Schwarzschild metric tensor

Provides the metric tensor of the Einstein equation's Schwarzschild solution in Schwarzschild coordinates where the Schwarzschild radius \(r_s\) is set to 1. $$ds^2 = - \left(1-\frac{r_s}{r}\right) dt^2 + \left(1-\frac{r_s}{r}\right)^{-1} dr^r + r^2 d\Omega^2$$

Usage

g_ss(n, coords = c("t", "r", paste0("ph", 1:(n - 2))))

Arguments

n

The dimension of the metric tensor.

coords

A character vector of coordinate names. The length needs to match the tensor dimensions.

Value

The covariant metric tensor as array imputed with coordinate names.

Details

Note that Schwarzschild coordinates become singular at the Schwarzschild radius (event horizon) \(r=r_s=1\) and at \(r=0\).

See also

Wikipedia: Schwarzschild metric

Other metric tensors: g_eucl_cart(), g_mink_cart(), g_sph(), metric_field()

Examples

g_ss(4)
#> <Covariant metric tensor field> (t, r, ph1, ph2)
#>      [,1]       [,2]        [,3]    [,4]            
#> [1,] "-(1-1/r)" "0"         "0"     "0"             
#> [2,] "0"        "1/(1-1/r)" "0"     "0"             
#> [3,] "0"        "0"         "r^2*1" "0"             
#> [4,] "0"        "0"         "0"     "r^2*sin(ph1)^2"
g_ss(4) %_% .(+i, +j)
#> <Labeled Array> [4x4] .(+i, +j)
#>      [,1]               [,2]                [,3]           
#> [1,] "(1) / (-(1-1/r))" "0"                 "0"            
#> [2,] "0"                "(1) / (1/(1-1/r))" "0"            
#> [3,] "0"                "0"                 "(1) / (r^2*1)"
#> [4,] "0"                "0"                 "0"            
#>      [,4]                    
#> [1,] "0"                     
#> [2,] "0"                     
#> [3,] "0"                     
#> [4,] "(1) / (r^2*sin(ph1)^2)"