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September 11 - September 22, 2021
In reality, the phenomenon we refer to as ‘gravity’ is deeply tied up with the nature of space and time. This means that a modern understanding of gravity doesn’t just tell us how objects in the Universe move, it also allows us to understand the behaviour of the very space and time that make up the fabric of the Universe itself.
not just how they move, but the behaviour of space and time
Gravity is by far the weakest of the four fundamental forces that exist in nature—the others being the electromagnetic force, and the strong and weak nuclear forces.
four fundamental forces
the meaning of the phrase ‘all objects accelerate under gravity at the same rate’ is not a statement about the motion of objects in our immediate environment. Rather, it is a statement about what should happen to an object if it were to fall under the influence of gravity alone. That is, if all other interactions are suppressed, then all objects should fall at the same rate.
Under influence of gravity alone
Unlike Aristotle, Newton didn’t try and seek an explanation for gravity. Instead he quantified its effects, and in the process deduced physical laws that accurately described the motion of not only objects on Earth, but also the motion of the Earth itself, as well as all other bodies in the Solar System.
Quantify the effect. Newton's theory of gravity
in Newton’s mechanics a given force causes a heavy body to accelerate more slowly than a light one. Consider this together with the fact that, according to Newton, the gravitational force must increase with mass. In Newton’s theory these two things happens at exactly the required rate to cancel each other out. A body being acted on by Newton’s gravity, and obeying Newton’s laws of motion, must therefore accelerate at a fixed rate, independent of its mass.
Cancel out
Maxwell says that the person at the side of the track sees the light propagate at the same speed as the person on the train. That is, Maxwell’s equations imply that velocities do not add linearly.
Light speed. Maxwell. Velocity
Time is not a universal concept, unfolding at the same rate for everyone. Time is a personal thing, and depends on our relative motion, with respect to others. Likewise, space is not the fixed backdrop that we think it is. What we think of as distances, and the lengths of objects, are actually dependent on how we are moving.
Relativity
Einstein’s solution to this problem was even more amazing. He hypothesized that gravity, instead of being a force that simply pulled things through space, was the result of the curvature of space-time. The fact that massive objects were drawn towards each other was then, according to Einstein, just a result of those objects following the shortest paths they could in the curved space-time in which they existed.
Curve
In this new picture, the only thing responsible for the attraction of massive bodies is space-time itself (which has to be there anyway). This is the fundamental idea behind the General Theory of Relativity.
General theory of relativity
in Einstein’s theory, there is no external force called gravity; the motion of every object is just a result of the curvature of space-time. But all objects are moving in the same space-time, so all objects must follow the same paths. In other words, all objects must fall at the same rate, just as Galileo had observed.
Hmm. The reason for the same velocity
the surface of a sphere. The shortest path between any two points on the surface of a sphere is called a great circle (the equator is an example of a great circle—on the globe).
Great circle
Two great circles, indicating the paths that particles might travel in a spherically curved space, if no external forces act upon them. The lines are no longer parallel forever, but meet at a point.
Two great circles
The curvature of space-time is usually more irregular than the surface of a sphere, but the basic idea is the same.
Complexity in reality
Normally, your intuitive description of your own motion would be that you are stationary. But again this is only because of our slavish regard to the Earth as the arbiter of what is at rest and what is moving. Free yourself from this prison, and you realize that you are, in fact, accelerating. You feel a force on the soles of your feet that pushes you upwards, in the same way that you would if you were in a lift that accelerated upwards very quickly. In Einstein’s picture there is no difference between your experience standing on Earth and your experience in the lift. In both situations you
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Upward accelerating
In the latter situation it is the lift that is responsible for your acceleration. In the former, it is the fact that the Earth is solid that pushes you upwards through space-time, knocking you off your free-fall trajectory. That the surface of the Earth can accelerate upwards at every point on its surface, and remain a solid object, is because it exists in a curved space-time and not in a flat space.
Upward accelerating to keep you stable on Earth surface. This acceleration comes from the shape of earth
The freely falling skydiver is brought to Earth because the space-time through which she falls is curved. It is not an external force that tugs her downwards, but her own natural motion through a curved space.
as a person standing on the ground, the pressure you feel on the soles of your feet is due to the rigidity of the Earth pushing you upwards. Again, there is no external force pulling you to Earth. It is only the electrostatic forces in the rocks below your feet that keep the ground rigid, and that prevents you from taking what would otherwise be your natural motion (which would also be free fall).
Electrostatic force that keeps the ground solid
So, if we free ourselves from defining our motion with respect to the surface of the Earth we realize that the skydiver is not accelerating, while the person who stands on the surface of the Earth is accelerating. Just the opposite of what we usually think.
Opposite of intuition
It wasn’t really the cannonballs that were accelerating away from Galileo at all, it was Galileo that was accelerating away from the cannonballs!
Hmmm
it is the curvature of space-time that is responsible for it all.
Main. Remember
mass is the quantity that tells us how much force we need to apply to an object in order to make it accelerate by a fixed amount. It’s thought to be a property of the object itself. This is different to weight, which is the name of the downwards force that an object exerts on your hand when you hold it, and which would be different if you held the same object while standing on a different planet.
Mass and weight
the best test of whether mass depends on position is to look at how light changes colour as it travels through a gravitational field. The basic idea behind this test is that photons (particles of light) should lose energy as they escape from the gravitational fields of massive objects such as stars or planets.
Mass & position. Photon
More recently, atomic clocks have been used to probe the same effect. The idea behind these experiments is that beams of light are themselves, in some sense, like clocks. The colour of light is determined by the wavelength of the photons that make up the light, which is itself related to the frequency at which they oscillate. If we were to take these oscillations to be the basis of a clock, by saying that each oscillation is one unit of time, then we can see that the redshifting of light is equivalent to the appearance of clocks at different positions running at different rates.
Atomic clock. Interesting connection. Unit of time. Oscillation
All we have to do is put two clocks at two different heights and arrange for them to transmit the time they display via radio signals. The difference in the rate at which they appear to tick, according to the radio signals that are observed, is then entirely equivalent to the rate at which a light signal between them should be redshifted.
Height. Rate of time
We can therefore take two highly accurate atomic clocks and put one on a rocket while leaving the other at our feet. If the clock in the rocket transmits its time via radio signals, then we can compare this signal to the time displayed by the clock we kept beside us. They should, in general, be different: an effect known as gravitational time dilation.
Gravitational time dilation
The current limiting factors in these experiments are the tiny rumblings from the Earth’s shifting tectonic plates, together with the tiny gravitational fields from other nearby objects (including from the experimenters themselves!). Space-based experiments are being considered that would remove some of these difficulties and increase the accuracy of the experiments still further.
Limitation
The challenges involved in testing Newton’s law of gravity in the laboratory arise principally due to the weakness of the gravitational force compared to the other forces of nature. This weakness means that even the smallest residual electric charges on a piece of experimental equipment can totally overwhelm the gravitational force, making it impossible to measure.
Limitation. Why is it so weak though
There are a large number of effects that result from Einstein’s theory. Here I am going to limit myself to describing four of them. These are the anomalous orbit of the planet Mercury; the bending of starlight around the Sun; the time delay of radio signals as they pass by the Sun; and the behaviour of gyroscopes in orbit around the Earth.
4 Relativistic gravitational effects
According to Einstein’s theory of gravity, there should be two new effects that can be observed when a gyroscope is put in orbit around the Earth. The first of these is a change in direction of the gyroscope’s axis of rotation, as it orbits the Earth. This effect, known as geodetic precession, is due to the curvature of space-time around the Earth.
Geodetic precession
The second effect is known as frame-dragging, and is due to the rotation of the Earth effectively pulling space around with it as it rotates. This is an entirely new type of gravitational interaction, and so it is of great interest to see if it can be confirmed experimentally.
Frame dragging
The LAGEOS satellite network provided an observation of the frame-dragging effect by measuring the change in the orbit of its satellites as they went around the rotating Earth. The long-awaited gyroscope experiment was performed by a mission called Gravity Probe B, in 2011. The geodetic precession and frame-dragging effects were measured by this experiment with accuracies of about 0.3 per cent and 20 per cent, respectively. The accuracy of the corresponding results from the LAGEOS satellites are estimated at between 5 and 10 per cent. All results were once again found to be consistent with
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Google
in Einstein’s theory, gravity is due to the curvature of space-time. Massive objects like stars and planets deform the shape of the space-time in which they exist, so that other bodies that move through it appear to have their trajectories bent. It is the mistaken interpretation of the motion of these bodies as occurring in a flat space that leads us to infer that there is a force called gravity. In fact, it is just the curvature of space-time that is at work.
curvature of space-time
if a group of massive bodies are in relative motion (such as in the Solar System, or in a binary pulsar), then the curvature of the space-time in which they exist is not usually fixed in time. The curvature of the space-time is set by the massive bodies, so if the bodies are in motion, the curvature of space-time should be expected to be constantly changing. The scientific way to describe this situation is to say that, in Einstein’s theory, space-time is a dynamical entity.
space-time in motion. constantly changing
The water in the pond was initially in a steady state, but the stone causes a rapid change in the amount of water at one point. The water in the pond tries to return to its tranquil initial state, which results in the propagation of the disturbance, in the form of ripples that move away from the point where the stone landed. Likewise, a loud noise in a previously quiet room originates from a change in air pressure at a point (e.g. a stereo speaker). The disturbance in the air pressure propagates outwards as a pressure wave as the air tries to return to a stable state, and we perceive these
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the disturbance propagates outwards as waves
sound waves travel slightly faster in warm air than they do in cold air.
ahh. hence quietness in winter
In fact, what the gravitational wave is really doing in this example is changing the amount of space that exists in the directions that are transverse to its direction of propagation. This means that although the atoms in the gas might be closer together (or further apart) than they were before the wave passed through them, it is not because the atoms have moved, but because the amount of space between them has been decreased (or increased) by the wave. The gravitational wave changes the distance between objects by altering how much space there is in between them, not by moving them within a
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Wow. Interesting. The gravitational wave change the space inbetween?
As the wave passes through, it stretches the circle smoothly until it reaches a maximum deformation, at which point it stops and then reverses the process until the direction in which it initially stretched is at a minimum. This stretching and squashing in different directions is illustrated in Figure 9, and continues until the wave has passed.
Example. Ring
the effect of the gravitational waves was enormously exaggerated in order to try and make it easier to visualize their effect. In reality, the fluctuations in the shape of the ring should only be expected to be at the level of about one part in a hundred million trillion.
Small effect in reality
That is, if we made a ring of particles that was a 1,000km across, we shouldn’t expect its shape to change by more than a trillionth of a centimetre. This is obviously very difficult to detect.
Scale of this small effect
Much of the early history of this work involved what are now called Weber bars. These instruments, named after Joseph Weber of the University of Maryland, consisted of large cylinders of metal. They were about a metre wide, and a couple of metres long. The idea was that if a gravitational wave passed through the Earth, and hence also through the detector, then it might cause the bar to start ringing, like a bell hit by a hammer.
Weber bars. Google
On 14 September 2015 it detected, for the first time in human history, direct evidence for the existence of gravitational waves from colliding black holes. It’s almost impossible to overstate the significance of this event, which will probably go down in history as one the greatest scientific achievements of the modern age. So let’s consider it in more detail.
Google
Second, and just as importantly, the LIGO scientists knew what shape the wave should take. This allowed them to scan the data, in a process known as match filtering, to look for their signal.
Interesting point
Friedmann was a pioneer, but his work was not widely recognized at first. He was initially criticized by Einstein, who thought he was in error. Einstein later introduced an alternative model of the Universe, which he forced to be static by introducing a new term into his equations that he called the cosmological constant.
Cosmological constant
Normally the tennis ball will reach a maximum distance from the surface of the Earth, before it starts falling back down. In the period before this, however, when the ball is travelling upwards, it is still being acted on by gravity, and it is by using the equations that govern the gravitational force that we can calculate the properties of its motion, such as how fast it will be moving at any given time in the future. Two nearby galaxies are very similar to this. The galaxies may be moving apart, but the rate at which they move, and whether or not they will fall back towards each other, is
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Hmm
This is a perfectly good question, and the answer can again be given by considering the tennis ball. If instead of throwing the tennis ball upwards we launch it at high speed, from some super-powered cannon, then it’s possible it might never come back down to Earth.
Re-colapse or not. Escape velocity
If the rate of recession is too low, then the galaxies will eventually fall back towards each other, and the Universe will start to collapse. The rate of recession between galaxies is known as the Hubble rate, and the velocity required to make them fly away from each other forever is known as the critical rate.
Hubble rate. Critical rate
When an object is very far away, however, the delay can become significant. If the Sun suddenly exploded, for example, it would take more than eight minutes for us to know anything about it, because that’s how long it takes the light emitted from the Sun to reach us (and nothing can move faster than light).
8 minutes
In a sense, we can see back in time by looking far away, and if we look far enough away we can see what the Universe looked like when it was very young.
Now, it’s a well-known result in thermodynamics that when you compress an object (like a balloon full of air), it gets hotter. Likewise, if you make the same object expand then it gets cooler. The Universe is no exception to this rule. If we think of the expanding Universe as playing on a movie reel, then if we run the reel backwards we should expect to see the Universe getting smaller and hotter, until at very early times it bursts into flames. Now recall that we can in fact see the early stages of the Universe’s evolution, and you might expect that we should see a fireball if we look far
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Interesting
They showed that the collapse due to gravity expected from Einstein’s theory took place just as expected, and that the amount of radiation in the early Universe was compatible with that required by the primordial nucleosynthesis calculations. They also found, however, that there appeared to be a large amount of matter in the Universe that didn’t interact with radiation in any way, other than through its gravitational field. This isn’t how normal matter behaves.
Interesting
