of the vector dot product, and when this is extended to 4-dimensional spacetime, a change of sign is introduced to either the spatial co-ordinates or the time co-ordinate depending on the convention used. This is due to the non-Euclidean nature of spacetime. In this article, we place a negative sign on the spatial coordinates (the time-positive metric convention ). The factor of (1/c) is to keep the correct unit dimensionality, [length], for all components of the 4-vector and the (−1) is to keep the 4-gradient Lorentz covariant. Adding these two corrections to the above expression gives the correct definition of 4-gradient: See also Four-vector Four-position Four-velocity Four-acceleration Four-momentum Four-force Four-current Four-potential Four-frequency Four-wavevector Four-spin Ricci calculus Index notation Tensor Antisymmetric tensor Einstein notation Raising and lowering indices Abstract index notation Covariance and contravariance of vectors References Note about References Regarding the use of scalars, 4-vectors and tensors in physics, various authors use slightly different notations for the same equations. For instance, some use for invariant rest mass, others use for invariant rest mass and use for relativistic mass. Many authors set factors of and and to dimensionless unity. Others show some or all the constants. Some authors use for velocity, others use . Some use as a 4-wavevector (to pick an arbitrary example). Others use or or or or or , etc. Some write the 4-wavevector as , some as or or or or or . Some will make sure that the dimensional units match across the 4-vector, others do not. Some refer to the temporal component in the 4-vector name, others refer to the spatial component in the 4-vector name. Some mix it throughout the book, sometimes using one then later on the other. Some use the metric , others use the metric . Some don't use 4-vectors, but do everything as the old style E and 3-space vector p. The thing is, all of these are just notational styles, with some more clear and concise than the others. The physics is the same as long as one uses a consistent style throughout the whole derivation. Further reading S. Hildebrandt, "Analysis II" (Calculus II), , 2003 L.C. Evans, "Partial differential equations", A.M.Society, Grad.Studies Vol.19, 1988 J.D. Jackson, "Classical Electrodynamics" Chapter 11, Wiley Four-vectors