The LISA Pathfinder spacecraft launched a few days. While the LISA Pathfinder won’t be making serious physics measurements, it will be testing new technologies that are expected to be used in the final LISA experiment. LISA is planned to be an orbital gravitational wave experiment, similar to LIGO but in space and with a much longer arm distance. It will be sensitive to different types of gravitational waves than LIGO.

# Tag Archives: Relativity

# Hundredth Anniversary of General Relativity

Since this year is the hundredth anniversary of Einstein’s theory of general relativity, various websites have been publishing articles on Einstein. The New York Times has one from earlier this week on the history of general relativity that you should read if you’re interested in that sort of thing.

# Advanced LIGO Turns On

Advanced LIGO, the newest iteration of the LIGO experiment has finally turned on. LIGO is a gravitational wave observatory using extremely high precision interferometers to look for signatures of gravitational waves hitting Earth. Advanced LIGO is meant to finally have enough sensitivity to have a decent likelihood of finding gravitational waves within a reasonable amount of time.

Finding direct evidence of gravitational waves would be a huge discovery, as that would confirm an important prediction of the theory of general relativity, which states that mass warps the shape of spacetime. Events such as black hole mergers are thought to generate measurable gravitational waves due to the huge gravitational forces governing their dynamics.

# A Brief, Qualitative History of the Universe: Part II

In the second part of my brief history of the universe, I will cover the evolution of the overall properties of the universe.

## The Constituents of the Universe

As I stated in the first part, the presence of energy (including mass) affects the shape of space and time, allowing that shape to change. Just as smoothing out the details lets us deduce the options for the global geometry of the universe, smoothing out all the details of where energy is lets us see how that energy affects the global shape of the universe. The average energy density of the universe won’t cause any local effects on shape, so the only thing it can do is cause space to expand or contract. How this occurs is governed by the Friedmann equations.

It turns out that different types of energy affect the evolution of the universe in different ways. Everything we know about can be broadly categorized into four types of energy:

- Nonrelativistic energy (matter): basically all its energy is from mass. Its density just tracks the expansion or contraction of the universe in the most naive way possible: Double the volume, halve the density. Double the linear size, reduce the energy density by 2x2x2=8
- Relativistic energy (radiation): basically all its energy is from momentum. Its density tracks the expansion or contraction but it is also redshifted or blueshifted. In this case, if you double the linear size, the energy density is reduced by a factor of 16.
- Curvature: Not really an energy density, but acts like one. In this case, if you double the linear size of the universe the effective energy density is reduced by a factor of 4.
- Cosmological constant: An allowed term in Einstein’s field equations. Can be interpreted as a constant energy density throughout the universe. More typically called “dark energy.”

Matter and radiation tend to slow expansion or even cause accelerating contraction, curvature will tend to lead to a constant rate of expansion/contraction, while dark energy will lead to exponential growth (positive energy density) or contraction (negative energy density). What happens depends on the amounts of these different species.

## Our Universe

Recent measurements from the Planck satellite tell us that currently, the energy content of our universe is around 68% dark energy, 32% matter, with tiny amounts of radiation and no measurable curvature. These percentages change since the densities of the types of energy change in different ways with respect to changes in the size of the universe (ignoring for now the fact that something can change from one type of energy to another). Plugging this into the equations of general relativity, we get the following overall history of the universe:

- The early universe was radiation dominated for around the first hundred thousand years
- Radiation was then replaced by matter as the dominant form of energy
- Dark energy became dominant fairly recently – a few billion years ago. The expansion of the universe has been seen to be accelerating, a telltale sign of dark energy.

## The Even Earlier Universe

These models only work so long as we understand the physics involved. At some point, we need to merge general relativity with quantum field theory to make sense of what’s going on. There is no working theory of quantum gravity yet. String theory is the main candidate but a working string theory describing our universe hasn’t yet been found. Once quantum gravity is needed we can basically only guess. So, the Big Bang isn’t necessarily the beginning of the universe. It’s just the start of where physics we understand applies.

There are some hints of things that occurred before the radiation dominated era. Recall that we have a finite horizon. If we look toward the horizon opposite directions, what we find is that the universe seems too uniform. The temperature of the relics of the early universe is nearly the same everywhere and looks characteristic of a system at thermal equilibrium, yet particles near the horizon in opposite directions are outside each others’ horizons. They cannot have ever been at thermal equilibrium at any time that we can observe.

One way to solve this is to say that those points really were at thermal equilibrium in the very early universe. The universe then must have expanded so fast (even faster than the speed of light – this is allowed since nothing is ever locally moving faster than the speed of light, things aren’t really moving but space is dragging everything along) that the horizon actually shrunk. This extremely fast expansion is known as inflation and was proposed by Alan Guth and Andrei Linde. One model for inflation proposes that at some point in the very early universe a field giving a constant energy density (like dark energy) dominated, leading to an exponential expansion by many orders of magnitude over a tiny fraction of a second. The expansion made the observable universe appear very flat and uniform, regardless of the initial conditions. This field, the inflaton, then mysteriously turned off, leading to the standard cosmological history briefly described above. So, we just add an inflationary step prior to the radiation dominated era ensuring the apparent flatness and uniformity of the universe.

## The Fate of the Universe

From the Friedmann equations, we can see that there are several possibilities for what will happen in the end. If there is too much matter and radiation, the observable universe will slow down its expansion and eventually contract back to a point (the Big Crunch). In some cases, instead of a crunch, a Big Bounce occurs, where the size oscillates indefinitely. If conditions are just right, the universe’s expansion will just level off to a constant value. In this case, everything just keeps getting further apart and the universe more or less cools off and everything dies (Big Freeze). If dark energy continues to dominate, the universe may start expanding exponentially, as if it reenters an inflationary stage. In this case, the horizon eventually ends up shrinking to nothing. At some point the universe will expand so fast that everything will be torn apart (the Big Rip).

In each of these cases, our understanding of physics will eventually break down, leaving room for wild speculation of things like the generation of new universes, the decay of a metastable vacuum to another state, colliding branes (higher dimensional objects) and more.

# A Brief, Qualitative History of the Universe: Part I

Returning to my Physics for Non-Physicists series, I will temporarily switch to a new topic: cosmology. I’d like to talk about dark matter next, and a little knowledge about modern cosmology is necessary to understand some things about dark matter. As the title suggests, this discussion is meant to be qualitative and pretty brief. I may return to the subject in the future to discuss it in further detail, but that wouldn’t be for a while.

## Introduction: The Expansion of the Universe

Many early physicists traditionally saw the universe as a static, fixed thing. However, by some time during the 20th century, astrophysicists were able to measure objects many millions of light years away from Earth. While the Milky Way galaxy contains huge numbers of stars gravitationally bound to one another, and the galaxy is even bound to other galaxies in a galactic cluster, it turned out that on large enough scales everything appeared to be moving away from us, as if in the distant past an enormous explosion sent all these things away from us.

That is not the whole story, though. It also seemed like on large enough scales, everything is moving away from everything else. In fact, the rate at which the distance between two objects is increasing is proportional to the distance between the objects. The proportionality constant is called Hubble’s constant. So, it almost seems like in the distant past this motion was set in motion by a huge explosion that existed everywhere all at once (the Big Bang).

It turns out that this is what we would expect if instead of what we might consider an explosion (a violent chemical reaction), the recession of everything from everything else were actually due to an expansion of space. So what does this mean? Consider an empty toy universe, where you can step outside the universe and place two test particles at some known distance. Also let these two test particles be noninteracting and have no relative velocity. If space expands or contracts uniformly, the distance between the particles will expand and contract as well, even with no forces acting on the particles. This may seem at first glance to violate Newton’s First Law (the Law of Inertia). It doesn’t. Many of the laws of physics describe local phenomena. It only matters that on a small enough (even infinitesimally small) scale the particles look stationary.

This expansion of space was something that was predicted by Einstein’s general theory of relativity (GR), first released in 1916. GR describes space and time as dynamical, rather than static, objects. The equations governing the dynamics of space and time are intrinsically related to the distribution of energy in the universe. In GR, the geometry of space and time give rise to gravity, which exerts forces on energy, which affects the geometry of space and time.

## The Shape of the Universe

Then how do we go from GR (a local theory whose dynamics depends only on knowledge of what’s going on in some finite region of space) to talking about the history of the whole universe?

We first invoke the principles homogeneity and isotropy. Homogeneity means that on large enough scales (hundreds of millions of light years), everywhere in the universe looks the same. Obviously, we exist so homogeneity does not hold on all scales, but on truly cosmological scales we assume this to be true. Isotropy means that on large enough scales every direction looks the same as well. Thus, the universe has no preferred position and no preferred direction.

With these assumptions, we can easily deduce that on large enough scales, the small structures in the universe from the uneven distribution of mass/energy smooth out, leaving us with a universe that is smooth, homogeneous and isotropic. Everything else is just minor details. It turns out that there are then only 3 possibilities for the shape of the universe. Now consider a two-dimensional universe. In two dimensions, the 3 possibilities are a sheet (a flat/Euclidean universe), the surface of a sphere (positive curvature), or a shape similar to a potato chip (negative curvature). The positive curvature universe has a finite size, while a flat or negative curvature universe is literally infinite in size. Our universe must be shaped like the 3-dimensional analog of one of these.

It’s actually in principle possible to directly differentiate the 3 types of universes. The “shape” is characterized mathematically by what’s known as the metric tensor. The metric tensor defines the length of an infinitesimal line segment given some set of coordinates. The different geometries basically just have different (and non-equivalent) definitions of a straight line. In the 2D case, the line on a sheet is what we wold term a line. The “line” on the surface of a sphere is a great circle. If you could draw a perfect triangle and perfectly measure its angles, you would see that in a flat universe, the sum of the three angles is 180 degrees, in a positive curvature universe, the sum is greater than 180, and in a negative curvature universe, the sum is less than 180. Universes with non-zero curvature have some characteristic length scale governing the curvature. On scales much less than this scale, everything looks flat. As far as we can tell, the universe is flat, but this really just means that the characteristic length scale of the universe is very large.

## The Observable Universe

Remember that two of the three global geometries of the universe are infinite in size. You may also know that we can only look so far away (a few tens of billions of light years). It seems to us like the universe really is finite. How do we reconcile these facts without just declaring the universe to have positive curvature?

Our best estimates of the age of the universe tell us that the Big Bang occurred 13.8 billion years ago. Light in a vacuum travels at speed of light (obviously), so in those 13.8 billion years, any given photon can only have traveled so far. Looking far away is also looking backward in time, so we can’t see farther away from us than the maximum distance light could have traveled throughout the whole age of the universe. It turns out that this distance is actually larger than 13.8 billion light years. The light traveled that far, but the objects have been pulled away from us since the light was emitted. We also can’t really see light from the Big Bang as at some point the universe was dense and hot enough to be opaque to light.

Basically, as long as the universe – or at least the universe as we know it – has a finite age, we will have a finite horizon beyond which we can never see.

That’s enough for today. I’ll continue later with discussions of the overall history of the universe and the thermal history of the universe.