Engineering
Lorentz Factor (gamma), Relativistic Time Dilation, Length Contraction & Kinetic Energy
Calculate Hendrik Lorentz & Albert Einstein's relativistic factor gamma = 1 / SQRT(1 - beta^2), dilated coordinate time, contracted length, and (gamma - 1)*mc^2.
Specify the frames and the measured quantities
For constant relative velocity between inertial frames in special relativity, beta=v/c and gamma=1/sqrt(1-beta²), with |beta|<1 for a massive object. Signed beta changes direction, not these scalar results. This worksheet does not include acceleration histories or gravitational clock effects.
Proper elapsed time is measured by the clock at rest in its own frame. The other inertial frame assigns gamma times that interval between the same clock events. Proper length is measured at rest; moving length requires simultaneous endpoint measurements in the observer frame and contracts only along the motion.
Rest energy E0=mc² and kinetic energy (gamma-1)*E0 use the same energy units. Here E0 is an illustrative rounded electron rest energy in MeV. The formula is not for a photon with a rest frame. GPS timing is not determined by this special-relativity calculation alone.
| Relativistic Quantity | Formula (beta = v / c in B2) | At v = 0.80c (gamma = 5/3) | At Rest v = 0c (gamma = 1) |
|---|---|---|---|
| Lorentz Factor gamma | 1 / SQRT(1 - B2^2) | 1 / 0.60 = 1.6667 | 1.0000 |
| Dilated Coordinate Time t | B3 / SQRT(1 - B2^2) (tau0 = 6 yr) | 6 / 0.60 = 10.00 years | 6.00 years |
| Contracted Length L | B4 * SQRT(1 - B2^2) (L0 = 100 m) | 100 * 0.60 = 60.00 meters | 100.00 meters |
| Relativistic KE (MeV) | (gamma - 1) * B5 (e- = 0.511 MeV) | (2/3) * 0.511 = 0.3407 MeV | 0.0000 MeV |
Guarded Lorentz Factor gamma = 1 / SQRT(1 - (v / c)^2)
Validate numeric beta and |beta|<1 before taking the square root. Values extremely close to one are sensitive to input and floating-point precision.
=IF(COUNT(B2)<>1,"Invalid inputs",IF(ABS(B2)>=1,"Subluminal |v/c| < 1 Required",ROUND(1/SQRT(1-B2^2),4)))Elapsed coordinate time from proper time
B3 is a nonnegative proper duration; this result assumes constant relative velocity. It does not by itself describe a round-trip accelerated journey.
=IF(COUNT(B2,B3)<>2,"Invalid inputs",IF(ABS(B2)>=1,"Subluminal |v/c| < 1 Required",IF(B3<0,"Invalid inputs",ROUND(B3/SQRT(1-B2^2),2))))Length measured parallel to motion
B4 is the nonnegative rest length in metres. Perpendicular dimensions are not contracted by this formula.
=IF(COUNT(B2,B4)<>2,"Invalid inputs",IF(ABS(B2)>=1,"Subluminal |v/c| < 1 Required",IF(B4<0,"Invalid inputs",ROUND(B4*SQRT(1-B2^2),2))))Kinetic energy in MeV
The displayed expression is algebraically equivalent to (gamma-1)*E0, using beta²/[s*(1+s)] with s=sqrt(1-beta²). It avoids subtracting two nearly equal numbers at low speed. Four-decimal rounding can still display a small nonzero energy as zero.
=IF(COUNT(B2,B5)<>2,"Invalid inputs",IF(ABS(B2)>=1,"Subluminal |v/c| < 1 Required",IF(B5<0,"Invalid inputs",ROUND((B2^2/(SQRT(1-B2^2)*(1+SQRT(1-B2^2))))*B5,4))))Try the example
Use the original values or change them to test your understanding.