In relativistic travel, which refers to traveling at significant fractions of the speed of light (c), we learn that many aspects of traditional spaceflight do not apply. This is due to Einstein's special relativity, which introduces the concept of dilation. In the context of relativistic spaceflight, this concept has various implications, including time dilation, mass dilation, and length contraction. While I won't go into detail about time dilation or length contraction in this write-up, it is important to note that all these concepts are connected through a critical formula known as the Lorentz factor. For the purpose of this scenario, we will be using its inverse:
The Lorentz factor is useful for various applications, but in this discussion, we will focus on its effects on mass dilation and, consequently, relative acceleration. Specifically, we will utilize it to correct the effective acceleration at relativistic speeds. In simple terms, the Lorentz factor characterizes the relative increase of things like mass with increasing velocity. When taken as the inverse, the Lorentz factor will be a number between 1 and 0, following a decaying trend similar to an exponential function. The relationship between the Lorentz factor and relativistic velocity is depicted in the image below:
To use the Lorentz factor, we can simply multiply it by a relevant value. For example, we can employ the following equation to determine relativistic mass as a function of velocity:
For mass, we must use the conventional, non-inverse. For simulating thrust, we use the inverse.
The Lorentz factor can be applied to various other aspects as well, with one notable example being the simulation of mass dilation by limiting thrust according to the function.
My study began with the goal of constraining the top speed of my fastest interstellar vehicles (ISV) to c. Some of the ISVs have delta-v reserves upward of 1.3c. One such vehicle is my Project Valkyrie Antimatter ISV replica. It would be unrealistic for these vehicles to exceed the speed of light.
To achieve this, I simulated mass dilation. Mass dilation is entirely governed by the Lorentz factor, and consequently, acceleration is as well. By the same logic, we can infer that thrust can directly scale with the Lorentz factor. However, in reality, the loss in acceleration is driven by mass dilation, not thrust dilation, as the latter is not a significant factor. This approach greatly simplifies the code required to simulate the phenomenon of mass dilation.
In fact, in the Vizzy code, we don't even need to calculate the effective thrust. We can simply set a throttle limiter that scales with the inverse of the Lorentz factor. It's that simple!
Make sure to set your engine throttle input to slider 1, otherwise, it will not work.
Well, that's all for now. I hope you've learned something new about this topic today. I know I certainly did!
Feel free to discuss this topic in the comments.
Further reading:
[1] https://en.wikipedia.org/wiki/Lorentz _ factor
[2] https://en.wikipedia.org/wiki/Inertial _ frame _ of _ reference
[3] https://galileoandeinstein.phys.virginia.edu/lectures/mass_increase.html
By the way: the formula concerning the relativisitc mass with the Lorentzfactor is also quite useful for calculation of electron trajectories in fields.