hits the row, a ball on the other
end flies outward, conserving momentum.
Why? Symmetry of space.