Relativity: the Special and General theories

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"I stand at the window of a railway carriage which is travelling uniformly, and drop a stone on the embankment, without throwing it. Then, disregarding the influence of the air resistance, I see the stone descend in a straight line. A pedestrian who observes the misdeed from the footpath notices that the stone falls to earth in a parabolic curve. I now ask: Do the "positions" traversed by the stone lie "in reality" on a straight line or on a parabola? Moreover, what is meant here by motion "in space" ? From the considerations of the previous section the answer is self-evident."

This is a quote from
Albert Einstein: Relativity
Part I: The Special Theory of Relativity, Space and Time in Classical Mechanics.

Could someone please explain to me why, to the pedestrian, the stone falls in a parabolic curve and what type of parabolic curve it is? thank you

Last edited:
Could someone please explain to me why, to the pedestrian, the stone falls in a parabolic curve
Well, the vertical motion is proportional to y**2 (for both observers) and the horizontal motion in the reference frame of the pedestrian is proportional to the speed of the train , which is linear.

SO ... if it is indeed a parabola (and I haven't checked that but if it's what you read, it's likely correct) then I can only assume that if you combine the two motions and calculate the loci, you get the equation for a parabola.
and what type of parabolic curve it is? thank you
What do you mean, what kind of parabolic curve?

FactChecker
Do the "positions" traversed by the stone lie "in reality" on a straight line or on a parabola?
You are assuming that one of the two frames of reference gives some kind of absolute truth, which is a false assumption. The answer to "how is the motion really described" is "it depends on your frame of reference." Neither one is absolute. All motion is relative.

To the pedestrian, the stone is initially traveling with some horizontal velocity, let's call it Vx, and zero vertical velocity, let's call it Vy. Gravity begins to accelerate the stone downwards at a rate of about 9.8 m/s2, while the horizontal velocity remains the same.

If we graph the velocities vs time, we see that since Vx remains constant and Vy accelerates at a rate proportional to time squared, the two combine to make a parabola. In other words, the equation that defines the trajectory is that of a parabola.

phinds said:
Well, the vertical motion is proportional to y**2 (for both observers)
I think that should be t2, not y2.

Drakkith said:
To the pedestrian, the stone is initially traveling with some horizontal velocity, let's call it Vx, and zero vertical velocity, let's call it Vy. Gravity begins to accelerate the stone downwards at a rate of about 9.8 m/s2, while the horizontal velocity remains the same.

If we graph the velocities vs time, we see that since Vx remains constant and Vy accelerates at a rate proportional to time squared, the two combine to make a parabola. In other words, the equation that defines the trajectory is that of a parabola.
ohh ok i got it!! thank you very much for the explanation! I'm not too good at physics.

what would t^2 and y^2 mean here?

what would t^2 and y^2 mean here?
Those are variables, where t is time and y is the vertical position coordinate.

Drakkith said:
I think that should be t2, not y2.

I'm not too good at physics.
It might be a good idea to study some classical kinematics before trying Einstein's book. You could look at something like the Khan Academy:

PeroK said:
It might be a good idea to study some classical kinematics before trying Einstein's book. You could look at something like the Khan Academy:

hmm ok, will do! thank you!!

This is about Newtonian dynamics/kinematics. We can explain this in two different ways:

(a) The point of view of Einstein in the train:

He uses Cartesian coordinates ##(x',y',z')## with the ##z'## axis pointing upwards. The equation of motion is
$$m \ddot{\vec{x}}'=-m \vec{g}=-m (0,0,g).$$
Einstein drops the stone initial at rest from ##\vec{x}_0'=(0,0,h)##. By integrating twice he finds
$$\vec{x}'(t)=-\vec{g}{2} t^2 + \vec{x}_0'=(0,0,h-g t^2/2).$$
The trajectory from Einstein's point of view is a straight line down.

How does the pedestrian on the embankment describe this? He uses coordinates
$$\vec{x}=\vec{x}'+\vec{v} t,$$
where ##\vec{v}=\text{const}=(v,0,0)## is the velocity of the train relative to him. So he describes the motion of the stone by
$$\vec{x}=(v t,0,h-g t^2/2),$$
which is a parabola. This becomes more obvious by describing the trajectory in the form ##z=f(x)##. To that end you simply eliminate time by ##x=v t## or ##t=x/v##:
$$z=h-g t^2/2 = h-g x^2/(2 v^2),$$
which obviously is a parabola in the ##(x,z)## plane.

(b) The point of view of the pedestrian

He describes the motion of the stone by the same equation as einstein,
$$m \ddot{\vec{x}}=-m \vec{g}=-m(0,0,g).$$
But now from his point of view the initial conditions are ##\dot{\vec{x}}(0)=\vec{v}=(v,0,0)##, ##\vec{x}(0)=\vec{x}_0=(0,0,h)##. Which gives
$$\vec{x}(t)=-\vec{g} t^2/2 + \vec{v} t +\vec{x}_0=(v t,0,h-g t^2/2),$$
which is the same result we got using Einstein's point of view.

The pedestrian will also conclude rightly what Einstein observes. He just uses the inverse transformation,
$$\vec{x}'=\vec{x}-\vec{v} t=(0,0,h-g t^2/2),$$
which is also the same as Einstein gets by solving the initial-value problem using his coordinates.

The parabola or straight line behavior that Einstein describes is an example of Galilean relativity (worth googling). Galilean relativity was, until Einstein developed special relativity, presumed to be the right mathematical relationship between positions described by observers moving relative to one another; as we might guess from the name, it was formalized by Galileo so has been around for a very long time.

Basically special relativity and the Lorentz transformation replaced Galilean relativity and the Galilean transform.

vanhees71

What is the main difference between Special Relativity and General Relativity?

Special Relativity deals with the physics of objects moving at constant speeds, particularly those close to the speed of light, and introduces the concept that space and time are intertwined into a single continuum known as spacetime. General Relativity, on the other hand, extends these concepts to include acceleration and gravity, describing gravity not as a force but as a curvature of spacetime caused by mass and energy.

What is time dilation and how does it work in Special Relativity?

Time dilation is a phenomenon predicted by Special Relativity where time passes slower for an object in motion compared to an object at rest. This effect becomes more pronounced as the object's speed approaches the speed of light. It occurs because, according to Special Relativity, the speed of light is constant for all observers, leading to the conclusion that time must adjust to preserve this constant speed.

What is the equivalence principle in General Relativity?

The equivalence principle is a key concept in General Relativity which states that locally (in a small region of spacetime), the effects of gravity are indistinguishable from the effects of acceleration. This means that an observer in a closed room cannot tell whether the force they feel is due to gravity or due to the room accelerating through space.

How does General Relativity explain the bending of light around massive objects?

General Relativity explains the bending of light around massive objects through the curvature of spacetime. Massive objects like stars and planets create a distortion in spacetime, and light follows the curved paths within this distorted spacetime. This phenomenon, known as gravitational lensing, has been confirmed through various astronomical observations.

What are some experimental confirmations of General Relativity?

Several key experiments and observations have confirmed General Relativity. One of the earliest confirmations was the observation of the bending of light from stars during a solar eclipse in 1919. Other confirmations include the precise measurements of the precession of Mercury's orbit, the detection of gravitational waves by LIGO, and the accurate functioning of GPS systems, which must account for the effects of both Special and General Relativity to provide precise location data.

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