Einstein's relativity does permits warping time (time dilation) to permit faster than light velocities(warp speed) with respect to (wrt) the ship accelerating at 1 g wrt the earth even though the velocity wrt the earth is less than C. Derived from the same equations warping space by accelerating at a constant 1 g with respect to the ship one warps both space and time also thereby obtaining warp speed wrt earth.
Using Einstein's equations placed in a
Java Calculator:
For a one way trip to the the closest star at a constant 1g wrt earth to mid trip point and decelerating at 1 g the remainder of the journey.
Trip length: 4.25 light years (ly)
Acceleration: 1.0 g wrt earth
Time on earth: 5.8780560467144 years.
Time on ship: 3.544401860293398 years
As Velocity = distance traveled/ time traveled
Velocity average wrt ship = 4.25 ly/3.5 year = 1.2 C =
warp speed 1.2 wrt ship
Velocity average wrt earth = .732 C
In the case of a 1 g acceleration measured wrt the ship so that the crew "feels" a constant artifical gravity field of 1 g the entire trip the rocket man returns to earth to find his left behind evil twin his same age having warped both space and time to obtain warp speed 1.2 wrt earth.
For a one way trip to the edge of the observable universe measuring a constant 1g wrt earth to midpoint and decelerating at 1 g the remainder of the journey.
Trip length: 1.7E10 light years.
Acceleration: 1.0 g.
Time on earth: 1.7004884192539843E10 years.
Time on ship: 45.71651222563561 years.
Velocity average with respect to (wrt) earth =
17E10 light years /1.7004884192539843 years =
.99971277707718905089055716347216 C
Velocity average wrt ship
1.7E10 light years/45.71651222563561 =
371856888.C
1 g acceleration wrt earth wrt earth Velocity = .99971277707718905089055716347216 C
Velocity average wrt ship = warp speed 371856888.7
of case one (1 g wrt earth).
In case two (1 g wrt ship)
This is also the velocity wrt earth when measuring the constant 1 g wrt ship.
Therefore, the rocket man returns to earth 90 years later to find his left behind evil twin his same age instead of ancient bones having warped both space and time to obtain:
warp speed 371856888.7 wrt earth.
If Christ was resurrected after death and did ascended to the heavens on a 1 g rocket ship to visit his father at a 1000 ly distant star he could return today Obeying Einstein's laws some 27 years older and he might be ticked off so behave as:
Trip length: 1000.0 light years.
Acceleration: 1.0 g.
Time on earth: 1002.2235407106124 years.
Time on ship: 13.453214568643295 years.
For the energy requirements and atomic rocket technology
for say a minimum distance to mars?
Minimum Trip length: 5.7605E-6 light years.
Acceleration: 1.0 g.
Time on earth: 0.006683627669698874 years. (1.73 days)
Time on ship: 0.00668361442242196 years.(1.73 days)
Velocity Max= 1g times time to mid distance = 687.960 km/sec
Estimating the ratio of mass final/mass initial >=.99 as atomic powered rockets are a minimum of 100,000,000 times as powerful as
chemical rockets of the same size when operating at near 100% efficiency like current chemical rockets so for same size payload to
given velocities the engine can be 100,000:1 for fission engines to 1,000,000,000:1 to one for fusion engines smaller by that factor.
Algebraic shorthand operands
: denotes defined as;
^denotes exponent;
~ approximate
delta denotes "change in"
Variables
V=velocity; Vr=velocity of rocket; Ve= velocity of exhaust
E=Energy; Ek = kinetic energy; Ep = potential energy
M=Mass; Mr=mass of rocket; Me = Mass of Exhaust
Constants
1 HorsePower = 33,000 Foot Pounds / 1 Minute
C = 186,000 miles/sec as constant velocity of light
g = 1gravity= 32.2 feet/sec^2
Gearth =earth gravitational constant = 6.67 x 10^-11 N*m^2 / kg^2 (Gmars ~1/3Gearth
r =equatorial radius of Earth =6378 km
mars distance =Minimum = 54.5Ee6 kilometers; Maximum 401.3e6 kilometers
Equations(eq)
(eq. 1) MeVe=MrVr :=momen6um conservative static science equation of rockets conserving momentum
(eq. 1a) Vrfinal = Vrinitial Ve * ln([Mr Me] / Mr := nonconservative engineering dynamic rocket equation
Not useful for atomic rockets as the Mf is much greater than 10 but very useful for chemical rockets
with typical Mf much less than 1 .(082 for space shuttle) so not used below as the difference in
calculated values as it is thought insignificant if not zero and 1a introduces errors with high Mf.
(eq.2) Ek = (MV^2)/2 := kinetic energy
(eq.3) Ep = MC^2 := potential energy
(eq.4) Ve=MrVr/Me := exhaust velocity
(eq.5) Vr = MeVe/Mr := rocket velocity
(eq.6) Vescape=square root ( 2GM/r)
(eq.7) Thrust =Mr * deltaVr/delta T = -Ve* delta Me/delta t
(eq. specifc impules(Isp) = Thrust /mass flow rate times g or = Mass flow rate* Ve; or
Isp = Ve/g
(eq.9) V=AT = acceleration times time
(eq.10) Mf= Mass fraction= (Mr Me)/Me
Calculated constants or Specified constants
Mr=100,000 tons
Me~=1000 tons
Thrust== -436 trillion horse power
Mass converted to energy ~= 1 ton
Total Impulse Momentum(Itot) =2887799039640 tons- feet/minute (deep space)
Isp = 1,494,200 seconds = specific impulse in earth's gravity field
Vescape from Earth = 11.2 km/sec; acceleration time at 1.2 g = 97.2 minutes
Vescape from Mars ~= 3.7 km/sec; acceleration time at .53 g = 32.4 minutes
Vrpeak at mid point=687.960 km/sec
Tmin = minimum distance one way trip time= 1.73 days 129.6 minutes = 1.82 days
Tmax = maximum distance coasting at zero mid journey for 1.45 days Tmin = 2.64 days
Round trip = 5.3 days (plus 1 day minus 3 days) depending on where Mars is at launch time using line of sight navigation
