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what are the implications of relativity? does it imply for example that there is no center to spacetime?
ie despite that the universe started from one point, there is no real center, every frame of reference can be considered as the center point vis a vis the laws of physics such as the speed of light.
Relativity has consequences for our understanding of space, time and motion: classic questions of a philosophy of nature from the time of the Ionian philosophers.
They haven't been central to modern philosophy however, though there has been some work.
In Rovellis view, the main insight is that causal influence, as it is understood physically, is local; this he traces from Relativity back to Maxwell through the concept of the field, and then to Faraday. It's also visible in Aristotle, in a particularly general form: 'change is by contact'.
Another possible consequence, at least to Rovelli, is that spacetime is emergent; and actual spacetime points disappear (Einsteins 'hole argument') - for he situates all movement in fields and their interaction: 'there are only fields on fields'.
(It's likely he takes this perspective as it is consistent with the character of the specific mode of QG he works on - LQG - in which spacetime is emergent, discrete and relational).
Yes, according to cosmology at a large scale the universe is homogeneous and isotropic, i.e. it looks alike in all space directions. As a consequence, no distinguished point in space exists, i.e. no cosmic center.
The statement that the universe started from one point, does not mean that a certain point in space was distinguished as the birth place of our universe. The start of our universe was also the birth of spacetime. One cannot locate this event in a space which existed before.
The above mentioned result about the homogenity and isotropy of our universe does not derive from the Theory of Relativity. Instead, it is a result of observation, mainly of the cosmic microwave background radiation (CMB).
But the most simple solutions of Eintein's field equations from the General Theory of Relativity are homogeneous and isotropic. Hence the observation is in accordance with theory.
Secondly, a deep implication of the Special Theory of Relativity is the relativity of space and time. Both concepts do not have an absolute meaning, only the combined notion spacetime has an absolute meaning. Absolute means that spacetime is independent from the observer and his coordinate frame. While the decomposition of spacetime into space and time depends on the selected coordinate frame - up to a certain degree. As a consequence simultaneouness depends on the observer. Two observers moving with a certain velocity relatively to each other observe different events as simultaneous.
According to my opinion, the latter set of philosophical implications has not yet been fully received by philosophers. See also a previous discussion in this blog Time and space – a subject of metaphysics?
For me its implication is re-affirm the importance of the "relative" nature of universal concepts, especially when it comes to whether abstract notion exists or not.
Historically there're too many unnecessary conflict and arguments even within philosophy itself, such as the debate of the "Problem of the Universals" between realism and nominalism. As long as we treat the definition of "exists" itself not absolute but relative to a certain layer of human perception and mind, all these conflicts disappear. More abstract concepts exist in more abstract layers, vice versa. Not that complicated and deep at all...
