Scalar invariants in gravitational and electromagnetic fields

The paper discusses some scalar invariants in the gravitational field and electromagnetic field by means of the characteristics of the quaternions. When we emphasize some definitions of quaternion physical quantities, the speed of light, mass density, energy density, power density, charge density, and spin magnetic moment density etc. will remain the same respectively in the gravitational and electromagnetic fields under the coordinate transformation. The results explain why there are some relationships among different invariants in the gravitational and electromagnetic fields.