Such a remarkable day with the announcement of the detection of primordial gravity waves from the earliest moments of the Universe. The nature of the detection may open up a lot of unexpected experimental physics.
The tensor fluctuations write quantum gravity on the sky - Liam McAllister earlier today
I'll stay away from details as seriously better writers aided by artists are probably on the task, but I had promised a continuation of pi day and Einstein's birthday.
The mention of Einstein conjures images of an odd looking older gentleman, relativity and a certain equation central to nuclear power. Reality is a bit different and much richer. There was relativity, but two flavors - special and general. The famous equation E = mc2 springs out of special relatively with a straightforward bit of reasoning. Special relativity forced a rethinking of how space and time are measured. General relativity came later and is a more dramatic leap. In it gravity is a property of spacetime.
I'm going to do something I swore I'd stay away from when I started the blog - I'm going to go into a bit of physics Not deeply - if you've seen Newton's gravitational force equation in high school or college you should be ok. I'll stay away from calculations and derivations. The task at hand is to show why the irrational number π is important in the general theory of relativity. If this seems a bit intimidating feel free to check out the recipe and find something of greater interest.
Let's start with Newton's equation for gravity: F = Gm1m2/r2. A very simple construction, it just says the force of gravity between two objects is the the product of their masses times some constant divided by the square of the distance between them. While you can hurt yourself falling, the force is very weak. You have a lot of mass and the Earth has a huge amount. Every bit of the Earth is pulling you towards it, but it is easy to jump and counteract that pull for awhile. The distance term is useful to think about - if you are twice as far from something the force is only a quarter as strong, if you are ten times farther it is a hundredth as strong.
Look up Einstein's equation you get something that looks a bit more intimidating:1
Conceptually it isn't too bad. All it is saying is gravity = 8 π G times (energy and momentum), or the amount of gravity is proportional to the amount of energy and momentum times some constant. The 8π part is a bit suspect - after all, why not incorporate it in a constant? But it turns out the G is the same gravitational constant that appears in Newton's equation. Einstein was nodding to history and making the connection clear.
So π is important, but why?
Let's go back to Newton and look at things a bit differently. Newton was bothered by the idea of action at a distance. How could one body effect another remotely? In physics 101 you can just ignore this, plug in numbers and get useful results. In fact for almost anything in engineering Newton's formula is just find. Einstein didn't replace it, but rather extended it adding a deeper understanding of what is going on.
In physics people play with fields. Think about the weather. You can imagine every point in the sky has its own temperature. This is just a field - a number attached to every point in space. You can talk about how this changes from one location to the next and can consider other measurables like the the velocity of the wind. Now each point in space has a number (speed) and direction (say northeast) giving the velocity at each point. Lots of fun and the basis for studying weather. A field of simple numbers is a scalar field, if you have vectors it is a vector field. Things can get more complicated (tensor fields), but just think of a field is some value attached to every point in space.
Not too bad so far, eh?
There is a gravitational potential field (don't worry about what potential implies) that is usually called Φ. It fills space and if you are around some mass M the field is just:
The gravitational constant shows up and the field decreases as you move away from it. If the mass is big, the field is big. You probably have noticed the square of distance is missing. The reason is this is just a field. To calculate the force you look at the rate of change of the field with distance - force is the derivative of the field (trust me on this one). Doing that brings in the 1/r2 term. Conceptually this is sweet. The bizarre action at a distance is now replaced by a field that is changing as you get closer or farther away.
There is still one more thing to consider. At this point we're just concerned with a single mass - not terribly practical in the real world. There are generally a lot of masses rather than a point mass and each has its own field. The trick is to use a mass density - the amount of mass per unit volume or ρ. Without derivation there is an equation that relates the gravitational field to the mass density - Poisson's equation:
The fancy pants triangle is a gradient operator. It is all about how the field changes in three dimensions. But before doing that note that Newton was all about forces between two objects. Now we want to sum everything up, but at a distance r. In three dimensions that just means you need to calculate the area of a sphere. The area of the square is proportional to 4πr2, so a factor of becomes part of the expression.
Einstein was working with fields rather than forces at the fundamental level. With Einstein keeping Newton's gravitational constant the area of the sphere surrounding the mass enters into the equation through Poisson's equation (trust me). Einstein's equation is way too hairy to detail further here - it has the geometry of space time embedded in it and calculations get difficult.
One of the features of science is while new discoveries sometimes completely overthrow what was known, more frequently they just add to it - often extending it. It is sometimes nice to think of the progress of science as sedimentary. In physics much of what we use comes from the classical physics you learn in physics. Fortunately for driving a car, building a skyscraper or flying an airplane you don't have to go much deeper. To get useful descriptions of other parts of the world around us you may require more modern and deeper understandings. The trick is to use an approach that is good enough for the job and not so unwieldy that you can't figure out how to use it.
To think about the Universe as a space willed with fields, Einstein needed a bit of pi.
And all of us, including Einstein in his day, can enjoy a little pie - so go ahead.
1 In reality it is much more intimidating. An undergrad physics major isn't ready for a general relativity course until she is in her junior or senior year and getting past the basics usually is grad school work and beyond.
Just go out and make or get some pie. Ice cream for it if you want. For the recipe this is what I made the other night - a super easy pressure cooker risotto. Not elegant, but delicious and vegan - but a simple recipe. I'm a big fan of pressure cookers.
° 1 cup dry lentils, soaked overnight and strained
° 1 tbl olive oil
° 1 medium yellow onion chopped
° 1 celery stalk, chopped
° 1 tbl shopped parsley stems and leaves (about 2 sprigs)
° 1 cup arborio rice
° 2 mashed garlic cloves
° 750 ml vegetable stock
° put pressure cooker pot over a medium burner and when hot add the olive oil and onions. Sauté until soft.
° add the celery and parsley and continue the sauté for another minute or so.
° add the rice and garlic and sauté until the rice is pear colored (about a minute to 90 seconds)
° add the stock and lentils
° seal the pressure cooker, burner to high. When pressure is reached throttle back to maintain pressure and cook for 7 minutes.
° release pressure and open
° mix and serve with a splash of quality extra virgin olive oil