How we can measure speed with time, when the rate at which time passes depends on speed?

Question

I was thinking about relativity, specifically about how we travel at the speed of light in 4 dimensions. The higher one’s velocity in space, the lower one’s velocity in time. Inversely, the lower one’s velocity in space, the higher one’s velocity i time. This led me to try to ‘calculate’ the velocity for different spatial speeds.

I started by taking a spatial speed, say 2. This is the speed of an object in 3 spatial dimensions. To find the speed at which time travels for the object, I would just subtract the speed, 2, from the speed of light, $c$. Though this is not the correct equation, I have derived a relation below. I then realized that this was doomed. The calculation for spatial speed, s=d/t, uses time in the equation. To calculate the ‘speed of time’, I must use an amount of time. The rate at which time travels is dependent on time itself! Thus, even though the principle is true, it cannot be calculated, at least not correctly.

$$c = \sqrt {s_s^2+s_t^2} \space\space\space\text{Pythagorean Theorem}$$ $$c = \sqrt{2^2 + s_t^2} \space\space\space\text{Substitution}$$ $$c = \sqrt{4 + s_t^2} \space\space\space\text{Simplifying}$$ $$c^2 = 4 + s_t^2 \space\space\space\text{Squaring Property of Equality}$$ $$\frac{c^2}{s_t^2}=4\space\space\space\text{Division Property of Equality}$$ $$\frac{c}{s_t}=2\space\space\space\text{Squaring Property of Equality}$$ $$c=2s_t\space\space\space\text{Multiplication Property of Equality}$$ $$\frac{c}{2}=s_t\space\space\space\text{Division Property of Equality }$$

I do recognize that this may not be the exact method for finding the speed of time. However, the exact method is irrelevant. Whatever the method, it does need to have an input of spatial speed, which involves time. The question is not a mathematical one, but more conceptual.

Perhaps this violates the Heisenberg uncertainty principle? It seems that by measuring speed, I make it impossible to calculate. This is eerily similar to what is found at small scales in quantum mechanics. Maybe that’s the problem.

Is what I’ve found possible? I can’t seem to wrap my mind around it. Am I doing something fundamentally wrong?

Answer 1

You are correct when you say that our speed (us who have rest mass) in the spatial dimensions affects our speed in the temporal dimension, and this is because you just have to accept that the universe is built up so and the four vector (velocity) is built up so.

In physics, in particular in special relativity and general relativity, a four-velocity is a four-vector in four-dimensional spacetime[nb 1] that represents the relativistic counterpart of velocity, which is a three-dimensional vector in space. Physical events correspond to mathematical points in time and space, the set of all of them together forming a mathematical model of physical four-dimensional spacetime. The history of an object traces a curve in spacetime, called its world line. If the object has mass, so that its speed is less than the speed of light, the world line may be parametrized by the proper time of the object. The four-velocity is the rate of change of four-position with respect to the proper time along the curve.

The value of the magnitude of the four velocity (quantity obtained by applying the metric tensor g to the four velocity U) is always c2.

https://en.wikipedia.org/wiki/Four-velocity

In SR, the path of an object moving relative to a certain reference frame is defined by four coordinate functions, where the timelike component is the time coordinate multiplied by the speed of light.

We just have to accept that the universe is built up so, and the four vector is built up so, that its magnitude is c always.

As you say, if you move faster in the spatial dimensions, your four velocity's magnitude has to stay the same constant, thus your speed in the temporal dimension hass to compensate, you will slow down in the temporal dimension (relatively).

Yes, you are correct, when you try to measure this speed in the temporal dimension, you say that the speed at which time travels (passes) is dependent on time itself. What is correct to say is that it passes at one second per second, but even that is not really a meaningful statement.

This is basically why there is no universal time. Every single different reference frame, traveling at different speeds, and in different gravitational zones, might measure time pass at a different rate. The only meaningful statement is, that time passes relatively faster or slower in one frame relative to the other frame. This is time dilation.

Answer 2

First: some terminology. A transformation that takes you from a stationary to a moving frame (or vice versa, or from a moving frame to a differently-moving frame) is called a "boost". We're talking only about inertial frames here.

In non-relativistic Physics, a boost by a velocity $v$ in the $x$-direction applied to the coordinates $(x,y,z)$ and the time $t$ yields the transform $$(x, y, z, t) → (x - vt, y, z, t).$$ On coordinate differentials, a similar transform also applies: $$(dx, dy, dz, dt) → (dx - v dt, dy, dz, dt).$$ So, for the velocity components, one has: $$\left(\frac{dx}{dt}, \frac{dy}{dt}, \frac{dz}{dt}\right) → \left(\frac{dx}{dt} - v, \frac{dy}{dt}, \frac{dz}{dt}\right).$$

For relativity, the boost transform also affects time, and is given by: $$(x, y, z, t) → \left(\frac{x - vt}{\sqrt{1 - (v/c)^2}}, y, z, \frac{t - vx/c^2}{\sqrt{1 - (v/c)^2}}\right),$$ with a similar transform on the coordinate differentials: $$(dx, dy, dz, dt) → \left(\frac{dx - v dt}{\sqrt{1 - (v/c)^2}}, dy, dz, \frac{dt - v dx/c^2}{\sqrt{1 - (v/c)^2}}\right).$$ So the transform on the velocity components is: $$\left(\frac{dx}{dt}, \frac{dy}{dt}, \frac{dz}{dt}\right) → \left(\frac{dx/dt - v}{1 - v/c^2 dx/dt}, \frac{\sqrt{1 - (v/c)^2} dy/dt}{1 - v/c^2 dx/dt}, \frac{\sqrt{1 - (v/c)^2} dz/dt}{1 - v/c^2 dz/dt}\right).$$

So the answer to your question of how you measure the velocity, when the boost transform also affects time is - by just simply taking it into account, as I just did! Just because something changes doesn't mean you can't measure or define it. That's the "Vaguefying Fallacy" - the false "it's relative, therefore it's vague" notion. It may be relative, but the relativeness of it is absolute and is not vague.

That'd be like saying you can't measure distance if you move to a different location. Of course you can. You just simply take that change in location into account.

Or, it'd be like saying that you can't make trades or purchases, when the value of a currency changes, because the prices are all relative. I don't know about you, but the last few times I went to the store, it was pretty crowded, even with the steady erosion of the currency brought on by the worldwide surge in inflation immediately (and predictably) following the massive 2020-2021 stimulus-driven hiking up of the sovereign debts of practically every nation-state on Earth.