The 4-D Laplace equation and wave equation

[member 141513]

In relativity, the scalar wave equation in the coordinate system (x,y,z,ict)
is
$$\frac{\partial^2\phi}{\partial x^2}+\frac{\partial^2\phi}{\partial y^2}+\frac{\partial^2\phi}{\partial z^2}+\frac{\partial^2\phi}{\partial (ict)^2}=0$$

In 3D classical mechanics, the Laplace equation is:{when the coordinate system is (x,y,z)+t}
$$\frac{\partial^2\phi}{\partial x^2}+\frac{\partial^2\phi}{\partial y^2}+\frac{\partial^2\phi}{\partial z^2}=0$$

And the ordinary wave equation is
$$\frac{\partial^2\phi}{\partial x^2}+\frac{\partial^2\phi}{\partial y^2}+\frac{\partial^2\phi}{\partial z^2}=\frac{1}{v^2}\frac{\partial^2\phi}{\partial t^2}$$

the first equation is similar to the third,and the second equation has the same format of dimensions as the first.

So, does this mean that a wave equation in 4 dimensions is a 4-D laplace equation in relativity,or something else,Thx very much.

 

[Altabeh]

In relativity, the scalar wave equation in the coordinate system (x,y,z,ict)
is
$$\frac{\partial^2\phi}{\partial x^2}+\frac{\partial^2\phi}{\partial y^2}+\frac{\partial^2\phi}{\partial z^2}+\frac{\partial^2\phi}{\partial (ict)^2}=0$$

In 3D classical mechanics, the Laplace equation is:{when the coordinate system is (x,y,z)+t}
$$\frac{\partial^2\phi}{\partial x^2}+\frac{\partial^2\phi}{\partial y^2}+\frac{\partial^2\phi}{\partial z^2}=0$$

And the ordinary wave equation is
$$\frac{\partial^2\phi}{\partial x^2}+\frac{\partial^2\phi}{\partial y^2}+\frac{\partial^2\phi}{\partial z^2}=\frac{1}{v^2}\frac{\partial^2\phi}{\partial t^2}$$

the first equation is similar to the third,and the second equation has the same format of dimensions as the first.

So, does this mean that a wave equation in 4 dimensions is a 4-D laplace equation in relativity,or something else,Thx very much.

Yes! In four dimensional spacetime of Minkowski with signature (+,-,-,-), the wave equation is

$$\square \phi=0$$

with $$\square = {\partial}^2_t-{\nabla}^2$$ being the famous D'Alembert operator, which is of course the field equation of Nordstrom's first theory of gravitation!

AB