File:Space Travel with Constant Acceleration.png

Summary[edit]

Licensing[edit]

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Description[edit]

A rocket leaves Earth and travels through space at a constant proper acceleration of 1 g. Inside the rocket, it feels like an artificial gravity with the same intensity as on Earth. After some time ${\displaystyle t/2}$, the rocket turns around and decelerates at 1 g for the remainder of the trip until it comes to a full stop at time ${\displaystyle t}$. The plot shows how far the rocket will have traveled as a comoving distance to Earth for different physics models:

• Newtonian: no speed limit. Absolute space and time.
• Relativistic (Earth): from an inertial frame of reference, the acceleration seems to decrease as the rocket gets closer and closer to the speed of light.
• Relativistic (traveler): from the traveler's perspective, everything around them seems to move faster and faster and eventually relativistic effects become noticeable. Length contraction makes the distance to the destination decrease much faster than in the non-relativistic scenario.
• Λ-CDM: Takes into account the expansion of the universe. This is pretty close to what would happen in the real world according to our best knowledge.

Values[edit]

Equations[edit]

Newtonian equation[edit]

${\displaystyle t=2{\sqrt {x/g}}}$

Relativistic equations[edit]

(Ignoring the expansion of the universe)

${\displaystyle t_{\mathrm {Earth} }={\frac {2c}{g}}{\sqrt {\left(1+{\frac {gx}{2c^{2}}}\right)^{2}-1}}}$

${\displaystyle t_{\mathrm {Traveler} }={\frac {2c}{g}}\mathrm {arccosh} \left(1+{\frac {gx}{2c^{2}}}\right)}$

Λ-CDM equations[edit]

(Relativistic with expansion of the universe)

${\displaystyle t_{\Lambda }={\frac {2}{3H_{0}{\sqrt {\Omega }}_{\Lambda }}}}$

${\displaystyle {\bar {a}}(t)={\sqrt[{3}]{{\frac {\Omega _{m}}{\Omega _{\Lambda }}}\mathrm {sinh} ^{2}\left({\frac {t+t_{0}}{t_{\Lambda }}}\right)}}}$

${\displaystyle a(t)={\bar {a}}(t)/a(0)}$

${\displaystyle x_{acc}(t)=\int _{0}^{t/2}{\frac {gt'}{a(t'){\sqrt {1+\left(gt'/c\right)^{2}}}}}dt'}$

${\displaystyle x_{dec}(t)=\int _{0}^{t/2}{\frac {gt'}{a(t-t'){\sqrt {1+\left(gt'/c\right)^{2}}}}}dt'}$

${\displaystyle x(t)=x_{acc}(t)+x_{dec}(t)}$

${\displaystyle \tau (t)={\frac {2c}{g}}\mathrm {arcsinh} \left({\frac {gt}{2c}}\right)}$

Parametric plot: ${\displaystyle \left(\tau (t),x(t)\right)}$

Images used[edit]

• 
  Proxima Centauri
• 
  Vega
• 
  Betelgeuse
• 
  Milky Way
• 
  Andromeda Galaxy
• 
  Hubble eXtreme Deep Field

Tools used[edit]

• Python (Matplotlib, NumPy, SciPy)
• GIMP
• Inkscape