Help me visualize this, boundary of the future

In summary, the boundary of the chronological future of a subset S in a spacetime (M, g_ab) is an achronal, 3-dimensional embedded manifold. This can be visualized by imagining a sphere with null geodesics emanating from it, forming the boundary of the chronological future. This boundary is a submanifold and not obvious. An argument by Proposition 6.3 and Definition 25 states that if a point x is on the boundary of the future of S, and a point y is in the future of x, then y is also in the future of S. This ensures that the future of the boundary is not also on the boundary.
  • #1
JasonJo
429
2
Let S be any subset of M, where (M, g_ab) is a spacetime.

Can you guys help me kind of visualize why the boundary of the chronological future of S is an achronal, 3 dimensional embedded manifold?

I am just having a hard time seeing why this is so. I'm picturing a sphere, and then having all the null geodesics emanating from it. The null geodesics form the boundary of the chronological future. Why is this thing achronal? Why is it a submanifold?
 
Physics news on Phys.org
  • #2
I can visualise it only for various flat space scenarios. I googled it and these guys give an argument which seems to make sense, and they also say it's not obvious (Proposition 6.3, and Definition 25): http://books.google.com/books?id=zJ5rPOpiKjYC&printsec=frontcover#PPA85,M1

I think it's roughly like this. Suppose x is on the boundary of the future of S, and suppose y is in the future of x. Then x is in the past of y. The past of y is open, so there must be some open region surrounding x that is in the past of y. Because x is the boundary of the future of S, every open region surrounding x contains a point z in the future of S. So y is the future of an open region surrounding x containing some point z in the future of S. So y, the future of x, is in the future of S. So we don't have to worry that future of the boundary is also on the boundary.
 
  • #3


Sure, let me try to explain this concept visually. First, let's define some terms. A spacetime is a mathematical model that combines space and time into a single entity. It is often represented as a 4-dimensional manifold, where each point represents a specific location in space and time. The metric g_ab describes the distance between two points in this manifold.

Now, let's imagine a subset S of this spacetime, which could represent any physical object or region. The boundary of the chronological future of S refers to the set of all points that can be reached from S by future-directed timelike or null curves. In simpler terms, it represents all the points that can potentially be affected by events in S.

To visualize this, let's use your example of a sphere. Imagine this sphere as representing the subset S in our spacetime. Now, imagine all the null geodesics (light rays) emanating from this sphere, forming a boundary around it. This boundary represents the chronological future of S. It is achronal because it contains only points that can be reached from S in the future, but not points that are simultaneous or in the past.

Now, why is this boundary a 3-dimensional embedded manifold? To answer this, we need to understand what a manifold is. In simple terms, a manifold is a mathematical space that locally looks like Euclidean space. In our case, the spacetime M is a 4-dimensional manifold, but the boundary of the chronological future of S is a 3-dimensional subset of M. This is because it is made up of null geodesics, which are 1-dimensional curves, and they form a 2-dimensional surface. So, the boundary is a 3-dimensional embedded manifold within the 4-dimensional spacetime.

I hope this helps you visualize the concept of the boundary of the future. It may seem abstract, but it is an important concept in understanding the behavior of spacetime and how it is affected by physical objects and events.
 

What is the "boundary of the future"?

The "boundary of the future" is a theoretical concept that refers to the limit of what is possible or likely to occur in the future, based on our current understanding of the world and its systems. It is often used in discussions about predicting future events or advancements.

Why is it important to visualize the boundary of the future?

Visualizing the boundary of the future allows us to better understand and prepare for potential future developments and challenges. It also helps us to identify gaps in our current knowledge and areas where we need to focus our research and efforts.

How can we visualize the boundary of the future?

There are several ways to visualize the boundary of the future, including using data analysis and modeling techniques, scenario planning, and creative thinking exercises. It is important to consider multiple perspectives and sources of information when visualizing the boundary of the future.

Is it possible to accurately predict the boundary of the future?

While we can make educated guesses and projections about the future, it is impossible to predict with 100% accuracy. The boundary of the future is constantly shifting and evolving as new information and advancements are made. However, by visualizing and preparing for potential scenarios, we can better navigate and adapt to future changes.

How can visualizing the boundary of the future impact decision making?

Visualizing the boundary of the future can help inform decision making by providing a broader perspective and considering potential future outcomes. It can also help identify potential risks and opportunities, allowing for more strategic and proactive decision making.

Similar threads

  • Special and General Relativity
Replies
6
Views
1K
  • Special and General Relativity
Replies
1
Views
977
  • Special and General Relativity
Replies
2
Views
1K
  • Special and General Relativity
Replies
1
Views
499
  • Special and General Relativity
Replies
1
Views
2K
  • Topology and Analysis
2
Replies
38
Views
4K
  • Special and General Relativity
Replies
8
Views
1K
  • Topology and Analysis
2
Replies
54
Views
5K
Replies
10
Views
2K
  • Special and General Relativity
Replies
5
Views
4K
Back
Top