Exploring the Light Cone in Big Bang Clock by Roger Penrose

[starkind]

Perhaps someone with a lot of patience would help me.

I am watching Roger Penrose Clocks at the Big Bang 09/30/2008 on PIRSA

http://streamer.perimeterinstitute.ca/mediasite/viewer/NoPopupRedirector.aspx?peid=940fba57-4cf9-4659-9c2d-1324d45cf4e4&shouldResize=False#

At about 37 minutes slide 43 shows a light cone. The slide is headed Clock in Relativity. The statement I would like to better understand is that the metric g(ab) has 10 components, one of which is scale, the other 9 specify location of the light cone.

I think I get it that the metric g(ab) is a 3x3 matrix, with nine co-efficients. But I don’t get the scale component.

Thanks for any guidance.

Richard

 

[George Jones]

The statement I would like to better understand is that the metric g(ab) has 10 components, one of which is scale, the other 9 specify location of the light cone.

I think I get it that the metric g(ab) is a 3x3 matrix, with nine co-efficients. But I don’t get the scale component.

The metric is symmetric and 4x4, and thus has ten independent components.

 

[starkind]

like this?

1 5 8 10
5 2 6 9
8 6 3 7
10 9 7 4

how did you know it was a 4x4, and how could I have known that?
is this an augmented matrix? So that the first row vector would be something like
"a +5b +8c = 10d"?
How does this lead to position and scale?

****later:

On reflection, I think I see that it must be an augmented matrix, and that the 10d does affect the size, or scale, of the resulting space. In the sample matrix above, I just numbered the coefficients consecutively. But if I increase the 10d to 20d, the size of the space described increases, so the 10d co-efficient represents scale.

I am less clear on the position. I suppose abc in my matrix above is the usual xyz.

Thanks for any hints.

r

 

[HallsofIvy]

Physics is concerned with "events" which means something that happens at a specific position at a specific time: to identify an "event" requires that you use 3 position coordinates and 1 time coordinate. That is why vectors in relativity are 4 dimensional. A second order tensor, as is the metric tensor is represented by a 4 by 4 matrix.

No, it is not an "augmented" matrix and there is no system of equations associated with it. In Euclidean, 3 dimensional, space, the "differential of distance" is ds= \sqrt{dx^2+ dy^2+ dz^2} or ds^2= dx^2+ dy^2+ dz^2. In a general 4 dimensional geometry, it is possible to show that the formula is still quadratic but may involve all possible products of two directions: ds^2= g_{11}dx^2+ g_{12}dxdy+ g_{13}dxdz+ g_{14}dxdt+ \cdot\cdot\cdot. The "metric tensor" is represented by a matrix having those coefficients as values. It is also true that, because there is no "preferred direction", the coefficents are symmertric: a_{ij}= a_{ji}.

Now, a 4 by 4 matrix has 16 entries. 4 of them are on the main diagonal, 6 above that diagonal and 6 below. You can choose the 4 entries on the main diagonal to be anything you want. You can choose, say, the 6 above the main diagonal to be anything you want, but then the 6 below are fixed- they must be the same as the 6 above. That is you can choose any 4+ 6= 10 numbers you want and the other 6 are fixed.

 

[starkind]

I am very disappointed to hear that there is no system of equations associated with the g(ab) metric, just when I thought I had made a connection between g(14) and scale.

Do you really mean the main diagonal entries can be anything one wants? I suspect there must be more limits than that. For example, it would surprise me to learn that g(14) could be a p-nut butter toast. However the depths of my gullibility are such that I have to consider that option. I had thought, at least, that any a(ij) should be a number.

I think I understand that the vertex of the light cone is an event. The upper cone (the arrow of time is usually up in these diagrams, per Dr. Penrose) represents the expansion of a sphere at light speed from the event, which encloses the space (xyz) which could be affected in the future of the event. Anything outside the cone could be affected only if a signal could travel faster than light. Similarly, the lower cone represents the space in which past events could have had an effect on the event at the vertex.

I still don’t get how the coefficients show position and scale of the event. Which of the a(ij) is the one affecting scale? More generally, if there is no system of equations, then how does the value of the coefficient affect anything at all?

 

[starkind]

I am sorry if my last post was offensive somehow. I wasn't trying to be sarcastic, but sometimes attempts at levity fall flat. Or, it could be that my ignorance is still so abysmal that no one here wants to waste energy trying to enlighten me. Maybe the answer to my question is a year of linear algebra, a study which I have by no means completed.

I am still wondering how symmetry in a 4x4 matrix is connected to the geometry of a light cone. How about giving me a clue?

 

[HallsofIvy]

I am very disappointed to hear that there is no system of equations associated with the g(ab) metric, just when I thought I had made a connection between g(14) and scale.

If this is in response to my post, I did not say there was "no system of equations associated with the g(ab) metric". The question was whether the 4 by 4 matrix was an "augmented" matrix for some system of equations. I said that there was no system of equations for which it is the augmented matrix. All I was trying to do was explain why a symmetric 4 by 4 matrix has 10 independent entries.

Do you really mean the main diagonal entries can be anything one wants? I suspect there must be more limits than that. For example, it would surprise me to learn that g(14) could be a p-nut butter toast. However the depths of my gullibility are such that I have to consider that option. I had thought, at least, that any a(ij) should be a number.

If your matrix is a matrix of numbers then, yes, all entries have to be numbers. I assumed that was what we were talking about. Are you saying I should have said explicitely that "the main diagonal entries can be any numbers you want"? I don't see anywhere in your posts where you specified that.

I think I understand that the vertex of the light cone is an event. The upper cone (the arrow of time is usually up in these diagrams, per Dr. Penrose) represents the expansion of a sphere at light speed from the event, which encloses the space (xyz) which could be affected in the future of the event. Anything outside the cone could be affected only if a signal could travel faster than light. Similarly, the lower cone represents the space in which past events could have had an effect on the event at the vertex.

I still don’t get how the coefficients show position and scale of the event. Which of the a(ij) is the one affecting scale? More generally, if there is no system of equations, then how does the value of the coefficient affect anything at all?

 

[HallsofIvy]

I am sorry if my last post was offensive somehow. I wasn't trying to be sarcastic, but sometimes attempts at levity fall flat. Or, it could be that my ignorance is still so abysmal that no one here wants to waste energy trying to enlighten me. Maybe the answer to my question is a year of linear algebra, a study which I have by no means completed.

If you were worried that your post could have been mistaken of sarcasm, you haven't seen me in action!

I am still wondering how symmetry in a 4x4 matrix is connected to the geometry of a light cone. How about giving me a clue?

I don't believe this is a linear algebra or even a mathematics question. I think you would do better in the "Special and General Relativity" section.

 

[starkind]

Thank you, HallsofIvy. I am grateful for your time and attention.