Example of a metric incompatible connection Watch

sunarcs
Badges: 0
Rep:
?
#1
Report Thread starter 3 years ago
#1
Hi,

I'm doing a Relativity course, and I'm a little confused by the concept of metric compatibility.Given (I assume these are correct):
Code:
gab  =  ea.ebea  =  gab eaΓabc  =  ea c eb
it seems to follow that
Code:
c gab  =  ebc ea  +  eac eb  (product rule)
 =  gbd Γdac  +  gad Γdbc
But this is supposedly only the case if the connection is metric compatible.So my question is, how could the connection not be metric compatible? (what incorrect assumptions have I just made?) An example of this would be helpful, but I can't seem to find one (since people generally assume metric compatibility)

Thanks,
0
reply
Gregorius
Badges: 14
Rep:
?
#2
Report 3 years ago
#2
(Original post by sunarcs)
Hi,

I'm doing a Relativity course, and I'm a little confused by the concept of metric compatibility.Given (I assume these are correct):
Code:
gab  =  ea.ebea  =  gab eaΓabc  =  ea c eb
it seems to follow that
Code:
c gab  =  ebc ea  +  eac eb  (product rule)
 =  gbd Γdac  +  gad Γdbc
But this is supposedly only the case if the connection is metric compatible.So my question is, how could the connection not be metric compatible? (what incorrect assumptions have I just made?) An example of this would be helpful, but I can't seem to find one (since people generally assume metric compatibility)

Thanks,
The basic issue here is that a connection and a metric are two different geometric concepts - and a-priori there is no reason for any compatibility between them. So the first thing to say is that a metric is compatible with a connection if the parallel transport of the connection preserves the inner product defined by the metric. So take any two vectors, take their inner product w.r.t the metric; you get the same answer if you parallel transport the vectors to another tangent space and then take their inner product. Equivalent to this is that the covariant derivative of the metric tensor is zero. Indeed the covariant derivative of a metric tensor (if it is non-zero) is sometime called the non-metricity of the connection.

What is sometimes confusing is that the connection that is often wheeled out first in relativity courses is the Levi-Civita connection. This is defined to be the unique torsion free connection that is compatible with the metric. If you trundle through the calculations, the Christoffel symbols of the connection can be expressed in terms of the dual of the metric tensor and of derivatives of the metric tensor. The Levi-Civita connection is then by definition compatible withe metric - but what you also get is that any other torsion free connection will not be compatible with the metric.
1
reply
X

Quick Reply

Attached files
Write a reply...
Reply
new posts
Latest
My Feed

See more of what you like on
The Student Room

You can personalise what you see on TSR. Tell us a little about yourself to get started.

Personalise

University open days

  • Edge Hill University
    Undergraduate and Postgraduate - Campus Tour Undergraduate
    Mon, 18 Feb '19
  • University of the Arts London
    MA Innovation Management Open Day Postgraduate
    Mon, 18 Feb '19
  • University of Roehampton
    Department of Media, Culture and Language; School of Education; Business School Undergraduate
    Tue, 19 Feb '19

Do you give blood?

Yes (44)
9.24%
I used to but I don't now (13)
2.73%
No, but I want to start (169)
35.5%
No, I am unable to (112)
23.53%
No, I chose not to (138)
28.99%

Watched Threads

View All