home | index | units | counting | geometry | algebra | trigonometry & functions | calculus
analysis | sets & logic | number theory | recreational | misc | nomenclature & history | physics

Final Answers
© 2000-2015   Gérard P. Michon, Ph.D.

Tensors

The general theory of relativity can be conceived only as a field theory.
It could not have
[been] developed if one had held on to the view   Gregorio Ricci-Curbastro 
1853-1925  Albert Einstein 
1879-1955
that the real world consists of material points which move 
 under the influence of forces acting between them.

Albert Einstein,  in his last scientific paper (December 1954).

Related articles on this site:

Related Links (Outside this Site)

The Rigid Rotating Disk in Relativity   by  Michael Weiss.
Introduction to General Relativity   by  Gerard. 't Hooft  (Utrecht University)
Tensors and Relativity   by  Peter Dunsby  (University of Cape Town, 1996).
General Relativity  by  David M. Harrison  (University of Toronto).
Reflections on Relativity  by  Kevin S. Brown.
General Relativity Tutorial  by  John Baez.
About Black Holes...  by  Chris Hillman.
General Relativity from Special Relativity Using Tetrads  by  Will M. Farr.
Riemannian Geometry  (PHY3033, University of Hong-Kong)
Einstein-Cartan theory  by  Heiko Herrmann  (2004).
Einstein-Cartan theory  by  Andrzej Trautman  (arXiv, June 2006).
Cartan-Einstein Teleparallelism  (TP)  by  Jose G. Vargas  &  Doug G. Torr.
On the History of Unified Field Theories  by  Hubert F.M. Goenner  (2004).
Did Einstein cheat?   by  John Farrell  (2000)
 
Wikipedia :   Introduction to General Relativity   |   General Relativity   |   Einstein field equations
Cartan formalism   |   Spin tensor   |   Einstein-Cartan theory   |   (Lack of) Nordtvedt effect
Tipler Machine (time travel)   |   Nordström's metric theory of gravitation   (1913, obsolete & falsified)
Variational methods in general relativity

Books :

Videos :

"Tensor Calculus"  (2014)  by  Pavel Grinfeld :   0 | 1 | 2 | 3 | 3a | 4s | 4 | 4a | 5? | 5b | 6a | 6b | 6c | 6d | 7a | 7b | 7c | 7d | 8 | 8b | 8c | 8d | 8e | 9a | 9b | 10a | 10b | 10c | 11a | 11b | 12s | 12 | 12a | 12b | 13b | 14a | 14b | 14c | 14d | 14e | 14f | 15 |
 
border
border

Tensor Calculus


(2015-01-27)   Definition of a Tensor

 Come back later, we're
 still working on this one...

Wikipedia :   Tensor Calculus


(2009-08-05)   Tensors in metric spaces
What tensors  really  are.

By definition, the scalars of a vector space are its  tensors of rank 0.

In any vector space, a linear function which sends a vector to a scalar may be called a  covector.  Normally, covectors and vectors are different types of things.  (Think of the  bras  and  kets  of quantum mechanics.)  However, if we are considering only  finitely many dimensions, then the space of vectors and the space of covectors have the same number of dimensions and can therefore be put in a  linear  one-to-one correspondence with each other.

Such a  bijective correspondence is called a  metric  and is fully specified by a nondegenerate  quadratic form, denoted by a  dot-product  ("nondegenerate" precisely means that the associated correspondence is bijective).

A metric is said to be Euclidean if it is "positive definite", which is to say that  V.V  is positive for any nonzero vector  V.  Euclidean metrics are nondegenerate but other metrics exist which are nondegenerate in the above sense without being "definite" (which is to say that  V.V  can be zero even when  V  is nonzero).  Such metrics are perfectly acceptable.  They include the so-called Lorentzian metric of four-dimensional spacetime, which is our primary concern here.

Once a metric is defined, we are allowed to blur completely the distinction between vectors and covectors as they are now in canonical one-to-one correspondence.  We shall simply call them here's only one such type, now).  A tensor of rank zero is a scalar.

More generally, a tensor of nonzero rank  n  (also called  nth-rank tensor, or n-tensor)  is a linear function that maps a vector to a tensor of rank n-1.

Such an object is  intrinsically  defined, although it can be specified by  either  its covariant  or  its contravariant coordinates  in a given basis  (cf. 2D example).


(2016-01-25)   Gradients
A tensor of rank 1 in covariant form:

 Come back later, we're
 still working on this one...

Tensor Analysis  by  Pavel Grinfeld  (video 2):  The Two Definitions of the Gradient

border
border
visits since January 27, 2015
 (c) Copyright 2000-2015, Gerard P. Michon, Ph.D.