Are Christoffel Symbols Considered Tensors?

  • Thread starter Thread starter Karl G.
  • Start date Start date
  • Tags Tags
    Symbols Tensors
Karl G.
Messages
40
Reaction score
0
I'm sort of confused about Christofel symbols being called tensors. I thought that to be considered a tensor, the tensor had to obey the standard component transformation law. For example: Ga'b' = Lca'Ldb'Gcd
But the Christofel symbols don't obey this transformation law; their components transform in a different way (I would like to show it, but I don't know how; see exercise 10.3 of MWT).
What gives?
 
Physics news on Phys.org
No, Christoffel symbols are not tensors.
 
Oh ... thought they were
 
Christoffel symbols make the covariant derivative a tensor. Try transforming one.
 
I haven't seen a single book on GR which introduces the Christoffel symbols without immediately pointing out that they're not tensors.
 
Karl G. said:
I'm sort of confused about Christofel symbols being called tensors. I thought that to be considered a tensor, the tensor had to obey the standard component transformation law. For example: Ga'b' = Lca'Ldb'Gcd
But the Christofel symbols don't obey this transformation law; their components transform in a different way (I would like to show it, but I don't know how; see exercise 10.3 of MWT).
What gives?

They transform inhomogeneously under general coordinates transformations, i.e., not tensors. However, the inhomogeneous term in the transformation law vanishes if the coordinate transformations are LINEAR. So, they do transform as tensors with respect to all linear coordinate transformations.

sam
 

Thread 'A sufficient condition for integrability of equation ##\nabla g=0##'

In Philippe G. Ciarlet's book 'An introduction to differential geometry', He gives the integrability conditions of the differential equations like this: $$ \partial_{i} F_{lj}=L^p_{ij} F_{lp},\,\,\,F_{ij}(x_0)=F^0_{ij}. $$ The integrability conditions for the existence of a global solution ##F_{lj}## is: $$ R^i_{jkl}\equiv\partial_k L^i_{jl}-\partial_l L^i_{jk}+L^h_{jl} L^i_{hk}-L^h_{jk} L^i_{hl}=0 $$ Then from the equation: $$\nabla_b e_a= \Gamma^c_{ab} e_c$$ Using cartesian basis ## e_I...
View full post »

Thread 'Relativator (Circular Slide-Rule): Simulated with Desmos - Insight'

Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. The Relativator was sold by (as printed) Atomic Laboratories, Inc. 3086 Claremont Ave, Berkeley 5, California , which seems to be a division of Cenco Instruments (Central Scientific Company)... Source: https://www.physicsforums.com/insights/relativator-circular-slide-rule-simulated-with-desmos/ by @robphy
View full post »

Thread 'Hawking After 54 Years Confirmed: BH surfaces don't shrink'

Abstract The gravitational-wave signal GW250114 was observed by the two LIGO detectors with a network matched-filter signal-to-noise ratio of 80. The signal was emitted by the coalescence of two black holes with near-equal masses ## m_1=33.6_{-0.8}^{+1.2} M_{⊙} ## and ## m_2=32.2_{-1. 3}^{+0.8} M_{⊙}##, and small spins ##\chi_{1,2}\leq 0.26 ## (90% credibility) and negligible eccentricity ##e⁢\leq 0.03.## Postmerger data excluding the peak region are consistent with the dominant quadrupolar...
View full post »

Similar threads

On the order of indices of the Christoffel symbol of the 1st kind
Replies
19
Views
2K
I Transformation of the Christoffel Symbols
Replies
4
Views
4K
A Tensors in null basis
Replies
9
Views
1K
I Calculating Covariant Derivative of Riemann Tensor in Riemann Normal Coordinates
Replies
5
Views
2K
I How to derive irreducible representations/tensors?
Replies
22
Views
3K
A Anti-symmetric Christoffel symbol
Replies
6
Views
3K
A Difference of Christoffel Symbols Transforms as Tensor
Replies
7
Views
2K
A Understanding Christoffel symbol in GR context
Replies
19
Views
5K
I Trying To Calculate Christoffel Symbols
Replies
16
Views
3K
I 3+1 split of the Electromagnetic Tensor and Maxwell's Equations
Replies
25
Views
2K

Hot Threads

Recent Insights

Back
Top