# What is distance, motion in space time?

To clarify things, I am completely new to special relativity. I apologize if this question does not make any sense at all, it may be due to my completely false understanding of basic concepts.

I understand that observers with different relative velocities can assign different space and time values for an event but spacetime is absolute:

$$s^2 = (ct)^2-x^2$$

$$s^2 = t^2(c^2-v^2)$$

$$s = t\sqrt{c^2-v^2}$$

So i thought that "velocity" in spacetime would be:

$$\dfrac{ds}{dt} = \dfrac{d(t\sqrt{c^2-v^2})}{dt} = \sqrt{c^2-v^2}$$

$$= \frac{c}{\gamma}$$

so if this is indeed correct, an object at rest is traveling at light speed in spacetime: $$\sqrt{c^2-0} = c$$. And the distance it travels in spacetime is:

$$\displaystyle\int{c \ dt} = ct$$

What I ask is: What does traveling in space-time mean? What is motion in space-time? Additionally, what I have done above is probably wrong and if you explain why I'd be more than grateful.

Your first equation can be also written: $$s^2 = (c\tau)^2 = (ct)^2-x^2$$

and the further equations are ways to express the relation between coordinate time and proper time:

$$\frac{dt}{d \tau} = \gamma$$

The intuitive concept of motion relates to change with respect to time. In this sense, we can say that if an object moves with a constant 3-velocity $$(v_x, v_y, v_z)$$ it also "moves" constant in time. But the only meaning of this expression is: the time dilation is constant, or "its clock is accumulating a difference (compared to my clock) at a constant rate".

And if an object is at rest in my frame, it is also not moving with respect to time because $$\gamma = 1$$

• It was my understanding that objects at rest in my frame are moving forward in time with me at c. May 16, 2021 at 2:59

Consider firstly the idea of the dimensions of space. When talking of space in everyday life we typically refer to three dimensions as left/right, forwards/backwards and up/down. In physics we tend to use the letters x,y,z to represent them.

An important point is that there is no absolute direction for any of the coordinates. You are free to pick a reference frame in which your coordinates are all rotated compared to mine. If you live in New York and I live in London, we will think of up and down to mean the direction normal to the ground where we each live, so that our vertical axes will be tilted relative to each other.

If you climb 100ft in an elevator in New York, you are rising vertically in your frame of reference, so your z coordinate increases by 100ft but your x and y coordinates are unchanged. From my perspective in London, your New York elevator is tilted, so your position along my z coordinate increases by less than 100 ft- your overall movement is still 100ft in my frame, but some of it is in my x and y directions.

Now imagine a fourth dimension, time. In your own reference frame, you are always moving forward through time, so your position on your time axis is always increasing at a certain rate. That, for you, is what is meant by moving through spacetime.

As with the x,y and z dimensions, there is no absolute fixed direction of the time axis. The time axes of two frames moving relative to each other are tilted, the degree of tilt increasing with speed, rather as the tilt of the vertical direction increases the further you move around the surface of the Earth.

This means that while you are always moving directly up your t axis in your frame, to someone else relative to whom you are in motion through space your t axis is tilted relative to theirs so you are progressing along their t axis more slowly.

So, you are never at rest in spacetime- you are always moving through it at a speed c. In your own frame of reference the movement is entirely in the t direction. In any frame of reference moving slowly relative to yours, your speed c will be mainly in the t direction but will be split across spatial components too, the size of the spatial components relative to the temporal will depend upon the speed at which the two frame of reference are moving relative to each other.