Peeling theorem

In <a href="/facts/General_relativity/jVsoIg8X">general relativity</a>, the peeling theorem describes the asymptotic behavior of the <a href="/facts/Weyl_tensor/lrXNLWHC">Weyl tensor</a> as one goes to <a href="https://en.wiktionary.org/wiki/null_infinity">null infinity</a>. Let 
 
 
 
 γ
 
 
 {\displaystyle \gamma }
 
 be a <a href="/facts/Null_geodesic/PEBX8FSm">null geodesic</a> in a <a href="/facts/Spacetime/uRIN1x4b">spacetime</a> 
 
 
 
 (
 M
 ,
 
 g
 
 a
 b
 
 
 )
 
 
 {\displaystyle (M,g_{ab})}
 
 from a point p to null infinity, with <a href="/facts/Affine_parameter/fbWR6l80">affine parameter</a> 
 
 
 
 λ
 
 
 {\displaystyle \lambda }
 
. Then the theorem states that, as 
 
 
 
 λ
 
 
 {\displaystyle \lambda }
 
 tends to infinity:

C
          
            a
            b
            c
            d
          
        
        =
        
          
            
              C
              
                a
                b
                c
                d
              
              
                (
                1
                )
              
            
            λ
          
        
        +
        
          
            
              C
              
                a
                b
                c
                d
              
              
                (
                2
                )
              
            
            
              λ
              
                2
              
            
          
        
        +
        
          
            
              C
              
                a
                b
                c
                d
              
              
                (
                3
                )
              
            
            
              λ
              
                3
              
            
          
        
        +
        
          
            
              C
              
                a
                b
                c
                d
              
              
                (
                4
                )
              
            
            
              λ
              
                4
              
            
          
        
        +
        O
        
          (
          
            
              1
              
                λ
                
                  5
                
              
            
          
          )
        
      
    
    {\displaystyle C_{abcd}={\frac {C_{abcd}^{(1)}}{\lambda }}+{\frac {C_{abcd}^{(2)}}{\lambda ^{2}}}+{\frac {C_{abcd}^{(3)}}{\lambda ^{3}}}+{\frac {C_{abcd}^{(4)}}{\lambda ^{4}}}+O\left({\frac {1}{\lambda ^{5}}}\right)}

where 
 
 
 
 
 C
 
 a
 b
 c
 d
 
 
 
 
 {\displaystyle C_{abcd}}
 
 is the Weyl tensor, and <a href="/facts/Abstract_index_notation/GjuskCay">abstract index notation</a> is used. Moreover, in the <a href="/facts/Petrov_classification/0Y4sNsgS">Petrov classification</a>, 
 
 
 
 
 C
 
 a
 b
 c
 d
 
 
 (
 1
 )
 
 
 
 
 {\displaystyle C_{abcd}^{(1)}}
 
 is type N, 
 
 
 
 
 C
 
 a
 b
 c
 d
 
 
 (
 2
 )
 
 
 
 
 {\displaystyle C_{abcd}^{(2)}}
 
 is type III, 
 
 
 
 
 C
 
 a
 b
 c
 d
 
 
 (
 3
 )
 
 
 
 
 {\displaystyle C_{abcd}^{(3)}}
 
 is type II (or II-II) and 
 
 
 
 
 C
 
 a
 b
 c
 d
 
 
 (
 4
 )
 
 
 
 
 {\displaystyle C_{abcd}^{(4)}}
 
 is type I.

<ul><li>Wald, Robert M. (1984), <a href="/facts/General_Relativity_(book)/lYIGX0gc">General Relativity</a>, University of Chicago Press, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-226-87033-2</li></ul>

Peeling theorem open-in-new

Peeling theorem