Gaussian polar coordinates

<h2 id="definition">Definition</h2>
<p>In a Gaussian polar chart (on a static spherically symmetric spacetime), the <a href="/facts/Metric_tensor_(general_relativity)/lJoCMdve">metric</a> (aka <a href="/facts/Line_element/asquI22v">line element</a>) takes the form
</p>

g
        =
        −
        a
        (
        r
        
          )
          
            2
          
        
        
        d
        
          t
          
            2
          
        
        +
        d
        
          r
          
            2
          
        
        +
        b
        (
        r
        
          )
          
            2
          
        
        
        
          (
          
            d
            
              θ
              
                2
              
            
            +
            sin
            ⁡
            (
            θ
            
              )
              
                2
              
            
            
            d
            
              ϕ
              
                2
              
            
          
          )
        
        ,
      
    
    {\displaystyle g=-a(r)^{2}\,dt^{2}+dr^{2}+b(r)^{2}\,\left(d\theta ^{2}+\sin(\theta )^{2}\,d\phi ^{2}\right),}

−
        ∞
        <
        t
        <
        ∞
        ,
        
        
          r
          
            0
          
        
        <
        r
        <
        
          r
          
            1
          
        
        ,
        
        0
        <
        θ
        <
        π
        ,
        
        −
        π
        <
        ϕ
        <
        π
        .
      
    
    {\displaystyle -\infty <t<\infty ,\,r_{0}<r<r_{1},\,0<\theta <\pi ,\,-\pi <\phi <\pi .}

<p>Depending on context, it may be appropriate to regard 
  
    
      
        a
      
    
    {\displaystyle a}
  
 and 
  
    
      
        b
      
    
    {\displaystyle b}
  
 as undetermined <a href="/facts/Function_(mathematics)/CdReEKCh">functions</a> of the radial coordinate 
  
    
      
        r
      
    
    {\displaystyle r}
  
.  Alternatively, we can plug in specific functions (possibly depending on some parameters) to obtain an isotropic coordinate chart on a specific Lorentzian spacetime.
</p>
<h2 id="applications">Applications</h2>
<p>Gaussian charts are often less convenient than Schwarzschild or isotropic charts.  However, they have found occasional application in the theory of static spherically symmetric perfect fluids.
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Static_spacetime/reDPaDQa">Static spacetime</a></li>
<li><a href="/facts/Static_spherically_symmetric_perfect_fluid/c09nSk2N">Static spherically symmetric perfect fluids</a></li>
<li><a href="/facts/Schwarzschild_coordinates/N50L86od">Schwarzschild coordinates</a></li>
<li><a href="/facts/Isotropic_coordinates/VVpstuNV">Isotropic coordinates</a></li>
<li><a href="/facts/Frame_fields_in_general_relativity/Lw90nUa6">Frame fields in general relativity</a> for more about frame fields and coframe fields.</li></ul>

Gaussian polar coordinates open-in-new

Gaussian polar coordinates