# Does Born rigidity describe particle creation?

• Naty1
In summary: This is because particles more than 1/aj behind the leading end of the rod are on the opposite side of the pivot event, and can only be part of the Born acceleration if they are accelerating in the opposite direction in both space and in time. This is shown clearly in the figure above.In summary, the diagram at Wikipedia seems to illustrate particle pair production with a Rindler horizon while describing Born rigidity, but the description at mathpages provides a better understanding of the relationship between the two concepts.
Naty1
I'd appreciate some help understanding Rindler coordinates better: I found a visual that seems to illustrate particle pair production with a Rindler horizon while describing Born rigidity. [Sounds like the Unruh effect to me.]

[As background, recent discussions on particle pair production associated with horizons of various sorts include these:

[Hubble sphere horizon] and

[Unruh Effect]

This discussion will relate.] At
http://en.wikipedia.org/wiki/Rindler_coordinates

Wikipedia says, in part:

...in relativistic physics, we see that the trailing endpoint of a rod which is accelerated by some external force (parallel to its symmetry axis) must accelerate a bit harder than the leading endpoint, or else it must ultimately break. ... This leads to a differential equation showing, that at some distance, the acceleration of the trailing end diverges, resulting in the Rindler horizon.

and I thought 'this sounds a lot like the 'particle pair' discussions in Physicsforums...

This Wikipedia 'half picture' [diagram] of Rindler coordinates is not so insightful as THIS source and description:

http://www.mathpages.com/home/kmath422/kmath422.htm

regarding Born rigidity... and check out this description:

...This is because particles more than 1/aj behind the leading end of the rod are on the opposite side of the pivot event, and can only be part of the Born acceleration if they are accelerating in the opposite direction in both space and in time. This is shown clearly in the figure above.

There is a lot of language in the overall mathpages description which I don't understand but the above description stood out to me. It seems just like the beginning of a description and visual representation of particle pair production!

What do you think? and why??

[a] old hat,
insightful,
[c] pathetically lame

Anybody know of other coordinate diagrams that address similar perspectives on Schwarzschild and FLRW descriptions for back holes, cosmology, respectively?? Hubble

[Isn't there something very similar inside a black hole horizon with Schwarzschild coordinates to this:..."accelerating in the opposite direction in both space and in time... ??" where signs change and 'x takes on the values of time'...moving towards the black hole singularity?]

thank you

Last edited:
I can offer some insights to my own question, if not a complete answer yet, from comments beginning with post #36 here:

A few key insights:

bapowell:
..All spacetimes have a particle horizon, on account of the finite speed of light.

One important distinction is that the de Sitter horizon is fundamental to the spacetime, while the Rindler horizon is observer dependent.

Chalnoth:
And just to be clear, if we're talking about particle creation, we're talking about an event horizon specifically.

Wikipedia:
Wikipedia says:

The particle horizon is the maximum distance from which particles could have traveled to the observer in the age of the universe.

Does the Hubble sphere meet that criteria??

Good catch, and it's an important one. I set up the case where the wavelength of the Fourier mode is larger than the particle horizon, and I chose this horizon because it marks the limit of causal separation. Now, in non-accelerating spacetimes, what happens? The Fourier mode increases in length less quickly than the particle horizon -- if it was once superhorizon, it eventually falls into the horizon: hardly a prescription for particle creation! What we need is to not only push the modes out of the particle horizon, we need to keep them out!...

But whether or just how the Hubble sphere eventually meets criteria is not yet clear from the discussion.

Seems I left out the conclusion part of post #2:

Coordinates, frames, type of acceleration, and type of horizon all affect apparent/ observed particle creation...

When I read your first post I was puzzled by the connection you made between Born rigidity and the existence of horizons. They seem to be independent concepts, each of a different kind. Born rigidity relates the worldlines of a bunch of accelerating partcles and looks at the case when they are part of a pice of matter, whereas Rindler horizons appear for any accelerating frame. The appearance of horizons lead to apparent particle creation for the accelerating observer, but I can't see how this relates to Born rigidity.

Mentz:
When I read your first post I was puzzled by the connection you made between Born rigidity and the existence of horizons. They seem to be independent concepts, each of a different kind.

They used to seem different to me, too, but see how the last link in the original post above, mathpages, begins the description of Born rigidity:

Consider a uniform distribution of particles at rest along some segment of the x-axis of an inertial coordinate system x,t at the time t = 0. Each particle is subjected to a constant proper acceleration (hyperbolic motion) such that...

In the diagram there, Born rigidity focuses on the rigid rod, in red, on the right while I am focusing on the particles 'disappearing/hidden' in the left hand side of the Rindler chart...beyond the 'horizon'...particles to the left 'disappear' from the accelerating observer ...

And the Wikipedia comment from above on rigidity:

This leads to ...at some distance, the acceleration of the trailing end diverges, resulting in the Rindler horizon.

seems to confirm such a horizon. This is the same thing going on with the Rindler horizon in the Unruh affect!

I am puzzled by this Mathpages quote mentioned in the OP:
...This is because particles more than 1/aj behind the leading end of the rod are on the opposite side of the pivot event, and can only be part of the Born acceleration if they are accelerating in the opposite direction in both space and in time. This is shown clearly in the figure above

As I understand it, Born rigid motion ensures that the proper distance between any two points on the rod remains constant from the point of view of observers accelerating with the rod. To be "part of the Born acceleration" surely this condition has to met and if the rod extends beyond the pivot point then the condition is not met. No physical rod can extend across the pivot point under these acceleration conditions without being torn apart. Even if we consider individual disconnected particles undergoing this acceleration pattern, it is not possible for observers co-accelerating with the particles on the right to communicate or compare measurements with observers co-accelerating with the particles on the left.

The two sides of the pivot point are separated by an effective event horizon but the analogy to a black hole is limited. Free falling observers inside a black hole can see events outside the black hole so neither side of the pivot point is equivalent to the inside of a black hole. The top region is the region most like the inside of a black hole because objects and information can enter that region but cannot leave. The equivalence between Rindler acceleration and a black hole also breaks down when we note that there is no equivalent of an absolute singularity that is at the centre of a black hole. This means we have to be careful about drawing conclusions about black holes from studying Rindler charts as there are obvious differences.

For an extended rod with points accelerating proportional to 1/x there can be no physical part of the rod at x=0 because that would require infinite acceleration. This is where the apparent Rindler horizon is located for the accelerating observers. While Unrah particle production is generally recognised as being a real effect, the Born rigid acceleration graph does not demonstrate in any tangible way how particle production occurs under these conditions. Reference to the "particle horizon" only mean the region from which particles could not have traveled in the lifetime of the universe and does not imply the region where particle pairs are produced. It is just saying "IF particles are emitted from behind this line then it is not possible for the accelerating observers to see them", which is different from saying "particles are produced here".

## What is Born rigidity and how does it relate to particle creation?

Born rigidity is a concept in physics that describes the rigidity of space-time. It suggests that the structure of space-time is fixed and unchanging, and that any changes or deformations are due to the interactions of particles. This concept is often used to explain the creation of particles in quantum field theory.

## What evidence supports the idea that Born rigidity describes particle creation?

One of the main evidences for Born rigidity is the success of quantum field theory in predicting the creation and interactions of particles. This theory is based on the concept of a rigid space-time structure, and its predictions have been confirmed through numerous experiments and observations.

## Does Born rigidity only apply to particle creation or does it have other implications?

Born rigidity has implications beyond just particle creation. It is also used to explain other phenomena such as the expansion of the universe, the behavior of black holes, and the effects of gravity. It is a fundamental concept in our understanding of the structure of space-time.

## Are there any alternative theories to Born rigidity that explain particle creation?

There are alternative theories to Born rigidity, such as loop quantum gravity and string theory, which propose different ways of understanding the structure of space-time and the creation of particles. However, these theories are still under development and do not have as much empirical evidence to support them as Born rigidity does.

## Can Born rigidity help us understand the fundamental nature of space and time?

Born rigidity is one of the key concepts in our understanding of the fundamental nature of space and time. It provides a framework for understanding the structure of space-time and its interactions with particles. However, it is not a complete theory and more research is needed to fully understand the nature of space and time.

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