Let A be a rocket moving with velocity v. Then the slope of its worldline in a spacetime diagram is given by c/v .Since it is a slope, c/v=tan(θ) for some θ>45 and θ<90. Does this impose a mathematical limit on v?

Ethen Blackwell

Ethen Blackwell

Answered question

2022-07-21

Let A be a rocket moving with velocity v.
Then the slope of its worldline in a spacetime diagram is given by c / v.
Since it is a slope, c / v = tan ( θ ) for some θ > 45 and theta < 90.
Does this impose a mathematical limit on v?
If so what is it?
As in, we know tan ( 89.9999999999 ) = 572957795131.
And c = 299792458.
Using tan ( 89.9999999999 ) as our limit of precision, the smallest v we can use is:
c / v = tan ( 89.9999999999 )
299792458 / v = 572957795131
Therefore, v = 1911.18   m / s
What is the smallest non zero value of v?

Answer & Explanation

Marisa Colon

Marisa Colon

Beginner2022-07-22Added 18 answers

Since a worldline along the time axis on Minkowski diagram is at rest, it is more intuitive to measure angles from that axis instead, as then 'slope' is (space)/(time), i.e., a velocity. Then we have the trigonometric relationship:
v c = tanh α
where Minkowski spacetime follows hyperbolic trigonometry because of the sign difference in the Minwkoski metric/distance formula compared to Euclidean metric/Pythagorean theorem.
The hyperbolic angle α can be any real number, and limit it imposes on speed under this restriction of real numbers is that | v | < c.
A lot of STR formula become rather intuitive in this form, e.g., Lorentz transformation is just a rotation with hyperbolic trigonometry, and the velocity addition formula is:
u v = u + v 1 + u v / c 2 tanh ( α + β ) = tanh α + tanh β 1 + tanh α tanh β ,
and so forth.
Greyson Landry

Greyson Landry

Beginner2022-07-23Added 5 answers

Note that in Euclidean space, the corresponding question is 'if you have three lines intersecting at a point, and the first makes a slope m with the second, while the second makes a slope l with the third, what slope does the first line make with the third?', and the answer to that also follows that pattern of the normal tangent addition formula.

Do you have a similar question?

Recalculate according to your conditions!

New Questions in Relativity

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?