Incompatability of GRT and Quantum Mechanics

[CarstenDierks]

Incompatibility of GRT and Quantum Mechanics

Brian Greene stated in "The Fabrics of Cosmos" (chap. 12) that the equations of GRT and quantum mechanics produce infinite results when used together (example: gravitational field and singularity of a black hole).

It would be great if someone can explain how this exactly happens (which equations at which point).

Thanks in advance,

Carsten

 

[marlon]

Basically this singularities mean that there is something wrong going on with the combination of those formula's. As to which formula exactly i can easily respond that you will find such problems with any formula that U use. For example in GTR you need to know the exact position of some object that will curve space time. But if you also want to apply QM then here is your first problem or singularity : If the position is exact, then via Heisenberg uncertainty, the momentum of this object needs to be infinite via $$\Delta p * \Delta x = \hbar$$

If you want to know more on this i suggest you look into my journal and especially in the entries on String Theory and Introduction to Loop Quantum Gravity
https://www.physicsforums.com/journal.php?s=&journalid=13790&action=view

regards
marlon

 

[CarstenDierks]

Thanks Marlon, I will look it up.

Do you also have formulas with which one can calculate the extent of space curvature?



Carsten

 

[marlon]

Thanks Marlon, I will look it up.

Do you also have formulas with which one can calculate the extent of space curvature?



Carsten

Well with the Riemanntensor you can evaluate whether some manifold is curved or not. The catch is that you can do this in a local flat reference frame. this is like being able to tell whether the Earth is flat or not just by looking in your local flat (you know : until the horizon) reference frame.

We can make estimates of the volume of the universe without knowing its boundaries. this is done with socalled Gaussian coordinates in which we work with 4 dimensions in stead of three (i mean four SPACE-like dimensions, time is not considered). This procedure yields a "nice" volume for our universe in the Friedmann models...

regards
marlon

 

[CarstenDierks]

Hi Marlon,

Well, the Riemann tensor was a plunge into deep water for me.

I have read about the Einstein tensor before but have never gotten into the metrics of it.

Looking into some web sites brought me once around the world: India, Iran, Japan, US, and Germany. I read about scalars, Christoffel-, Jacobi- and Ricci-tensors, covariant and contravariant tensors etc. ...

For those of you in the forum who like to get into it, I will quote the better links in the end of this reply.

However, I have not found a real good explanation or equation that could tell me something like:

At the distance x of a mass M the space curves with dx, dy, dz [meters]. (A 2 dimensional approach in the beginning would be great also!)

Can you, Marlon, or anybody else help me out?

Thanks a lot,

Carsten


Links to the Riemann tensor:

http://www.bun.kyoto-u.ac.jp/~suchii/Einstein/riemann.curv.html
A first quick non-mathematical approach

http://mathworld.wolfram.com/RiemannTensor.html
Short mathematical description

http://khwarizmi.phy.iitb.ac.in/~surhud/seminarhtml/node8.html
Equations (this page: energy momentum and space curvature)

http://www.inb.uni-luebeck.de/~eebarth/demos/R/
Nice animations of the Riemann equations

http://rcswww.urz.tu-dresden.de/~lohrmobs/RCMA2/rcma2.html
Geometry of space manifolds

And maybe you know even better links...

 

[CarstenDierks]

As to which formula exactly i can easily respond that you will find such problems with any formula that U use.

Marlon,

Do you have some other examples? Maybe 1 or 2 important ones just for clarification.

Thanks,

Carsten

 

[marlon]

Hi Marlon,

However, I have not found a real good explanation or equation that could tell me something like:

At the distance x of a mass M the space curves with dx, dy, dz [meters]. (A 2 dimensional approach in the beginning would be great also!)

Basically what this means is that the actual curvature of spacetime is described in terms of dx,dy,dz. These "things" are used in what we call the metric-tensor. This tensor describes distances on some manifold (curved of flat, both are possible), and therefore it describes the "geometry", to some extent, of the space-time-continuum. generally this tensor is written as
$$ds^2 = g_{\mu \nu} dx^{\mu}dx^{\nu}$$

The mu and nu take values from 0 to 3 and for example a dx corresponds to the value mu = 1. When mu = 0 you have the time-coordinate. The g is the actual tensor (eg a diagonal matrix with first element 1 and the rest -1 on the diagonal ) and ds² expresses the distance. The clue is to find THAT form of g, which respects the laws of general relativity. The ds² is also called the metric and depending on the actual equations you have different possibilities for such a metric (remeber they need to respect GTR or the Einstein-equation in cosmology). For example for black holes you have the Schwardzschild metric if the black hole doesn't rotate or you can have the Kerr-metric if the black hole DOES rotate. In the last case, this metric also describes the socalled ergosphere. A region around the black hole where you can never stand still. So even if your momentum is zero, you rotate because the black hole actually makes the entire space time continuum rotate. In this reagion the famous Penrose-process can take place. Particles of certain energy are injected into the ergosphere and they can pop up back out of it with HIGHER energy. it looks like these particles have extracted energy from the space time continuum.

Read the entry on black holes in my journal for more indept info...


regards
marlon

 

[marlon]

Marlon,

Do you have some other examples? Maybe 1 or 2 important ones just for clarification.

Thanks,

Carsten

Well, another big problem is this : what about superposition in GTR ? In QM you can say that the position of cat is the superposition of here and there. Now once we look this wavefunction will have entangeled with our wavefunction and only one state will remain because here and there are orthogonal states. They cannot occur at the same time. Well, there are other interpretations for looking at this superposition thing (entanglement (the one i gave you here), many worlds, standard Copenhagen, decoherence)but rest asure the SUPERPOSITION itself exist in QM and it is fundamental there. How are you going to describe the fact that a planet is here and there? Remember, you need the two components of this wave function in order to give the adequate explanation of the physical behaviour, just like in QM


regards
marlon

 

[Nereid]

Hi Carsten.

Have you had a look at the Special &General Relativity section of PF (in Astronomy & Cosmology)? There are several members who hang out there and are very au fait with GR, tensors, etc.