General Questions about Special Relativity

[soy252]

I just need some clarification but if one person goes . 5c relative to Earth in one direction and another person goes .5 c in the other direction then wouldn't each person be going to speed of light relative to the other person? I'm sure that there is some sort of flaw in this thought experiment but could someone explain this to me?

 

[Ibix]

I just need some clarification but if one person goes . 5c relative to Earth in one direction and another person goes .5 c in the other direction then wouldn't each person be going to speed of light relative to the other person? I'm sure that there is some sort of flaw in this thought experiment but could someone explain this to me?

Velocities do not add that way. The immediate cause of that is that a moving object's rulers are length contracted, their clocks are time-dilated, and also de-synchronised due to the relativity of simultaneity. So its notions of time and distance and, hence, velocity do not match up with your intuition. If there are two objects with velocities u and v in a given frame then according to the first one the second has velocity $$v'=\frac {v-u}{1-uv/c^2} $$In your case, with velocities of ±0.5c, you get 1/1.25=0.8c. Note that for everyday velocities, $1-uv/c^2\simeq 1$, and the formula reduces to what you expect.

 

[Mister T]

I just need some clarification but if one person goes . 5c relative to Earth in one direction and another person goes .5 c in the other direction [...]

Earth would indeed see the distance between those two people increase at the speed of light.

Each of them would, on the other hand, see Earth moving at a speed of $0.5c$.

But, each of them would see the other moving at a speed of $0.8c$.

 

[pr3dator]

It is true. The tailllings question you can tell why nature works this way?
I try.
When you draw it in spacetime diagram you will see how it is works.
Whatever your speed is , the "lightspeed" related to you will be on lightcone always.
The two moving observers with +-0.5c have same lightcone. (pinned into common/starting point of spacetime)
Now we see that it is impossible the c value of difference between.

 

[Ebeb]

Adding velocities on spacetime diagram.

Let's consider 3rd observer: black reference system.
Relative to black reference frame the orange observer travels .4513c per 1 time unit
Relative to black reference frame the green observer travels -.4513c per 1 time unit

Adding velocities (Newton way) would mean relative velocity between orange and green frame = 0.4513c + 0.4513c = .9c
However, by using the relativistic equation for adding velocities
U = (v-(-u))/((1- (-vu))/c2) = .9026/1.2036 = 0.75c

On diagram same scale for green and orange x and t units (because symmetrical to orthogonal black axes).
Different scale for black units (smaller than green and orange x and t units).

(PS. v and gamma top right of dwg is for green relative to orange (and orange relative to green))

 

[robphy]

Let me suggest this tick-supplemented Minkowski spacetime diagram for velocity-addition.
[It's taken from my article described in my Insight although it's not in the Insight itself.]
(I think Loedel-type diagrams okay, but are of limited value.)

I think you can clearly see that the relativity of simultaneity is an important feature of velocity-addition.

 

[NoahsArk]

Thank you very much Ebeb and robphy for the diagrams, and to everyone who's helped me with this.

I plan to study the diagrams further, and to also work out more the example that you gave Ebeb.

Earlier I was wondering why, in a space time diagram, both the time and space axis are tilted inwards for the moving frame compared to the stationary frame's set of axis. Now, after looking at the diagrams, I see the reason I think: the angle between the beam of light and the space axis has to be equal to the angle between the beam of light and the time axis. In the stationary frame, the angle is 45% from both the space and time axis. For someone traveling at half the speed of light, the angle has to be 22.5% from each axis (which is similar to the orange reference frame). If only the space axis were tilted but not the time axis, it would mean that the beam was going faster than light I think.

 

[Orodruin]

If only the space axis were tilted but not the time axis, it would mean that the beam was going faster than light I think.

Note that in Galilean relativity only the time axis is tilted. The space axis being tilted is what is new in SR.

 

[robphy]

Careful... For a slope of 0.5 (with respect to the t-axis), the angle in degrees is
arctan(.5) = 26.5650512 degrees
(Google "arctan(.5) in degrees". Similarly, you can compute
tan(22.5 degrees)=0.41421356237.)

 

[robphy]

The geometric reason for the tilting of the "space axes" is due that axis being tangent to the "circle" ( a hyperbola in special relativity ).

Consult this post from a past thread...
https://www.physicsforums.com/threads/minkowski-diagram.876359/#post-5503686
Play with the interactive desmos visualization.

 

[Orodruin]

The geometric reason for the tilting of the "space axes" is due that axis being tangent to the "circle" ( a hyperbola in special relativity ).

You mean normal, i.e., with a tangent vector orthogonal to the tangent vector of the hyperbola at the point they cross. This is just like a rotation in Euclidean space, the axes cross a circle at right angles.

 

[robphy]

The geometric reason for the tilting of the "space axes" is due that axis being tangent to the "circle" ( a hyperbola in special relativity ).

You mean normal, i.e., with a tangent vector orthogonal to the tangent vector of the hyperbola at the point they cross. This is just like a rotation in Euclidean space, the axes cross a circle at right angles.

I think I really did mean "tangent [to the circle]",
where
the space-axis being "tangent to the circle"
defines
what "normal [to the worldline]" means.
I will add some words to my original statement for clarity:

The geometric reason for the tilting of the "space axes" is due to that space axis being parallel to the tangent to the "unit circle" ( a hyperbola in special relativity formed from the tips of the future unit-timelike vectors) at the event where that observer's 4-velocity meets the circle. This diagram (from the past thread I referenced...
https://www.physicsforums.com/threads/minkowski-diagram.876359/#post-5503686 ) should be useful.

https://www.desmos.com/calculator/ti58l2sair [my t axis is horizontal ]

 

[Orodruin]

I think I really did mean "tangent [to the circle]",
where
the space-axis being "tangent to the circle"
defines
what "normal [to the worldline]" means.
I will add some words to my original statement for clarity:

The geometric reason for the tilting of the "space axes" is due to that space axis being parallel to the tangent to the "unit circle" ( a hyperbola in special relativity formed from the tips of the future unit-timelike vectors) at the event where that observer's 4-velocity meets the circle. This diagram (from the past thread I referenced...
https://www.physicsforums.com/threads/minkowski-diagram.876359/#post-5503686 ) should be useful.

That really looks like the spatial axis being normal to the hyperbola ...

 

[robphy]

That really looks like the spatial axis being normal to the hyperbola ...

In that diagram, it is noted that my t-axis is horizontal... and the future-"circle" (whose events are timelike related to the origin) is shown in the right half. The worldline through the origin meets the hyperbola at an event. At that event, the tangent line to the circle is drawn. That tangent line to the circle is declared to be "normal" to the worldline [thought of as a radius vector].

This is in line with Minkowski, "Space and Time" p. 84-85

We divide up any vector we choose, e.g. that from O to x, y, z, t, into the four components x, y, z, t.
If the directions of two vectors are, respectively,
that of a radius vector OR from O to one of the surfaces (+/-) F = 1,
and that of a tangent RS at the point R of the same surface,
the vectors are said to be normal to one another.

In the context of my diagram,
the "F=1"-surface is my future-"circle" centered at O,
"OR" is the future-unit-timelike vector of the observer, where R is on that circle.
Then the tangent "RS" is normal to "OR" (along the observer's time axis).
Thus, the observer's space axis at event O will be defined as the parallel to this tangent-"RS"-to-the-circle drawn through the origin O.

 

[Orodruin]

In that diagram, it is noted that my t-axis is horizontal...

Ok, so your spatial axis is the dashed line. Yes, this line is tangent to the hyperbola, but it is not the spatial axis used in the standard Lorentz transformation - that spatial axis is a normal line to the other hyperbola (the hyperbola that has space-like separated from the origin). I think it is less confusing to talk about the axes which the standard configuration of the Lorentz transform relate - just as it is easier to talk about just rotations in Euclidean space instead of rotations+translation to the circle.

 

[robphy]

Ok, so your spatial axis is the dashed line. Yes, this line is tangent to the hyperbola, but it is not the spatial axis used in the standard Lorentz transformation - that spatial axis is a normal line to the other hyperbola (the hyperbola that has space-like separated from the origin). I think it is less confusing to talk about the axes which the standard configuration of the Lorentz transform relate - just as it is easier to talk about just rotations in Euclidean space instead of rotations+translation to the circle.

Yes, the dashed line is that observer's spatial axis.

At a glance, it might be confusing. But I thought I was explicit about my construction and my conventions.

Pedagogically, I think my presentation is better because

• my "circle" (the hyperbola that is timelike separated from the origin) arises from an operational construction: an experiment starting at event O, where observers traveling with various velocities marked when (say) 1 second has elapsed on their watch since event O.
  
• Once this circle is determined, orthogonality (being "normal" or "perpendicular") to a radius vector (along an inertial observer's worldline) in this geometry is then defined by tangency to this circle.
  (Then the parallel to this tangent can be used to construct the spatial-axis through event O.)
  
• This construction works in the Minkowski and Galilean spacetime geometries and in Euclidean space (using odometers in the plane, with surveyors traveling in different spatial directions).
  
• In the desmos visualization ( https://www.desmos.com/calculator/ti58l2sair ), you can see this by tuning the signature of the metric by tuning the y-coefficient of the "circle". [Here, I didn't want x along the vertical.] I use the parametrization $t^2-Ey^2=1$, where $E=1,0,-1$ is Minkowski, Galilean, and Euclidean, respectively.
  
  Furthermore, by tuning the two velocities, you can see how the observers generally disagree on these tangent lines (that is, what each declares the "line of constant t=1" is, demonstrating the relativity of simultaneity for the Minkowski case and the absoluteness in the Galilean case).
• [I kept the t-axis horizontal so that we obtain the usual PHY 101 position-vs-time graph in the Galilean case (although most are unaware of this nonEuclidean geometry underlying the usual position-vs-time graph).]

From this viewpoint, the "hyperbola that is timelike separated from the origin" is physically more fundamental (i.e. more primitive)
than the "hyperbola that is spacelike separated from the origin" (which can then be derived later, if needed).