**9. Acknowledgment**

I thank Yosuke Nakata for helpful discussions. This research was supported by a Grant-in-Aid for Scientific Research on Innovative Areas (No. 22109004) and the Global COE program "Photonics and Electronics Science and Engineering" at Kyoto University.

**3** 

Wei-Tou Ni1,2

*1Taiwan, ROC* 

*2China* 

*1Center for Gravitation and Cosmology* 

*2Shanghai United Center for Astrophysics Shanghai Normal University, Shanghai,* 

**Foundations of Electromagnetism, Equivalence** 

Standard electromagnetism is based on Maxwell equations and Lorentz force law. It can be derived by a least action with the following Lagrangian density for a system of charged

where *Fij ≡ Aj,i - Ai,j* is the electromagnetic field strength tensor with *Ai* the electromagnetic 4 potential and comma denoting partial derivation, *ηij* is the Minkowskii metric with signature (+, -, -, -), *mI* the mass of the *I*th charged particle*, sI* its 4-line element, and *jk* the charge 4 current density. Here, we use Einstein summation convention, i.e., summation over repeated indices. There are three terms in the Lagrangian density *LEMS* – (i) *LEM* for the electromagnetic field, (ii) *LEM-P* for the interaction of electromagnetic field and charged

The electromagnetic field Lagrangian density can be written in terms of the electric field **E** [*≡* (*E1*, *E2*, *E3*) *≡* (*F01*, *F02*, *F03*)] and the magnetic induction **B** [*≡* (*B1*, *B2*, *B3*) *≡* (*F32*, *F13*, *F21*)] as

 *LEM* = (1/8π)[**E**2-**B**2]. (2) This classical Lagrangian density is based on the photon having zero mass. To include the

needs to be added (Proca, 1936a, 1936b, 1936c, 1937, 1938). We use *ηij* and its inverse *ηij* to raise and lower indices. With this term, the Coulomb law is modified to have the electric potential *A0*,

 *A0* = *q*(*e-μr*/*r*), (4) where *q* is the charge of the source particle, *r* is the distance to the source particle, and *μ*

*LProca* = (*mphoton2c2/8*π*ħ2*)(*AkAk*), (3)

*LEMS=LEM+LEM-P+LP=-*(1/(16π))[(1/2)*ηikηjl-*(1/2)*ηilηkj*]*FijFkl-Akjk-*Σ*I mI*[(*dsI*)/(*dt*)]*δ*(*x-xI*), (1)

**1. Introduction** 

particles in Gaussian units (e.g., Jackson, 1999),

particles and (iii) *LP* for charged particles.

effects of nonvanishing photon mass *mphoton*, a mass term *LProca*,

(*≡mphotonc/ħ*) gives the inverse range of the interaction.

**Principles and Cosmic Interactions** 

*Department of Physics, National Tsing Hua University, Hsinchu,* 

#### **10. References**

Burke, W. L. (1985). *Applied Differential Geometry*, Cambridge University Press.

