If the field effects such as gravity and charge, where, spreaded out in a region of space, a body experience acceleration without a contact force, is due only to time variation expressed through the conservation of energy equation across the time and space dimensions,
qv2t+12qv2c=12qc2=constant
2v2t+v2c=c2
The factor of two before time velocity, vt, is due to the fact that the kinetic energy along the time dimension at speed is not
E=12mv2t but
E=mv2t
Differentiating with respect to x,
4vtdvtdx+2vcdvcdx=0
2vtdvtdx+vcdvcdtdtdx=0
2vtdvtdx+ac=0 ----(1)
2vtdvtdtdtdx+ac=0,
2vtvcdvtdt+ac=0
Differentiating (1) again with respect to x,
2(dvtdx)2+2vtd2vtdx2+dacdx=0
−dacdx=2(dvtdx)2+2vtd2vtdx2
From the post "Calculus We Have A Problem",
vtd2vtdx2=vtv2cd2vtdt2+vtacdvtdt
So,
−dacdx=2(dvtdtdtdx)2+2vtv2cd2vtdt2+2vtacdvtdt
−dacdx=2(1vc)2(dvtdt)2+2vtv2cd2vtdt2+2vtacdvtdt ----(*)
Differentiating the conservation of energy equation with respect to time t twice,
4vtdvtdt+2vcdvcdt=0,
2vtdvtdt+vc.ac=0
2(dvtdt)2+2vtd2vtdt2+a2c+vcdacdt=0
2vtv2cd2vtdt2+2v2c(dvtdt)2=−a2cv2c−1vcdacdt
Substitute into (*)
−dacdx=2vtacdvtdt−acv2c−1vcdacdt
−dacdx=−dacdtdtdx=−1vcdacdt
2vtacdvtdt=a2cv2c
since, v2c=c2−v2t
(c2−v2t)2vtdvtdt=a3c
(c2−v2t)d(v2t)dt=a3c
(c2−v2t)d(v2t)dxdxdt=a3c
(c2−v2t)vcd(v2t)dx=a3c
(c2−v2t)√c2−v2td(v2t)dx=a3c
but, from (1)
2vtdvtdx=−ac
(c2−v2t)32(−ac)=a3c
a2c=−(c2−v2t)32
a2c=−c3(1−γ2)32
ac=ic32(1−γ2)34
which is true because ac is perpendicular to vt.
|acmax|=c32 when γ=0
ac=ic32(1−γ2)34
which is true because ac is perpendicular to vt.
|acmax|=c32 when γ=0
Maximum ac not c2 but c32.