Itty Bitty Bang

 Another question this week from Papou: Since a Black Hole can continuously acquire mass (except those cases wherein it loses matter per S. Hawking), does it follow that those Black Hole’s Event Horizon is also continuously getting larger. If that were not the case and the Event Horizon continuously reduced its boundary, does it not follow that Black Hole would become a point mass followed by a Big Bang. If that were the case, then it would be irrational that there was only one Big Bang and we are the product of that singular Big Bang. It is more likely, then, that there may have been other Big Bangs and there are other Universes out in Space. Is there anywhere in space where the Red Shift is not consistent with our Big Bang; which would then imply that there may have been multiple Big Bangs.

I think you get my drift ..... basically I am saying:   “Can a Black Hole become a Big Bang? What is the latest Red Shift evidence?

There are a couple different issues at play here, so let's address them one by one. First off, the event horizon of a black hole: A black hole is a region of space where matter has become so dense, light cannot escape its gravitational pull. The size of that space, called the Schwartzschild radius, is proportional to the amount of mass inside it:

where G is the gravitational constant, M is the mass, and c is the speed of light. You can actually find this yourself by looking for when the escape velocity is equal to c. This radius is sometimes called the event horizon, since in Special Relativity, events are described as points in space and time that are observed through light. If light cannot escape the black hole, we cannot observe events within it.

That brings us to the next part of the question: What happens to a black hole over time? As the equation above states, the event horizon radius is directly proportional to the mass within it, so if it loses mass due to Hawking radiation, or gains mass due to objects falling it, the radius can shrink or grow, but for fixed mass, the event horizon should stay fixed. For small black holes, Hawking radiation can eventually reduce the mass to zero, which is believed to result in the black hole evaporating. As the black hole shrinks, it will cross between the theories of General Relativity, and Quantum Mechanics. In their current forms, these theories are incompatible, but it's believed the evaporating black hole would release a burst of gamma rays as it vanished.

Still, there is a connection between event horizons and big bangs: In 2013, a group of scientists proposed that our universe could exist as the event horizon of a black hole in 4 spacial dimensions. In our 3 spatial dimensions, an event horizon is the surface of a sphere, which is 2D. A 4 dimensional black hole though would result in a 3D event horizon. Of course, that implies the possibility of a 2D universe on the event horizons of our universe.

Finally, the connection to red shift: The universe is expanding at every point, which means every point is moving away from every other point. I often find it helpful to imagine a big rubber sheet being stretched outward; any two points drawn on the sheet will get farther apart. As light moves through the universe, its wavelength gets stretched too, making it "redder", i.e. lower frequency. If you point a radio telescope at an empty part of the sky, as Arno Penzias and Robert Wilson did in 1965, you'll find a constant signal in the microwave band of light, called the Cosmic Microwave Background (CMB). This light is distributed in the blackbody spectrum, the range of photons emitted by objects of a given temperature. That temperature is from 380,000 years after the Big Bang, when things had cooled enough for protons and electrons to combine into hydrogen, about 3000 Kelvin. Over the billions of years that light has travelled, it's been red shifted down to around 2.725 Kelvin, in the microwave range.

If you look at a picture of the CMB, you may notice that it's not entirely uniform:

NASA

These anisotropies are mainly due to gravity pulling particles into clumps, which cool differently. Some have suggested the CMB also contains evidence of "bruises" from collisions between our universe and others existing in a larger multiverse. However, no such collisions have been detected so far.

Thanks for another great question, Papou!

Frog Blast the Vent Core!

 [Title thanks to the Marathon game series.]

We've (mostly) moved in to our new home in Gainesville, FL! This week's post is actually inspired by something I've noticed about the house's A/C. A few of the vents are mounted near the ceiling, and I can feel where the cold air is hitting the ground a few feet away:

This wouldn't be too surprising for something heavier than air, which would follow a parabolic trajectory, but I assumed the air resistance of air would be pretty high. Since cold air is denser than warmer air, I supposed I could try to find the change in energy as the cold air dropped. Using the gravitational potential energy, we can imagine swapping a bit of cold air for a bit of warm:

where m is the average mass of an air molecule, d is the vertical distance between the two bits, g is the acceleration of gravity, and ρ 1 and 2 are the number of air molecules per unit volume for the cold and warm air respectively. Using the Ideal Gas Law, we can write

where T 1 and 2 are the temperatures (in Kelvin) of the two gases. We can combine these two equations, and use

to get the force on the bit of cold air:

What this says is that the force on the air will be a fraction of the normal gravitational force, determined by the ratio of absolute temperatures. Since F = ma, we can cancel the mass density terms, and find that the acceleration is simply g times 1 minus the temperature ratio.

Let's put some numbers to this to see what kind of ratio we might expect. According to this site, air conditioners typically cool the air in your house by 16-22°F at a time. If we suppose the house is at 75°F when the A/C turns on, we can expect it to put out air at around 55°F. Converting these to Kelvin, we have 297 K and 286 K. That means the cold air will be dropping at a rate of 0.037 * g = 0.36 m/s^2. Assuming our ceilings are about 10 ft high, the air will hit the ground after 4.1 seconds. Without air resistance, it would take only 0.79 seconds for an object to fall, but I had imagined the air mixed long before it ever got to the ground.

We still need to know the speed of the air coming out of the A/C. This site gives the total capacity of a 1-ton A/C unit as 400 ft^3/minute (I have no idea what the tonnage is for ours, but let's go with 1). The vents are about 1 ft x 0.5 ft, and there are 7 in the house, so we get a horizontal speed of about 0.58 m/s. That means in the 4.1 seconds of descending time, the air goes about 2.4 meters from the vent. Without resistance, that would be 46 cm! Often in physics we begin a problem by neglecting friction, air resistance, and/or higher order terms, so it was interesting to take a closer look at a situation where that's not possible.

Not So Oblious

Marika and I have been getting ready to head down to Florida this coming week, and while paying a farewell visit to her grandparents, Papou posed another of his fascinating questions: Given the Earth is an oblate spheroid (flattened at the poles and bulging in the middle), what direction does gravity point relative to the surface?

If we take a cross-section of the Earth through the poles, we get an ellipse, which is given by the equation

where a and b give the half-width and half-height of the ellipse respectively. We want to try varying these two values to see what happens to the gravity. If we assume a constant density, we can keep the total mass constant by fixing the area of the ellipse:

We can also consider the eccentricity of the ellipse, given by

Using these, we can rewrite a and b as

Newton's Law of Gravity says the force between two masses is

where m1 and m2 are the two masses, r is the distance between them, and G is Newton's constant. For our elliptical planet, we can add up little bits of mass throughout the volume to make a complicated integral:

where ρ is the mass density and A is the space covered by the ellipse. Initially, I thought I could avoid evaluating this nasty object by using Gauss's Law, which would suggest that the pull was always toward the center, but the ellipse doesn't offer the necessary symmetry. I decided to prove this to myself by throwing the integral at Python to solve for me:

The red line goes to the origin, and the black arrow points in the direction of gravity, as found from the integral above. You can see that for large eccentricities, the arrow begins to diverge. We can plot the angular difference:

We get up to around 12° difference, which suggests that an assumption of radial gravity won't cut it. Unfortunately, the numerical integration is both slow, and has inaccuracies for certain points. That leads to some uninformative plots:

Here, theta is the angle from the x-axis, and the horizontal gravity is in arbitrary units. The integrator has problems when it gets to the edges of the ellipse, which leads to some of the force arrows going nuts. If we disregard the deviation discussed above, and assume the gravity really does point toward the center, we can get a much smoother graph, which roughly matches the true solution:

As a check on this, it's often useful in Physics to consider the most extreme case (e.g. zero or infinite mass, length, energy). For this system, we could imagine a planet that was flattened into a disk. The mass ends up concentrated in the center, tapering off toward the edges. If you stand in the center, gravity pulls straight down, but moving away, it would begin to pull back toward the center. If you were standing on the edge of the disk, the center would be straight down, so there's no horizontal force. This leads to the 4 zeros on the plot above: the centers of the top and bottom, and the right and left edges. Thanks for another great question, Papou!