# How can the energy applied to an object differ depending on the reference frame

This is a question about energy being relative. To accelerate an object travelling at very fast (relativistic) speeds requires more and more energy (as I understand that more and more energy translates into mass rather than speed). But for the object itself it will be at rest in its own frame of reference so from “its” perspective a force applied to it will not have relativistic implications. How can this be explained ?

• Are you asking for an intuitive explanation?
– TLDR
Aug 8, 2021 at 23:37

You don't need to take special relativity into account to get a scenario where different observers will disagree about how much energy is applied to an object. Here's why.

In a simple scenario where the force vector is parallel to the object's velocity, work is the amount by which an object's kinetic energy changes, and is equal to $$F\Delta x$$, where $$F$$ is the force applied to an object, and $$\Delta x$$ is the total distance the object travels as the force is being applied. Even if the force and the duration of the period over which the force is applied are constant, the starting velocity can still affect how much kinetic energy the object can appear to gain or lose. If the object appears to be moving quickly to one observer, $$\Delta x$$ will be very large, the force will be applied over a great distance, so that observer will see a lot of energy added to or taken from the object. If an observer sees the object initially at rest, however, $$\Delta x$$ will be smaller, because the force will be applied over a smaller distance, so less work will be done on it.

• I don't understand what you're talking about. What I'm describing here is a real thing, and it's responsible for the Oberth Effect. en.wikipedia.org/wiki/… Aug 9, 2021 at 0:28
• @Viesik that is incorrect. Energy is frame variant
– Dale
Aug 9, 2021 at 0:38
• @Veisik you're thinking of momentum, not kinetic energy. Aug 9, 2021 at 0:40

Kinetic energy, or energy, is not an invariant quantity, but is frame dependent, both in relativity and classical mechanics.

But for the object itself it will be at rest in its own frame of reference so from “its” perspective a force applied to it will not have relativistic implications. How can this be explained ?

But there is no meaningful concept as the kinetic energy "in its own frame of reference". You need to specify a frame of reference, from which you can measure the objects kinetic energy, momentum etc.

As measured from within the same frame of reference of the object, or any other frame moving with constant velocity relative to it, its kinetic energy and momentum will be zero.

• @Viesik that is not correct. KE is frame variant in SR, just as it is in Newtonian mechanics
– Dale
Aug 9, 2021 at 0:36
• @Viesik No, it most certainly is not. If velocity $v$ depends on frame of reference, then so to does kinetic energy, $1/2mv^2$ Aug 9, 2021 at 0:36
• Right. And relative velocity is different for different observers (moving with different velocities). Therefore, so to is kinetic energy different, for observers moving with different velocities. Aug 9, 2021 at 0:45
• @Viesek you are wrong here. You should listen to Joseph h
– Dale
Aug 9, 2021 at 0:47
• @Cleonis the energy that is available for creating particles etc is specifically the energy in the center of momentum frame, which is the invariant mass of the system of particles. That is not the same as the energy in any other frame. There is no if’s and’s or but’s here. Energy is frame variant. It is the timelike component of the four momentum, so its frame variance is indisputable
– Dale
Aug 9, 2021 at 4:23

My explanation is quite simple. You are standing still somewhere on earth, therefore someone observes your velocity is zero. But at the same time an alien in the space, outside the earth, sees you are moving with a speed of 1470.2 km/h ! Because earth is rotating and you are also moving with that speed : )

P.S. : Forgot to mention about kinetic energy. But I am sure you know $$KE=\frac 12mv^2$$

I think you have answered your own question. In the frame in which an object is moving at non-relativistic speeds, the result of applying a force to it will be an increase in its speed in accordance with Newton's second law. When viewed from another frame moving at a relativistic speed compared with the first, the increase in the speed of the object will be appear to be less. It is simply a manifestation of the relativistic addition of velocities.