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I don't understand the concept of a frame of reference. I've read from online sources and they defined it as an abstract coordinate system. What is the coordinate in this system then and how do they all connect to each other? Then I've read the mentioning of an observer and the observer's state of motion and I don't understand how that relates to a frame of reference. I don't understand how these concepts all work together to define a frame of reference. Please help.

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The modern concept of a frame of reference did not exist in Newtonian mechanics; the phrase reference frame was not used until the late nineteenth century. In Newton's time the fixed stars were invoked as a reference frame supposedly at rest relative to absolute space, but that is as near as it gets. Of course we now know that the "fixed" stars are not fixed. Newton resorted to the notions of absolute space and absolute time, as a starting point from which he could give meaning to his laws precisely because no better concept existed. Thus the precise answer to this question is that in the context of Newtonian mechanics you must first assume absolute space and absolute time.

Of course this was changed by Einstein. A reference frame does not refer to a coordinate system, such as absolute space (as some authors seem to think). A frame refers to physical matter. A reference frame is the matter relative to which a coordinate system is defined. We can talk of the Earth frame, the frame of the fixed stars, and when travelling by car it is natural to think of the car as the reference frame. The minimum requirement for a reference frame is that it must contain a clock, a ruler (or equivalent apparatus for measuring distance), and a physical definition for coordinate axes.

In general relativity Newtonian mechanics is re-expressed within the context of a reference frame (not a reference frame in the context of Newtonian mechanics). First we can use Newton's first law to define an inertial reference frame (replacing the need to use Newton's first law to determine "absolute space"). Then inertial frames are necessarily local and Newtonian mechanics holds in inertial reference frames.

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    $\begingroup$ A reference frame is the matter relative to which a coordinate system is defined.” - Sadly this is rarely understood. So often people use the Kruskal and other “creative” coordinates to describe black holes without understanding that there are no reference frames in these coordinates. +1 $\endgroup$
    – safesphere
    Jun 23, 2020 at 4:14
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A reference frame is a coordinate system that defines the position of any body and if the body is at rest or moving.

A good example is the longitude and latitude given by a GPS device. We assume that if that numbers change we are moving, the direction and speed can be related to how, and how fast they move.

From an observer out of the surface of the Earth but close to it, maybe the same frame is still useful. The ISS, at 400 km of altitude can track itself using the same system, only adding the altitude.

But from an hypothetical lunar base, it would be not pratical. They would probably use lunar coordinates to define position and movement.

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A simple but incomplete answer, by means of an example.

A train carriage moves with uniform speed and clockwise on a circle:

Frame of reference

The reference frame $(x,y)$ certainly allows adequate and easy description of the velocity vector and forces (vectors) acting on the carriage.

But assume that inside the carriage is a person who is throwing a ball, maybe upward or downward or forward or backward (it doesn't matter really).

It should be intuitive that $(x,y)$ is a nightmare to describe the ball's physics (kinetics, forces) with. Another reference frame, $(x',y')$, this one attached to the moving carriage is far more suited to that.

This situation is common: we all sit on a sphere rotating about the Earth's $\text{NS}$ axis and trains with people in them move on it in all directions.

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