Thursday, September 27, 2012

Gravity - The Love Story I: Black Holes Are Not 'Universal' Drains

This semester I am teaching a conceptual physics class at LSU that uses minimal mathematics to understand how the Universe works.  Yesterday, we covered the chapter on gravity and my closing question to my students was, "What would happen to the Earth's orbit if the Sun were to become a black hole instantly?"  Assume that it simply changes in size from what it is now to how big a black hole with the same mass would be and the center of mass never changes.

I'm not going to make you wait...  Nothing would happen to the Earth's orbit!

This is one of the most dramatic examples of simply using an equation to tell a story that I have come across.  I suspect that much of the drama comes from the misconception that black holes WILL consume EVERYTHING, turning most people's mental picture of a black hole into a universal drain.

(I know the following analogy is a bit corny, but it makes the point that equations can tell stories and aren't just recipes to combine numbers into new numbers...)


In order to resolve this misconception, consider Newton's law of universal gravitation:

Now, don't worry overly that this is an equation because we will be making no calculations.  Instead, we are going to use it as a script for a play.  This play just so happens to be a love story... 


On the left side of "=" we don't have a character, but the ending of our story: F.  (This is the gravitational force that will be felt between two masses.)  We can also think of F as the attraction between our characters.  Therefore, the larger the attraction F, the better the 'Happily ever after...' ending.

The story is told by our characters on the right side of "=": G, m1, m2, and r:
  • G is a VERY small constant that is fundamental to the Universe.  That is, there is no way to derive its value from any theory, we simply determined this value from measurements.  Since G doesn't change, it is more of a background prop than a character; we don't need to worry about it since the moral of our story will be the same with or without it.
  • Next we have our two lovers: masses m1 and m2.  I call them lovers because they are attracted to each other (literally since gravity tends to pull mass together).  
  • Finally, we have our villain, r, who keeps our lovers apart.  (This is the distance between each of our lovers' center of mass.)  
That is the complete cast of characters in this story!  There are no extras milling around in the background.


When you multiply G, m1, and m2 together and then divide by r2 (which is equivalent to r*r), you are able to determine the ending to our story which is the attraction (F) between our lovers.  Now we are able to establish some plot points:
  • The more massive either of our lovers (m1 or m2) are, the more they will be attracted to each other.
  • The farther apart (r) they are, the less they will be attracted to each other; the bigger the number you divide by, the smaller your result.  (The square on r only serves to make the reduction in attraction between our lovers less even faster.  For example, if you double the distance between the lovers, you quarter their attraction.)


Now let's take a look at some of the more subtle plot points, specifically the properties that determine the attraction of our lovers (m1 and m2):
  • No unrequited lovem1 and m2 are always equally attracted to each other.  It doesn't matter if one is more massive than the other.  
  • Love is blind:  There is nothing in our script which describes the size or shape of our lovers.  Assuming m1 and m2 stay the same distance apart and their masses don't change, they will always be equally attracted to each other.  m1 will love m2 the same regardless of whether its mass is made up of dense muscle or voluminous blubber. 


Now that we have the script to our play, let's see how the ending turns out when we cast the Sun as m1 and the Earth as m2.  The scene opens the with Earth orbiting the Sun a fixed distance r away (this is called an astronomical unit, AU, and it is about 93 million miles).  We sit and watch the Sun and the Earth be attracted to each other, but the villain of distance keeps them apart.  In an attempt to overcome our villain, the Sun decides to implode on itself, sucking all of its mass into a ball less than about 3.72 miles across.  Now it is a black hole but, according to our script, the Earth felt no change since its love it blind!  The mass of the Sun didn't change and its center is still in the same place.  Drat, the Sun didn't succeed in increasing its attraction with the Earth!

~ FIN ~

♥  Stay tuned for the next installment of "Gravity - The Love Story"!  We will find out what properties our lovers need to have to come together (that is, what properties a mass needs to have to actually get "eaten" by a black hole).