Build Connections for Interstellar Engines
Inspired by The Portal, episode 20, with the host Eric Weinstein and the guest Sir Roger Penrose.
Make life multiplanetary
Today, we are going to talk about how to make life multi-planetary, but with two conditions: A. within our lifetimes and B. at a destination much cooler than Mars. Ask any astronomer. There are planets, plenty of them, as good as Earth. We’re going to see how to achieve that goal.
Journey to a distant star
A group of Earthers decide to embark on a journey to a distant star, in order to build a better future for their kids. And they were hopeful that speeding up, close to the speed of light, they’re going to experience time dilation and they will make it far enough within a few generations. And that’s due to the lorentzian metric.
In order to measure length and distances, they came up with a differential operator which measures the rise over the run. Since spacetime is curved, as the source of gravity, we can have measurements along a worldline and that’s how we can come up with a metric.
Einstein’s bad news!
Then, they hop on their rocket ships. But, Einstein hits them with the bad news. That if they’re going to take this idea further, when they make it there and come back to Earth we’re going to end up with two clocks that tick at different rates. And according to the postulates of quantum physics that can’t happen!
So, they take this in. They have a moment and then try to develop a more advanced theory. In a principal bundle, we can have transportations on the base base such that the orientation and the magnitude doesn’t change. But, then we’re going to need a more advanced differential operator in order to measure where we end up.
Taking a different path
Taking a different path helps us observe things in the universe that wouldn’t be observable otherwise. And that’s due to the phenomenon of curvature. And taking vectors in the fiber space and pulling them back to the base base we can have the Yang-Mills field where our operator works.
Now, what does this have to do with Penrose steps? Choosing a path and completing a loop, we’re going to end up higher or lower in the fiber space. And this is best illustrated by Penrose steps. You go up and up and up, and when you get back where you started in the base space you’re going to end up at a different point in the fiber space.
Visualizing different types of curvature, we have two types: the Weyl curvature and the Ricci curvature. The Ricci curvature describes gravity. But, the Weyl curvature also has a significance, of stretching the plane in one direction and compressing it in the other.
We’ve known since the time of Newton, that the Ricci curvature is responsible for the gravity that we experience. It has the effect of reducing volume and it makes geodesics get closer to each other. That’s how the gravity hole of Earth operates. And in order to compensate for that we need to speed up in the orbit.
Associated vector bundles
Then, there’s this other picture of the Yang-Mills field, where we have a principal bundle and we can associate with each point on the base base a vector space, and we call it an associated vector bundle. There, we can pull back the connection 1-form on the principal bundle and apply our differential operator.
The first person who proposed this idea was Maxwell. He connected the idea of light and electromagnetism. He used connection 1-forms and then, he built curvature 2-forms and it was the beginning of the Yang-Mills theory, which explains force carriers other than gravity.
Path dependence due to curvature
Now, taking a non-trivial path, due to curvature we’re going to displace within the fiber space, and it’s closely related to the path that we choose. So it’s important to note that in order to get out of the flatland we need to be “conscious” about our path.
The Aharonov-Bohm effect
In 1959, Aharonov and Bohm had this experiment setup where the electromagnetic field strength is zero. However, the connection 1-form has a very important role and so, that’s experimentally surprising.
In a principal bundle, we can have the effect of curvature visualized by the Hopf fibration. GDP is the market value of finished goods and services produced within a year in a country. It’s a good measure for comparing the standard of living between two countries.
Sensing the effect of curvature as the change in volume
If we take sections using the connection, we divide the total space of the principal bundle into the base base and the fiber space. Next, we can see the effect of curvature as decreasing or increasing the volume of curvature 2-forms. Therefore, we have something like the figure where the volume of each country changes based on the structure action.
Finding The Portal
Taking these ideas together opens a portal, where we’re going to have to find a very particular path in order to escape this Einsteinian prison. It requires building new mathematical and physical concepts, a more advanced connection.
When we move along say the border of a country, the boundary in the base base, when we get back to where we started we’re going to have a displacement in the fiber space. That is the quantity of the curvature 2-form which is related to the Penrose steps, the degree of Escheredness or Penroseness.
The plan to build an interstellar engine
Our earthers found the portal. Played with these ideas and then decided to come up with a more unifying theory where they can use the concept of curvature and find the non-trivial paths and build interstellar engines that operate based on new physics. So that was the plan.
Curvature 2-forms describe the degree that given any exotic section makes it possible to have a parallel transport. Curvature 2-forms give us information about how to course correct in order to keep the magnitude and the orientation of our velocity intact.
This is the most interesting idea in quantum physics. We have operators such as Pauli matrices. When when we go from the Lie algebra to the Lie group, say a three-sphere, we continuously apply these operators and see how the quantum particle propagates through spacetime. Applying an operator for 2π continuous rotation we’re going to end up with the negative identity operator, which is strange because in ordinary experience objects go back to themselves after a 2π rotation or 360°. But, we apply this operation again and we get our identity back.
Sensing the effect of curvature as a lens for light rays
Finding new hope, they try to understand the effect of curvature as a lens, having an effect on the light rays from the past to the future. The observer can see this best, where light rays come to the them through the celectial sphere.
To build such an interstellar engine, we play with these ideas and for example come up with things that come to us naturally, such as reinventing robotics. Because it’s all about rigid-body transformations and also taking advantage of the covariant derivative and the left-invariance property of Lie groups.
The difference between Weyl and Ricci curvatures
Spreading out into the universe, locally it’s about the Ricci curvature which has the effect of a magnifying lens for the observer. For example, you can see it by looking at the sun through a camera that can detect neutrinos. Nighttime would be a good time to observe that. But, the Weyl curvature has also a significant role.
Feeling the vanishing of the arrow of time
Taking the journey in such a vehicle has the feeling of, first there’s a sharp contrast between the past and the future. Then, as you speed up, you’re going to feel like the difference between the past and the future vanish, which is the arrow of time. By then, you have a much easier time to imagine complex lengths and distances.
These ideas come together in a theoretical framework best visualized by the Graph, the Wall and the Tome, the GWT project, which stands for: the Graph by Edward Witten, the Wall at the Stony Brook university and The Road to Reality by Sir Roger Penrose.
We see what’s in the Graph. There are three fundamental observations about reality. The first is the Einsteinian spacetime, which has four degrees of freedom. Second, we measure lengths, distances and angles and so we come up with the differential operator which is covariant. It compensates for the curvature. Third, at each point of spacetime we have the Yang-Mills field where we use the connection in order to explain force carriers other than gravity, for example electromagnetism. Next, we have associated vector bundles that are responsible for the existence of matter. Finally, sections of complex line bundles on these associated vector bundles are the right way to think about quantum physics in curved spacetime.
On the Wall we have the Yang-Mills field. And we see that the connection a plays such an important role. We also have curvature in the Einstein equation where there is Ricci curvature on the left side and matter on the other side, and these two relate by the phenomenon of curvature. Then, we have the experiment of Aharonov-Bohm where the gauge potentials are the crucial components. And the Maxwell equations that are the seeds that started this whole line of thinking. That’s how we can connect these theories and then think about what’s the most complex connection that can unify these ideas.
In order to build tools and automate away some of these concepts, to play with these ideas and find the right connection, we have sections in the Tome:
- Vector fields and 1-forms
- Manifolds and coordinate patches
- Parallel transport
- Covariant derivative
There are other subsections that are important in order to build the tools:
- Curvature and torsion
- Bundle curvature
- Lorentzian orthogonality; “the clock paradox”
- Maxwell field as gauge curvature
These are the most basic elements that we need in order to theorize about how to escape this planet.
Connecting gravity to electromagnetism
Connecting these ideas, on the left side we have gravity and on the right we have electromagnetism. There are analogous terms between the two. As the source of gravity we have matter and energy, E = mc², and we have as the source of electromagnetism the charge. We have the gauge fields in the electromagnetism and the analog to that would be the metric, which is which begs the question: is this the metric that’s primary or is it the connection? Nevertheless, as the degrees of freedom of gravity we have the Weyl curvature, which plays an important role since the early universe, and that’s best explained by the electromagnetic field strength in the Yang-Mills field.
Make a pull request!
How to contribute? Well, how about buildings stuff using Porta.jl and then submitting pull requests. If you’re not a user of Porta.jl, I don’t see how it makes sense to contribute, because ultimately it’s about the tools that we make.
What are curvature 2-forms geometrically?
Here, we argued about what is the curvature 2-form geometrically, which is denoted by F_A.
What are Ricci and scalar curvatures geometrically?
Also, we argued about what is R_μν and R geometrically, which are the Ricci curvature and the scalar curvature respectively. In order to make computations convenient, we take the trace of the Ricci curvature and break it into the trace-free and the scalar components.
How do they relate?
And these two relate by the connection. So we see that the idea of connection is very fundamental in order to connect these ideas and unify them.
What does this have to do with Penrose steps?
And the Penrose steps are the best visualization of connecting these two ideas together in a geometric way.
What are “Horizontal Subspaces” and what do they have to do with vector potentials and gauge fields?
Then, we demonstrated the the “horizontal subspaces” in a principal bundle where the connection determines how we move across these horizontal subspaces. Using parallel transport it’s inevitable that we are taken from one horizontal subspace to the other and so it’s also about the connection.
Not betting against Elon
This has been fun, and I look forward to have metrics that are complex in nature. This is not about betting against Elon. More power to his team. But, we’re going to do something different here. Then again, this is a Middle-Eastern kid that doesn’t know how to drive! So, until next time!
Please click on the image below to watch the statement video!