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**invariant "spacetime velocity"**

This is related to this thread https://www.physicsforums.com/showthread.php?t=207251". To make responding easier, I have marked questions in red. I have tried to address some concerns pre-emptively. These I have marked in silver.

If you would like to respond, please respond primarily to the core questions, or the explanation in standard font. If you are responding to a silver clarification, please note that, so that I understand that you are aware that you are not addressing something central. Thanks.

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Is there any validity in considering an invariant "spacetime velocity"?

Let me try to explain. According to me, in spatial terms, I am stationary, so I traverse a spacetime "distance" of ct' over a period of t' (this is my proper time, the time we expect to be dilated when compared to someone travelling relative to me). My buddy helps me out in a little experiment by not remaining stationary relative to me but rather having a velocity of v.

During the period of t', according to me, my buddy travels a distance of vt' to reach an event E, which is simultaneous (in my frame) with the event I reach after a period of t'.

According to my buddy, he travels a timespace distance of ct before reaching the event E.

This step may upset some people, but watch closely:

My total spacetime distance travelled is

**ct'**.

The magnitude of my buddy's total spacetime distance travelled is

**sqrt((ct)^2 + (vt')^2)**- this is just the hypoteneuse of the triangle with ct (temporal component) and vt' (spatial component).

Equating these:

**ct' = sqrt((ct)^2 + (vt')^2)**

or

**(ct')^2 = (ct)^2 + (vt')^2**

then rearranging:

**(ct)^2 = (ct')^2 - (vt')^2**

and solving:

**t' = t / sqrt(1 - v^2 / c^2)**

I acknowledge that it is not the simplest way to arrive at the equation for time dilation, but is there a problem other than that?

Note that this is based on the assumption that relative to me, everything travels at an invariant "spacetime velocity" of c, including myself.

I am not assuming that I am at rest, but that I have a purely "temporal velocity" of c and anyone who does not have a purely "temporal velocity", but rather has a spatial velocity as well, will have a reduced "temporal velocity" as a result. I

**am**assuming that spatial velocity and temporal velocity are orthogonal.

Note further that I am talking about "spacetime traversed" by my buddy not the spacetime interval, relative to me, between two events - both of which are inhabited by my buddy. Conceptually, my buddy needed to cut a corner to arrive at a future event (ie event E) after traversing less time than I needed to arrive at an event which was simultaneous with that event

**in my "rest" frame**. So my buddy travelled through less time due to the need to travel through some space.

(I am aware that, from my buddy's point of view, it is I who cuts a corner to arrive at a future event which is simultaneous with event E in his "rest" frame. I am aware that we will disagree on which events are simultaneous with event E, but that I can work out which events he will perceive as simultaneous.)

Is it a standard understanding that, relative to me (or any given observer), any inertial thing travels with an invariant "spacetime velocity" of c?

thanks,

neopolitan

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