How does matter accelerate in a gravitational field

[49ers2013Champ]

Wannabe:

One last question regarding your last sentence: "These geodesics are the geodesics of the space-time geometry generated by some other mass-energy distribution acting as the source."

The pronoun "These" is referring to geodesics in flat spacetime--the aforementioned scenario--or in spacetime where bodies are moving along lines being created by energy-matter distributions acting as the source?

 

[WannabeNewton]

The pronoun "These" is referring to geodesics in flat spacetime--the aforementioned scenario--or in spacetime where bodies are moving along lines being created by energy-matter distributions acting as the source?

In flat space-time "these" geodesics refers to the straight lines of Minkowski space-time. When there is mass-energy present then yes "these" geodesics refers to what you said in the latter part of your sentence.

 

[Naty1]

These two already posted statements together with DrGreg' charts summarizes all this nicely for me:

[No gravity:]

In flat space-time ... geodesics refers to the straight lines of Minkowski space-time.

[With gravity:]

... objects move in {curved} geodesics in curved "space-time". Why that is true, we don't know- any more than we knew why, in Newtonian physics, "gravity" caused all mass to attract one another.

To clarify these statements further: A geodesic in GR generalizes the notion of a "straight line" to include curved spacetime...so a geodesic IS a straight line path in flat spacetime and a curved path [or worldline] in curved spacetime [where gravity] is present. In other words, a freely falling particle [a particle not subject to other non gravitational forces like electromagnetic attraction or repulsion] always moves along a geodesic.

What 'causes' spacetime curvature: sources captured in the stress energy tensor [like pressure, energy, mass] If the graph paper of DrGreg's illustrations remains 'flat'...as in his first three illustrations...that is a 'no gravity' condition; gravity is illustrated via a 'curved' graph paper as his last illustration.

The curvature of the coordinate grid, which we seem to call in these forums 'apparent curvature' [for lack of a better term] it isn't spacetime curvature. This is not gravitational spacetime curvature, is not sourced from the stress energy tensor; GRAVITATIONAL spacetime curvature is frame invarient. [I mention 'apparent curvature' because I recently got people inadvertently confused in another thread by not carefully distinguishing between it and gravitational spacetime curvature.]

The introduction here is a decent overview:
http://en.wikipedia.org/wiki/Geodesics_in_general_relativity

and discusses that massive particles follow time like geodesics while photons [massless] follow null geodesics...

 

[Nugatory]

Wouldn't the particle be stationary in space--not in observable relative motion and not being influenced by gravity--and so wouldn't it be incorrect to say it was moving along a geodesic?

No. Instead, you've just identified about the simplest example of a geodesic in four-dimensional space-time.

I'm floating motionless in an otherwise completely empty and curvature-free flat universe. My x, y, and z coordinates are not changing. But what's happening to my t coordinate? It's steadily increasing... So my path through space-time is a straight line oriented along the t axis.

It's sort of like how in a one-dimensional number line $x=3$ specifies a point, but in the two-dimensional Cartesian plane it specifies a line parallel to the y-axis. Likewise, a point (me at rest) in three-dimensional space is a line in four-dimensional spacetime.

 

[Naty1]

Wouldn't the particle be stationary in space...

In absolutely flat spacetime, a particle would theoretically 'sit still' in SPACE...but Nugatory points out it still moves through the time dimension of SPACETIME.

 

[49ers2013Champ]

Naty, WN, others:

I guess what I'm struggling with here is that I thought there had to be relative motion for time to be present. I've already emailed Mr. Donis about this, and while he struck it all down, I'm still missing something. I completely understand that the mathematics suggests that the point is perhaps stationary on the space (spatial?) side of the graph but is advancing through, or perhaps curving, the time side of the graph--and this is all based off the particle or mass in deep, deep space and far away from the influence of gravity. I thought all notions of time were based off relative motion--our orbiting the sun being the most obvious example. Another example, perhaps a silly one, is this: Albert, Isaac, and Peter are all far removed from each other in deep, deep space, but in the same universe. They all have rocket boosters. They all hit them and eventually cross paths, and they start hanging out. They've never heard of the concept of time, aging, etc. But now, since they are seeing each other moving around and talking to each other, they start to record these events in their minds--a memory develops--and they begin to experience the notion of time. "Yesterday", Albert's mustache was long; today it's trimmed. You get my point.

I'm sure I'm completely wrong, but I just don't know how a material point (mass, particle, or body) can be said to be curving spaceTIME. To me, time seems to be a human-constructed concept that can be applied to science only under the right conditions--planets orbiting stars and other types of relative motion.

Everyone, thanks for your thoughts. In no way am I saying I'm right--please don't misunderstand me. I'm simply trying to communicate that the time part of spacetime curvature in the aforementioned scenario doesn't make sense to me.

 

[Drakkith]

Why would relative motion be required for time? Let's say your particle in this otherwise empty universe is a muon, an elementary particle similar to an electron. It has a half life of 2.2 microseconds. There is no reason that I know of to believe that it wouldn't decay just because there is no relative motion. Besides, a muon in the real world has no way of telling time. It can't see a clock or watch as planets orbit stars, yet it still obeys time.

 

[Naty1]

49er's...

I guess what I'm struggling with here is that I thought there had to be relative motion for time to be present.

Keep an eye out for any source that claims that. See what the 'logic' might be. It's just not mainstream science.

What we know, so far, for example is that relative motion affects the relative passage of time and that differences in gravitational potential also do. Maybe there is something else we haven't discovered yet. [We rarely know all we don't know.]

No one knows before the big bang if time existed or not. Maybe everything emanated from that source, but how did anything get started if there was no time? What would that even mean?

... I just don't know how a material point (mass, particle, or body) can be said to be curving spaceTIME. To me, time seems to be a human-constructed concept that can be applied to science only under the right conditions--planets orbiting stars and other types of relative motion.

It took an 'Einstein' to figure that out, so don't feel bad...it IS rather crazy! [Did you read the first quote in my signature below??] For thousands of years nobody knew how any forces could act at a distance. Even Newton did not know. Then we created field theories, like electromagnetic and gravitational, and voila we had a means of studying and predicting physical phenomena. The 'right conditions' you mention for gravity...that is, spacetime curvature...so far seems to be everywhere except black hole singularities and the big bang itself. Quantum mechanics is not applicable at these two extremes. But not bad for a start!