Energy required to accelerate from different reference frames

Question

So I've recently been studying relativity a lot trying to understand it and I feel like I grasp most things conceptually but I have one issue I've been trying to understand for the last couple days and I just can't seem to find an answer anywhere.

Let's say you're traveling very fast, somewhere close to the speed of light. At this point you're in an inertial reference frame. From this frame you shouldn't be aware that you are anywhere near the speed of light. Now you want to accelerate by 5 m/s to a new reference frame that is even closer to the speed of light. Is there any difference in the amount of energy required to accelerate to the new reference frame in this scenario from the amount required to accelerate by 5 m/s from a slower reference frame like here on Earth?

I sort of understand the concept of relativistic energy but if that somehow applies here I don't get how. If you are in an inertial reference frame, then it seems that the amount of energy required to accelerate to a new reference frame should be the same no matter what your frame is. However, from various things I've read I get the impression that the energy required to accelerate at a constant rate is not constant, which seems to make sense from a static reference from but not from an accelerating one.

I'm sure I'm missing something or there's a flaw in my thinking somewhere. I hope this all makes sense since I'm still very new to this. Also, if there's a way to explain the concept with little to no math that would be very helpful since I still don't grasp a lot of the math involved in relativity. I'll take what I can get though. This has been eating at me too much.

Update: For anyone wondering about this same thing I finally found the answer explained in a way I grasped it here.

Answer 1

In special relativity the transition from one frame to another is given by the Lorentz boosts. This is not quite the same as an acceleration, but a transformation that relates observations on one frame with another. We can think of an acceleration as being a succession of infinitesimal Lorentz boosts that map one frame to another.

The infinitesimal distance in flat spacetime $ds$ for a particle is given by $$ ds^2 = c^2dt^2 - dx^2 - dy^2 - dx^2 $$ where this distance $ds – cd\tau$, for $\tau$ the time as measured by a clock on the frame of the particle. Let us consider the motion of this particle in the $x$ direction. Now divide through by $ds^2$ to get $$ 1 = \left(\frac{dt}{d\tau}\right)^2 - \frac{1}{c^2}\left(\frac{dx}{d\tau}\right)^2. $$ We can write this according to four-velocity $U_t = \frac{dt}{d\tau}$, $U_x = \frac{dx}{d\tau}$ $$ 1 = U_t^2 - U_x^2. $$ Now take the derivative of this with respect to $\tau$ so that $$ 0 = \left(\frac{dU_t}{d\tau}\right)U_t - \frac{1}{c^2}\left(\frac{dU_x}{d\tau}\right)U_x. $$ This leads to the interesting observation that in spacetime the four-acceleration is perpendicular to the four-velocity.

This system of equations leads to a solution for the four-velocity $$ U_t = cosh(g\tau),~U_x = c~sinh(g\tau), $$ for $g$ the acceleration. We can see that the equation defines a hyperbola in $t, x$ coordinates. For large $\tau$ that the hyperbola is approximately $U_t^2 = U_x^2$, and it is not hard to get these in the $t, x$ coordinates. We can also see that the coordinate based velocity is $$ \frac{dx}{dt} = \frac{U_x}{U_t} = c~tanh(gt), $$ which indicates this particle asymptotes to the speed of light as $\tau\rightarrow\infty$.

When it comes to energy we appeal to the four-momentum interval in special relativity $$ m^2 = E^2 - p^2 $$ where $E = mU_t$ and $p = mU_x$ are energy and spatial momentum respectively. Using the properties of hyperbolic trigonometric functions this can be seen. We can see right off that energy is given by a hyperbolic cosine function that diverges enormously as $\tau\rightarrow\infty$.

Answer 2

The fundamental principle in relativity (both special and general) is that the laws of physics are the same for all observers. We express this most naturally using the principle of general covariance, which means that the laws of physics are expressed using tensor (including vector and scalar) quantities. Then we can understand the laws of physics most easily in terms of proper quantities, that is to say in terms of properties as measured by an observer moving with the object being measured.

If you are moving very fast compared to another observer, you are not moving close to the speed of light in your own reference frame, because the speed of light is a fundamental property in physics and is the same for all observers. You apply a proper acceleration and it will require a certain amount of energy, but the speed of light remains constant.

Now consider it from the point of view of the other observer. He does not measure your proper quantities directly, but if he has expressed the laws of physics correctly in terms of tensors he can calculate your proper acceleration, and he can calculate the amount of energy you used. He will think you are moving nearer to the speed of light, but he will not think you have increased your speed by much. The apparent acceleration which he sees is not a proper quantity, and he is misleading himself if he uses it in a calculation of energy without first calculating the appropriate proper quantities.