E = γmc2
I2 = (ct)2 – x2 = (ct´)2 – x´2 =…
NM: Vwrt x=0 = (x´/t) = (x´/I)c
SRT: vwrt x=0=(x´/t´)=(x´/H)c
NM: Vwrt* = (H/t) = (c´t)/t = c´
SRT: vwrt * = (H/t´) = (ct´/t´) = c
Abstract
All of Special Relativity (SRT) is based on and can be derived from its Interval Equation
I2 = (ct)2 – x2 = (ct´)2 – x´2 = (ct´´)2 – x´´2 = (ct´´´)2 – x´´´2 … #(1)
If this is multiplied by (mc/t)2 you get the well known
total energy squared equals Pc squared plus rest mass energy squared where P is momentum:
E2 = (γmc2)2 = (Pc)2 + (mc2)2 #(2)
This analysis shows that, because all the terms in the Interval Equation are either distance, time, or their ratio, velocity, inserting any other type of quantity, such as temperature or density, or even mass, results in an equation that makes no physical sense.
Discussion
Consider an object that moves the distance x´ from its point of origin in a ‘non-moving’ (unprimed) reference frame during the time t . A beam of light simultaneously leaves the point of origin perpendicularly in the unprimed frame the distance ct where c is the speed of light. The distance ct is called the Interval (I) which is where the name for the equation comes from. The Interval Equation is about the relationship after the time t of the distances between the light and its source, the light and the object, and how far the object has moved. The following is then true:
x´2 + (ct)2 = H2 #(3)
where H2 is the distance from the light to the object after the time t .
The velocity (V) of the object is x´/t which is the distance the object moved divided by the distance the light moved times the speed of the light, i.e., (x´/I)c . Since H is greater than ct=I (the distance the light moved from its origin), then velocity H/t is greater than c . Let’s call it c´ , yielding the NM Interval Equation
(c´t)2 – x´2 = (ct)2 #(4)
But according to Special Relativity (SRT) (Einstein’s Second Postulate) the light speed must be the same in all reference frames. Since the distance H is greater than the distance I , the ‘time’ in the moving frame, call it t´, is increased to get the correct distance H , i.e., H=ct’ . So now we have, instead of (c´t)2 , we have
x´2 + (ct)2 = H2 = (ct´)2 #(5)
the Interval Equation of SRT, often written as
I2 = (ct´)2 – x´2 #(6)
which is the source of everything in SRT. In SRT, velocity v of the object becomes v=x´/t´, the ratio of distance of object from source to distance of light from object times c : (x´/ct´)c vs. V=(x´/ct)c of NM.
The gamma function (γ) , used very frequently in SRT, is usually expressed as
γ = (1–(v/c)2)–½ #(7)
However, γ can be expressed in many different ways. The formula below shows this:
γ 2 = 1/(1 – (v/c)2) = (ct´)2 / ((ct´)2 – (ct´)2(v/c)2) =
(ct´)2 / ((ct´)2– x´2) = (ct´)2 /I 2 = (x´2 + I2)/I2 = (V/c)2 + 1 #(8)
γ is simply the ratio of the distance the light is from the object to the distance the light is from its source after the time t , as shown by the third last term in Equation #(8). Another useful form is the last term.
NM has the same Interval Equation as SRT except that the final distance between the light and the object is c´/t rather than the newly invented ct´ of SRT. The c´t and ct´ have the same numerical value.
Any sum of squares can be represented as the hypotenuse of a triangle as shown below. This will be used to aid in showing what can be learned from manipulating the Interval Equation in both SRT and NM. Apply the following manipulations to the Interval Equations. Note well that the only difference between the NM and SRT equations is the c´t vs. the ct´ and both have the same numerical value.
Multiply the equations #(11) and #(12) by c4 and you get
(γmc2)2 – (mx´/t)2c2 = (mc2)2 #(13)
otherwise known as
E2 – (Pc)2 = (mc2)2 #(14)
for both SRT and NM. However, the preceding discussion shows that putting mass in the equation is invalid for both SRT and NM.
Experiment shows that mass and energy can exchange. But it isn’t due to SRT. NM has essentially the same equations.
The relationships between the variables of SRT and NM are as follows:
x´ = γvt = γ(v/c)ct = γ(v/c)I = v(γt) = vt´ = Vt = (V/c)ct = (V/c)I #(15)
400m (5/3)(4/5)300m (4/3)300m
The variables x´, t , c , and I are the same in both SRT and NM. The variables v , γ , and t´ are exclusive to SRT. The variable V is exclusive to NM. v=V/γ . Any physical situation can be described in either NM or SRT and each can be converted to the other.
Mass (m) in Special Relativity (SRT)
Dr. Sherwood Kaip
El Paso, TX
<skaip799@gmail.com>; cell: 1 (915) 309-6340
This material may be reproduced if author attribution is given.
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