Exploring the Curvature of Space: Tensor Analysis & Beyond

In summary: To observe what happens".:confused:In summary, Tensor analysis was developed because all the other math fell apart. This new type of math measures the curvature of space. It is used in various areas of physics, including general relativity and crystal structure.
  • #1
misskitty
737
0
When Enstein developed tensor analysis because all the other math fell apart, there was another type of math that was developed to measure the curvature of space. What is it? How does it work?
 
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  • #2
misskitty said:
When Enstein developed tensor analysis because all the other math fell apart, there was another type of math that was developed to measure the curvature of space. What is it? How does it work?

Connection? Old name Christoffel symbols. Out of these are built the covariant derivatives and the Riemann-Christoffel tensor (now called the curvature tensor; I don't know what physicists have against old Christoffel).
 
  • #3
How did he derive the covariant part? He obviously didn't make it up.
 
  • #4
Einstein didn't develop anythnig into the fields of mathematics (diff.geom.to be exact).He used the knowledge acquired from his pals H.Minkowski (his ex math tech at Zürich),H.Weyl and especially D.Hilbert (there's a long story with Hilbert).

He simply postulated the equations of gravity for a free space and space with matter...
Alongside other axioms,inluding the famous Equivalence Principle.

Daniel.

P.S.The math was not "done" by 1915.Tullio Levi-Civita (1917) and Elie Cartan brought tensor calculus to a complete form.
 
  • #5
Ah ok. Is this a widely used level of math?
 
  • #6
It is now. In addition to general relativity (and related fields like string theory), one of the main areas where tensor math is used is in solid state physics and things like crystal structure. Every legitimate college and university that offers a degree in physics or mathematics teaches it. Tensor analysis is the calculus based big brother of maxtrix algebra which you may have encountered already.

Here is an archieved physics forum thread on tensors:
https://www.physicsforums.com/archive/t-35920_Math_"Newb"_Wants_to_know_what_a_Tensor_is.html

They are used among other things to evaluate earthquake data:

http://seismo.berkeley.edu/seismo/annual_report/ar99_00/node12.html
 
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  • #7
In what sense...?"Widely used".:confused:General Relativity is one branch of physics which "widely used" tensor calculus (algebra & analysis),differential geometry to be fair.

Daniel.
 
  • #8
Hmmm, sounds very complicated. This is going to sound dumb; could someone please tell me what string theory is because I don't know what it is or what it does and stuff.
 
  • #9
In short,it is a theory aimed to complete the dream of theoretical physics:unifying the 4 interactions:2 long ranged:electromagnetism & gravity and 2 short ranged:weak & strong...If possible,supply a new view on cosmology.

The best model in physics (also applied to fundamental interactions) is the "pointlike particle".It's the simplest,the best and so far very successful.This "string theory" doesn't use "pointlike particles",it doesn't use particles at all,actually.It uses strings,unidimensional objects immersed into a higher dimensional space.Particles have 0 dimension,while strings have only one...

This theory,unlike the famous Standard Model of Particles and Interactions (short,SM),is not completed and is under debate and study for more than 20 years...

Daniel.
 
  • #10
Does it provide any clues about how the universe works or is it all just theory?
 
  • #11
It does.And it is just theory.Unfortunately we cannot (at this point) experimentally check its predictions.Heck,we didn't even find the Higgs boson...

Daniel.
 
  • #12
What experimentation was used to try to find Higgs boson?
 
  • #13
I'm not a specialist at this,i think someone else might tell u more about what kind of experiments will be done at CERN once they get the LHC running.

Daniel.
 
  • #14
So there hasn't been anything done yet? So theoretically Higgs' boson could exsist.
 
  • #15
Tevatron is looking. They are hoping to get something - either a mass for it or a definite no to a small mass - before LHC comes and rains on their parade.
 
  • #16
LHC? How would you go about looking for something like that?
 
  • #17
LHC=Large Hadron Collider is a collider,that is a facility in which preaccelerated particles (in this case large hadrons and even nuclei) are being crushed...

Daniel.
 
  • #18
Do they accelerate these particles and launch them at a wall to observe what happens? :bugeye:
 
  • #19
You accellerate the particles into a target often a thin sheet of gold or magnetically contained particle, which is surrounded by film or other observation technology that shows you where the bits the broke apart landed and you back track from there.
 
  • #20
Particle physicists have violent tendencies - they are the academic equivalent of NASCAR fans. They have a reputation for being especially fond of head on collisions between objects traveling in opposite directions at nearly the speed of light.
 
  • #21
Chronos said:
Particle physicists have violent tendencies - they are the academic equivalent of NASCAR fans. They have a reputation for being especially fond of head on collisions between objects traveling in opposite directions at nearly the speed of light.

I guess I ahve it twice as bad then. I like particle and the physics involved and I'm a NASCAR fan. :biggrin:

Why launch the particle at gold specifically?
 
  • #22
misskitty said:
I guess I ahve it twice as bad then. I like particle and the physics involved and I'm a NASCAR fan. :biggrin:

Why launch the particle at gold specifically?

Heh,hold on,you're on a dangerous ground here,at this rate,you might even fall in love with a physicist... :wink:

Are u referring to the famous Rutherford-Marsden-Geiger experiment...?They needed a solid,chemically pure substance,made up of one species of atoms,with a high # of electrons in the atom,so that the alpha particles would have plenty of interactions.His theory proved that significant scattering at large angles was noticed with heavier atoms...

Listen to this guy
http://www.mhhe.com/physsci/chemistry/essentialchemistry/flash/ruther14.swf

and read this guy

http://galileo.phys.virginia.edu/classes/252/Rutherford_Scattering/Rutherford_Scattering.html


Daniel.
 

1. What is tensor analysis?

Tensor analysis is a branch of mathematics that deals with the properties and operations of tensors, which are mathematical objects that describe the relationships between different coordinate systems. In the context of space exploration, tensor analysis is used to understand the curvature of space and its effects on the movement of objects.

2. How does tensor analysis help us explore the curvature of space?

Tensor analysis allows us to describe and quantify the curvature of space by using mathematical equations to represent the relationships between different points in space. This helps us to understand how space is distorted by the presence of massive objects, such as planets and stars.

3. Can you give an example of how tensor analysis is used in space exploration?

One example is the use of tensors in Einstein's theory of general relativity, which describes how the curvature of space is related to the distribution of matter and energy. The famous equation E=mc² is a result of this theory, and it explains the relationship between mass and energy in the context of gravitational fields.

4. What are some practical applications of tensor analysis in space exploration?

Tensor analysis is used in the design and navigation of spacecraft, as it allows us to account for the effects of gravity and the curvature of space on their trajectories. It is also used in studying the effects of black holes and other massive objects on the fabric of space.

5. Is tensor analysis a difficult concept to understand?

Tensor analysis can be a challenging concept to grasp, as it involves advanced mathematical concepts and equations. However, with a strong foundation in mathematics and physics, it can be understood and applied to explore the curvature of space. It is a complex yet fascinating area of study that continues to advance our understanding of the universe.

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